Matconvnet manual

MatConvNet
Convolutional Neural Networks for MATLAB
Andrea Vedaldi Karel Lenc Ankush Gupta
i

ii
Abstract
MatConvNet is an implementation of Convolutional Neural Networks (CNNs)
for MATLAB. The toolbox is designed with an emphasis on simplicity and flexibility.
It exposes the building blocks of CNNs as easy-to-use MATLAB functions, providing
routines for computing linear convolutions with filter banks, feature pooling, and many
more. In this manner, MatConvNet allows fast prototyping of new CNN architec-
tures; at the same time, it supports efficient computation on CPU and GPU allowing
to train complex models on large datasets such as ImageNet ILSVRC. This document
provides an overview of CNNs and how they are implemented in MatConvNet and
gives the technical details of each computational block in the toolbox.

Contents
1 Introduction to MatConvNet 1
1.1 Getting started . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 MatConvNet at a glance . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.3 Documentation and examples . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Speed . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.5 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2 Neural Network Computations 9
2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2 Network structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.1 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Directed acyclic graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Computing derivatives with backpropagation . . . . . . . . . . . . . . . . . . 12
2.3.1 Derivatives of tensor functions . . . . . . . . . . . . . . . . . . . . . . 12
2.3.2 Derivatives of function compositions . . . . . . . . . . . . . . . . . . 13
2.3.3 Backpropagation networks . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.4 Backpropagation in DAGs . . . . . . . . . . . . . . . . . . . . . . . . 15
2.3.5 DAG backpropagation networks . . . . . . . . . . . . . . . . . . . . . 18
3 Wrappers and pre-trained models 21
3.1 Wrappers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.1 SimpleNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1.2 DagNN . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Pre-trained models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
3.3 Learning models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.4 Running large scale experiments . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.5 Reading images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 Computational blocks 27
4.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.2 Convolution transpose (deconvolution) . . . . . . . . . . . . . . . . . . . . . 29
4.3 Spatial pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
4.4 Activation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.5 Spatial bilinear resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.6 Region of interest pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
iii

iv CONTENTS
4.7 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7.1 Local response normalization (LRN) . . . . . . . . . . . . . . . . . . 33
4.7.2 Batch normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.7.3 Spatial normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.7.4 Softmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
4.8 Categorical losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8.1 Classification losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
4.8.2 Attribute losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.9 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.9.1 p-distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
5 Geometry 39
5.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 Simple filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.2.1 Pooling in Caffe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3 Convolution transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
5.4 Transposing receptive fields . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
5.5 Composing receptive fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.6 Overlaying receptive fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
6 Implementation details 45
6.1 Convolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
6.2 Convolution transpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.3 Spatial pooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4 Activation functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4.1 ReLU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.4.2 Sigmoid . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.5 Spatial bilinear resampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.6 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.6.1 Local response normalization (LRN) . . . . . . . . . . . . . . . . . . 48
6.6.2 Batch normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
6.6.3 Spatial normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.6.4 Softmax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.7 Categorical losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.7.1 Classification losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.7.2 Attribute losses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
6.8 Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.8.1 p-distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.9 Other implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.9.1 Normal sampler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
6.9.2 Euclid’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Bibliography 55

Chapter 1
Introduction to MatConvNet
MatConvNet is a MATLAB toolbox implementing Convolutional Neural Networks (CNN)
for computer vision applications. Since the breakthrough work of [8], CNNs have had a
major impact in computer vision, and image understanding in particular, essentially replacing
traditional image representations such as the ones implemented in our own VLFeat [13] open
source library.
While most CNNs are obtained by composing simple linear and non-linear filtering op-
erations such as convolution and rectification, their implementation is far from trivial. The
reason is that CNNs need to be learned from vast amounts of data, often millions of images,
requiring very efficient implementations. As most CNN libraries, MatConvNet achieves
this by using a variety of optimizations and, chiefly, by supporting computations on GPUs.
Numerous other machine learning, deep learning, and CNN open source libraries exist.
To cite some of the most popular ones: CudaConvNet,1
Torch,2
Theano,3
and Caffe4
. Many
of these libraries are well supported, with dozens of active contributors and large user bases.
Therefore, why creating yet another library?
The key motivation for developing MatConvNet was to provide an environment par-
ticularly friendly and efficient for researchers to use in their investigations.5
MatConvNet
achieves this by its deep integration in the MATLAB environment, which is one of the most
popular development environments in computer vision research as well as in many other areas.
In particular, MatConvNet exposes as simple MATLAB commands CNN building blocks
such as convolution, normalisation and pooling (chapter 4); these can then be combined and
extended with ease to create CNN architectures. While many of such blocks use optimised
CPU and GPU implementations written in C++ and CUDA (section section 1.4), MATLAB
native support for GPU computation means that it is often possible to write new blocks
in MATLAB directly while maintaining computational efficiency. Compared to writing new
CNN components using lower level languages, this is an important simplification that can
significantly accelerate testing new ideas. Using MATLAB also provides a bridge towards
1
https://code.google.com/p/cuda-convnet/
2
http://cilvr.nyu.edu/doku.php?id=code:start
3
http://deeplearning.net/software/theano/
4
http://caffe.berkeleyvision.org
5
While from a user perspective MatConvNet currently relies on MATLAB, the library is being devel-
oped with a clean separation between MATLAB code and the C++ and CUDA core; therefore, in the future
the library may be extended to allow processing convolutional networks independently of MATLAB.
1

2 CHAPTER 1. INTRODUCTION TO MATCONVNET
other areas; for instance, MatConvNet was recently used by the University of Arizona in
planetary science, as summarised in this NVIDIA blogpost.6
MatConvNet can learn large CNN models such AlexNet [8] and the very deep networks
of [11] from millions of images. Pre-trained versions of several of these powerful models can
be downloaded from the MatConvNet home page7
. While powerful, MatConvNet re-
mains simple to use and install. The implementation is fully self-contained, requiring only
MATLAB and a compatible C++ compiler (using the GPU code requires the freely-available
CUDA DevKit and a suitable NVIDIA GPU). As demonstrated in fig. 1.1 and section 1.1,
it is possible to download, compile, and install MatConvNet using three MATLAB com-
mands. Several fully-functional examples demonstrating how small and large networks can
be learned are included. Importantly, several standard pre-trained network can be immedi-
ately downloaded and used in applications. A manual with a complete technical description
of the toolbox is maintained along with the toolbox.8
These features make MatConvNet
useful in an educational context too.9
MatConvNet is open-source released under a BSD-like license. It can be downloaded
from http://www.vlfeat.org/matconvnet as well as from GitHub.10
.
1.1 Getting started
MatConvNet is simple to install and use. fig. 1.1 provides a complete example that clas-
sifies an image using a latest-generation deep convolutional neural network. The example
includes downloading MatConvNet, compiling the package, downloading a pre-trained CNN
model, and evaluating the latter on one of MATLAB’s stock images.
The key command in this example is vl_simplenn, a wrapper that takes as input the
CNN net and the pre-processed image im_ and produces as output a structure res of results.
This particular wrapper can be used to model networks that have a simple structure, namely
a chain of operations. Examining the code of vl_simplenn (edit vl_simplenn in MatCon-
vNet) we note that the wrapper transforms the data sequentially, applying a number of
MATLAB functions as specified by the network configuration. These function, discussed in
detail in chapter 4, are called “building blocks” and constitute the backbone of MatCon-
vNet.
While most blocks implement simple operations, what makes them non trivial is their
efficiency (section 1.4) as well as support for backpropagation (section 2.3) to allow learning
CNNs. Next, we demonstrate how to use one of such building blocks directly. For the sake of
the example, consider convolving an image with a bank of linear filters. Start by reading an
image in MATLAB, say using im = single(imread('peppers.png')), obtaining a H × W × D
array im, where D = 3 is the number of colour channels in the image. Then create a bank
of K = 16 random filters of size 3 × 3 using f = randn(3,3,3,16,'single'). Finally, convolve the
6
http://devblogs.nvidia.com/parallelforall/deep-learning-image-understanding-planetary-scien
7
http://www.vlfeat.org/matconvnet/
8
http://www.vlfeat.org/matconvnet/matconvnet-manual.pdf
9
An example laboratory experience based on MatConvNet can be downloaded from http://www.
robots.ox.ac.uk/~vgg/practicals/cnn/index.html.
10
http://www.github.com/matconvnet

1.1. GETTING STARTED 3
% install and compile MatConvNet (run once)
untar(['http://www.vlfeat.org/matconvnet/download/' ...
'matconvnet−1.0−beta12.tar.gz']) ;
cd matconvnet−1.0−beta12
run matlab/vl_compilenn
% download a pre−trained CNN from the web (run once)
urlwrite(...
'http://www.vlfeat.org/matconvnet/models/imagenet−vgg−f.mat', ...
'imagenet−vgg−f.mat') ;
% setup MatConvNet
run matlab/vl_setupnn
% load the pre−trained CNN
net = load('imagenet−vgg−f.mat') ;
% load and preprocess an image
im = imread('peppers.png') ;
im_ = imresize(single(im), net.meta.normalization.imageSize(1:2)) ;
im_ = im_ − net.meta.normalization.averageImage ;
% run the CNN
res = vl_simplenn(net, im_) ;
% show the classiﬁcation result
scores = squeeze(gather(res(end).x)) ;
[bestScore, best] = max(scores) ;
ﬁgure(1) ; clf ; imagesc(im) ;
bell pepper (946), score 0.704
title(sprintf('%s (%d), score %.3f',...
net.classes.description{best}, best, bestScore)) ;
Figure 1.1: A complete example including download, installing, compiling and running Mat-
ConvNet to classify one of MATLAB stock images using a large CNN pre-trained on
ImageNet.

image with the filters by using the command y = vl_nnconv(x,f,[]). This results in an array
y with K channels, one for each of the K filters in the bank.
While users are encouraged to make use of the blocks directly to create new architectures,
MATLAB provides wrappers such as vl_simplenn for standard CNN architectures such as
AlexNet [8] or Network-in-Network [9]. Furthermore, the library provides numerous examples
(in the examples/ subdirectory), including code to learn a variety of models on the MNIST,
CIFAR, and ImageNet datasets. All these examples use the examples/cnn_train training
code, which is an implementation of stochastic gradient descent (section 3.3). While this
training code is perfectly serviceable and quite flexible, it remains in the examples/ subdirec-
tory as it is somewhat problem-specific. Users are welcome to implement their optimisers.
1.2 MatConvNet at a glance
MatConvNet has a simple design philosophy. Rather than wrapping CNNs around complex
layers of software, it exposes simple functions to compute CNN building blocks, such as linear
convolution and ReLU operators, directly as MATLAB commands. These building blocks are
easy to combine into complete CNNs and can be used to implement sophisticated learning
algorithms. While several real-world examples of small and large CNN architectures and
training routines are provided, it is always possible to go back to the basics and build your
own, using the efficiency of MATLAB in prototyping. Often no C coding is required at all
to try new architectures. As such, MatConvNet is an ideal playground for research in
computer vision and CNNs.
MatConvNet contains the following elements:
ˆ CNN computational blocks. A set of optimized routines computing fundamental
building blocks of a CNN. For example, a convolution block is implemented by
y=vl_nnconv(x,f,b) where x is an image, f a filter bank, and b a vector of biases (sec-
tion 4.1). The derivatives are computed as [dzdx,dzdf,dzdb] = vl_nnconv(x,f,b,dzdy)
where dzdy is the derivative of the CNN output w.r.t y (section 4.1). chapter 4 de-
scribes all the blocks in detail.
ˆ CNN wrappers. MatConvNet provides a simple wrapper, suitably invoked by
vl_simplenn, that implements a CNN with a linear topology (a chain of blocks). It also
provides a much more flexible wrapper supporting networks with arbitrary topologies,
encapsulated in the dagnn.DagNN MATLAB class.
ˆ Example applications. MatConvNet provides several examples of learning CNNs with
stochastic gradient descent and CPU or GPU, on MNIST, CIFAR10, and ImageNet
data.
ˆ Pre-trained models. MatConvNet provides several state-of-the-art pre-trained CNN
models that can be used off-the-shelf, either to classify images or to produce image
encodings in the spirit of Caffe or DeCAF.

1.3. DOCUMENTATION AND EXAMPLES 5
epoch
0 10 20 30 40 50 60
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
dropout top-1 val
dropout top-5 val
bnorm top-1 val
bnorm top-5 val
Figure 1.2: Training AlexNet on ImageNet ILSVRC: dropout vs batch normalisation.
1.3 Documentation and examples
There are three main sources of information about MatConvNet. First, the website con-
tains descriptions of all the functions and several examples and tutorials.11
Second, there
is a PDF manual containing a great deal of technical details about the toolbox, including
detailed mathematical descriptions of the building blocks. Third, MatConvNet ships with
several examples (section 1.1).
Most examples are fully self-contained. For example, in order to run the MNIST example,
it suffices to point MATLAB to the MatConvNet root directory and type addpath ←
examples followed by cnn_mnist. Due to the problem size, the ImageNet ILSVRC example
requires some more preparation, including downloading and preprocessing the images (using
the bundled script utils/preprocess−imagenet.sh). Several advanced examples are included
as well. For example, fig. 1.2 illustrates the top-1 and top-5 validation errors as a model
similar to AlexNet [8] is trained using either standard dropout regularisation or the recent
batch normalisation technique of [4]. The latter is shown to converge in about one third of
the epochs (passes through the training data) required by the former.
The MatConvNet website contains also numerous pre-trained models, i.e. large CNNs
trained on ImageNet ILSVRC that can be downloaded and used as a starting point for many
other problems [1]. These include: AlexNet [8], VGG-S, VGG-M, VGG-S [1], and VGG-VD-
16, and VGG-VD-19 [12]. The example code of fig. 1.1 shows how one such model can be
used in a few lines of MATLAB code.
11
See also http://www.robots.ox.ac.uk/~vgg/practicals/cnn/index.html.

model batch sz. CPU GPU CuDNN
AlexNet 256 22.1 192.4 264.1
VGG-F 256 21.4 211.4 289.7
VGG-M 128 7.8 116.5 136.6
VGG-S 128 7.4 96.2 110.1
VGG-VD-16 24 1.7 18.4 20.0
VGG-VD-19 24 1.5 15.7 16.5
Table 1.1: ImageNet training speed (images/s).
1.4 Speed
Efficiency is very important for working with CNNs. MatConvNet supports using NVIDIA
GPUs as it includes CUDA implementations of all algorithms (or relies on MATLAB CUDA
support).
To use the GPU (provided that suitable hardware is available and the toolbox has been
compiled with GPU support), one simply converts the arguments to gpuArrays in MATLAB,
as in y = vl_nnconv(gpuArray(x), gpuArray(w), []). In this manner, switching between CPU
and GPU is fully transparent. Note that MatConvNet can also make use of the NVIDIA
CuDNN library with significant speed and space benefits.
Next we evaluate the performance of MatConvNet when training large architectures
on the ImageNet ILSVRC 2012 challenge data [2]. The test machine is a Dell server with
two Intel Xeon CPU E5-2667 v2 clocked at 3.30 GHz (each CPU has eight cores), 256 GB
of RAM, and four NVIDIA Titan Black GPUs (only one of which is used unless otherwise
noted). Experiments use MatConvNet beta12, CuDNN v2, and MATLAB R2015a. The
data is preprocessed to avoid rescaling images on the fly in MATLAB and stored in a RAM
disk for faster access. The code uses the vl_imreadjpeg command to read large batches of
JPEG images from disk in a number of separate threads. The driver examples/cnn_imagenet.m
is used in all experiments.
We train the models discussed in section 1.3 on ImageNet ILSVRC. table 1.1 reports
the training speed as number of images per second processed by stochastic gradient descent.
AlexNet trains at about 264 images/s with CuDNN, which is about 40% faster than the
vanilla GPU implementation (using CuBLAS) and more than 10 times faster than using the
CPUs. Furthermore, we note that, despite MATLAB overhead, the implementation speed is
comparable to Caffe (they report 253 images/s with CuDNN and a Titan – a slightly slower
GPU than the Titan Black used here). Note also that, as the model grows in size, the size of
a SGD batch must be decreased (to fit in the GPU memory), increasing the overhead impact
somewhat.
table 1.2 reports the speed on VGG-VD-16, a very large model, using multiple GPUs. In
this case, the batch size is set to 264 images. These are further divided in sub-batches of 22
images each to fit in the GPU memory; the latter are then distributed among one to four
GPUs on the same machine. While there is a substantial communication overhead, training
speed increases from 20 images/s to 45. Addressing this overhead is one of the medium term
goals of the library.

1.5. ACKNOWLEDGMENTS 7
num GPUs 1 2 3 4
VGG-VD-16 speed 20.0 22.20 38.18 44.8
Table 1.2: Multiple GPU speed (images/s).
1.5 Acknowledgments
MatConvNet is a community project, and as such acknowledgements go to all contributors.
We kindly thank NVIDIA supporting this project by providing us with top-of-the-line GPUs
and MathWorks for ongoing discussion on how to improve the library.
The implementation of several CNN computations in this library are inspired by the Caﬀe
library [6] (however, Caﬀe is not a dependency). Several of the example networks have been
trained by Karen Simonyan as part of [1] and [12].

Chapter 2
Neural Network Computations
This chapter provides a brief introduction to the computational aspects of neural networks,
and convolutional neural networks in particular, emphasizing the concepts required to un-
derstand and use MatConvNet.
2.1 Overview
A Neural Network (NN) is a function g mapping data x, for example an image, to an output
vector y, for example an image label. The function g = fL ◦ · · · ◦ f1 is the composition
of a sequence of simpler functions fl, which are called computational blocks or layers. Let
x1, x2, . . . , xL be the outputs of each layer in the network, and let x0 = x denote the network
input. Each intermediate output xl = fl(xl−1; wl) is computed from the previous output xl−1
by applying the function fl with parameters wl.
In a Convolutional Neural Network (CNN), the data has a spatial structure: each xl ∈
RHl×Wl×Cl is a 3D array or tensor where the first two dimensions Hl (height) and Wl (width)
are interpreted as spatial dimensions. The third dimension Cl is instead interpreted as
the number of feature channels. Hence, the tensor xl represents a Hl × Wl field of Cl-
dimensional feature vectors, one for each spatial location. A fourth dimension Nl in the
tensor spans multiple data samples packed in a single batch for efficiency parallel processing.
The number of data samples Nl in a batch is called the batch cardinality. The network is
called convolutional because the functions fl are local and translation invariant operators
(i.e. non-linear filters) like linear convolution.
It is also possible to conceive CNNs with more than two spatial dimensions, where the
additional dimensions may represent volume or time. In fact, there are little a-priori re-
strictions on the format of data in neural networks in general. Many useful NNs contain a
mixture of convolutional layers together with layer that process other data types such as text
strings, or perform other operations that do not strictly conform to the CNN assumptions.
MatConvNet includes a variety of layers, contained in the matlab/ directory, such
as vl_nnconv (convolution), vl_nnconvt (convolution transpose or deconvolution), vl_nnpool
(max and average pooling), vl_nnrelu (ReLU activation), vl_nnsigmoid (sigmoid activation),
vl_nnsoftmax (softmax operator), vl_nnloss (classification log-loss), vl_nnbnorm (batch nor-
malization), vl_nnspnorm (spatial normalization), vl_nnnormalize (local response normal-
9

10 CHAPTER 2. NEURAL NETWORK COMPUTATIONS
ization – LRN), or vl_nnpdist (p-distance). There are enough layers to implement many
interesting state-of-the-art networks out of the box, or even import them from other tool-
boxes such as Caffe.
NNs are often used as classifiers or regressors. In the example of fig. 1.1, the output
ˆy = f(x) is a vector of probabilities, one for each of a 1,000 possible image labels (dog, cat,
trilobite, ...). If y is the true label of image x, we can measure the CNN performance by a
loss function y(ˆy) ∈ R which assigns a penalty to classification errors. The CNN parameters
can then be tuned or learned to minimize this loss averaged over a large dataset of labelled
example images.
Learning generally uses a variant of stochastic gradient descent (SGD). While this is an
efficient method (for this type of problems), networks may contain several million parameters
and need to be trained on millions of images; thus, efficiency is a paramount in MATLAB
design, as further discussed in section 1.4. SGD also requires to compute the CNN derivatives,
as explained in the next section.
2.2 Network structures
In the simplest case, layers in a NN are arranged in a sequence; however, more complex
interconnections are possible as well, and in fact very useful in many cases. This section
discusses such configurations and introduces a graphical notation to visualize them.
2.2.1 Sequences
Start by considering a computational block f in the network. This can be represented
schematically as a box receiving data x and parameters w as inputs and producing data y
as output:
x f y
w
As seen above, in the simplest case blocks are chained in a sequence f1 → f2 → · · · → fL
yielding the structure:
x0 f1 f2
... fL xL
w1 w2 wL
x1 x2 xL−1
Given an input x0, evaluating the network is a simple matter of evaluating all the blocks
from left to right, which defines a composite function xL = f(x0; w1, . . . , wL).

2.2. NETWORK STRUCTURES 11
f1 x1
x0 f3 x3
f2 x2 f5 x7
x5
x4 f4
x6
w1
w2
w4
w5
Figure 2.1: Example DAG.
2.2.2 Directed acyclic graphs
One is not limited to chaining layers one after another. In fact, the only requirement for
evaluating a NN is that, when a layer has to be evaluated, all its input have been evaluated
prior to it. This is possible exactly when the interconnections between layers form a directed
acyclic graph, or DAG for short.
In order to visualize DAGs, it is useful to introduce additional nodes for the network
variables, as in the example of Fig. 2.1. Here boxes denote functions and circles denote
variables (parameters are treated as a special kind of variables). In the example, x0 and x4
are the inputs of the CNN and x6 and x7 the outputs. Functions can take any number of
inputs (e.g. f3 and f5 take two) and have any number of outputs (e.g. f4 has two). There
are a few noteworthy properties of this graph:
1. The graph is bipartite, in the sense that arrows always go from boxes to circles and
from circles to boxes.
2. Functions can have any number of inputs or outputs; variables and parameters can
have an arbitrary number of outputs (a parameter with more of one output is shared
between diﬀerent layers); variables have at most one input and parameters none.
3. Variables with no incoming arrows and parameters are not computed by the network,
but must be set prior to evaluation, i.e. they are inputs. Any variable (or even param-
eter) may be used as output, although these are usually the variables with no outgoing
arrows.

4. Since the graph is acyclic, the CNN can be evaluated by sorting the functions and
computing them one after another (in the example, evaluating the functions in the
order f1, f2, f3, f4, f5 would work).
2.3 Computing derivatives with backpropagation
Learning a NN requires computing the derivative of the loss with respect to the network
parameters. Derivatives are computed using an algorithm called backpropagation, which is
a memory-eﬃcient implementation of the chain rule for derivatives. First, we discuss the
derivatives of a single layer, and then of a whole network.
2.3.1 Derivatives of tensor functions
In a CNN, a layer is a function y = f(x) where both input x ∈ RH×W×C
and output
y ∈ RH ×W ×C
are tensors. The derivative of the function f contains the derivative of
each output component yi j k with respect to each input component xijk, for a total of
H ×W ×C ×H ×W ×C elements naturally arranged in a 6D tensor. Instead of expressing
derivatives as tensors, it is often useful to switch to a matrix notation by stacking the input
and output tensors into vectors. This is done by the vec operator, which visits each element
of a tensor in lexicographical order and produces a vector:
vec x =











x111
x211
...
xH11
x121
...
xHWC











.
By stacking both input and output, each layer f can be seen reinterpreted as vector function
vec f, whose derivative is the conventional Jacobian matrix:
d vec f
d(vec x)
=

















∂y111
∂x111
∂y111
∂x211
. . . ∂y111
∂xH11
∂y111
∂x121
. . . ∂y111
∂xHW C
∂y211
∂x111
∂y211
∂x211
. . . ∂y211
∂xH11
∂y211
∂x121
. . . ∂y211
∂xHW C
...
... . . .
...
... . . .
...
∂yH 11
∂x111
∂yH 11
∂x211
. . .
∂yH 11
∂xH11
∂yH 11
∂x121
. . .
∂yH 11
∂xHW C
∂y121
∂x111
∂y121
∂x211
. . . ∂y121
∂xH11
∂y121
∂x121
. . . ∂y121
∂xHW C
...
... . . .
...
... . . .
...
∂yH W C
∂x111
∂yH W C
∂x211
. . .
∂yH W C
∂xH11
∂yH W C
∂x121
. . .
∂yH W C
∂xHW C

















.
This notation for the derivatives of tensor functions is taken from [7] and is used throughout
this document.

2.3. COMPUTING DERIVATIVES WITH BACKPROPAGATION 13
While it is easy to express the derivatives of tensor functions as matrices, these matrices
are in general extremely large. Even for moderate data sizes (e.g. H = H = W = W =
32 and C = C = 128), there are H W C HWC ≈ 17 × 109
elements in the Jacobian.
Storing that requires 68 GB of space in single precision. The purpose of the backpropagation
algorithm is to compute the derivatives required for learning without incurring this huge
memory cost.
2.3.2 Derivatives of function compositions
In order to understand backpropagation, consider ﬁrst a simple CNN terminating in a loss
function fL = y:
x0 f1 f2
... fL
w1 w2 wL
xl ∈ R
x1 x2 xL−1
The goal is to compute the gradient of the loss value xL (output) with respect to each network
parameter wl:
df
d(vec wl)
=
d
d(vec wl)
[fL(·; wL) ◦ ... ◦ f2(·; w2) ◦ f1(x0; w1)] .
By applying the chain rule and by using the matrix notation introduced above, the derivative
can be written as
df
d(vec wl)
=
d vec fL(xL−1; wL)
d(vec xL−1)
× · · · ×
d vec fl+1(xl; wl+1)
d(vec xl)
×
d vec fl(xl−1; wl)
d(vec wl )
(2.1)
where the derivatives are computed at the working point determined by the input x0 and the
current value of the parameters.
Note that, since the network output xl is a scalar quantity, the target derivative
df/d(vec wl) has the same number of elements of the parameter vector wl, which is moder-
ate. However, the intermediate Jacobian factors have, as seen above, an unmanageable size.
In order to avoid computing these factor explicitly, we can proceed as follows.
Start by multiplying the output of the last layer by a tensor pL = 1 (note that this tensor
is a scalar just like the variable xL):
pL ×
df
d(vec wl)
= pL ×
d vec fL(xL−1; wL)
d(vec xL−1)
(vec pL−1)
× · · · ×
d(vec xl)
×
d(vec wl )
= (vec pL−1) × · · · ×
d(vec xl)
×
d(vec wl )
In the second line the last two factors to the left have been multiplied obtaining a new
tensor pL−1 that has the same size as the variable xL−1. The factor pL−1 can therefore be

explicitly stored. The construction is then repeated by multiplying pairs of factors from left
to right, obtaining a sequence of tensors pL−2, . . . , pl until the desired derivative is obtained.
Note that, in doing so, no large tensor is ever stored in memory. This process is known as
backpropagation.
In general, tensor pl is obtained from pl+1 as the product:
(vec pl) = (vec pl+1) ×
d(vec xl)
.
The key to implement backpropagation is to be able to compute these products without
explicitly computing and storing in memory the second factor, which is a large Jacobian
matrix. Since computing the derivative is a linear operation, this product can be interpreted
as the derivative of the layer projected along direction pl+1:
pl =
d pl+1, f(xl; wl)
dxl
. (2.2)
Here ·, · denotes the inner product between tensors, which results in a scalar quantity.
Hence the derivative (2.2) needs not to use the vec notation, and yields a tensor pl that has
the same size as xl as expected.
In order to implement backpropagation, a CNN toolbox provides implementations of each
layer f that provide:
ˆ A forward mode, computing the output y = f(x; w) of the layer given its input x
and parameters w.
ˆ A backward mode, computing the projected derivatives
d p, f(x; w)
dx
and
d p, f(x; w)
dw
,
given, in addition to the input x and parameters w, a tensor p that the same size as y.
This is best illustrated with an example. Consider a layer f such as the convolution operator
implemented by the MatConvNet vl_nnconv command. In the “forward” mode, one calls
the function as y = vl_nnconv(x,w,[]) to apply the ﬁlters w to the input x and obtain the
output y. In the “backward mode”, one calls [dx, dw] = vl_nnconv(x,w,[],p). As explained
above, dx, dw, and p have the same size as x, w, and y, respectively. The computation of large
Jacobian is encapsulated in the function call and never carried out explicitly.
2.3.3 Backpropagation networks
In this section, we provide a schematic interpretation of backpropagation and show how it
can be implemented by “reversing” the NN computational graph.
The projected derivative of eq. (2.2) can be seen as the derivative of the following mini-
network:

x f ·, · z ∈ R
w p
y
In the context of back-propagation, it can be useful to think of the projection p as the
“linearization” of the rest of the network from variable y down to the loss. The projected
derivative can also be though of as a new layer (dx, dw) = df(x, w, p) that, by computing
the derivative of the mini-network, operates in the reverse direction:
dfdx
dw
wx
p
By construction (see eq. (2.2)), the function df is linear in the argument p.
Using this notation, the forward and backward passes through the original network can
be rewritten as evaluating an extended network which contains a BP-reverse of the original
one (in blue in the diagram):
x0 f1 x1
w1
f2 x2
w2
. . . xL−1 fL xL
wL
dfL dpL
dwL
dxL−1
. . .df2 dx2
dw2
df1 dx1
dw1
dx0
2.3.4 Backpropagation in DAGs
Assume that the DAG has a single output variable xL and assume, without loss of generality,
that all variables are sorted in order of computation (x0, x1, . . . , xL−1, xL) according to the

DAG structure. Furthermore, in order to simplify the notation, assume that this list contains
both data and parameter variables, as the distinction is moot for the discussion in this section.
We can cut the DAG at any point in the sequence by fixing x0, . . . , xl−1 to some arbitrary
value and dropping all the DAG layers that feed into them, effectively transforming the first l
variables into inputs. Then, the rest of the DAG defines a function hl that maps these input
variables to the output xL:
xL = hl(x0, x1, . . . , xl−1).
Next, we show that backpropagation in a DAG iteratively computes the projected derivatives
of all functions h1, . . . , hL with respect to all their parameters.
Backpropagation starts by initializing variables (dx0, . . . , dxl−1) to null tensors of the
same size as (x0, . . . , xl−1). Next, it computes the projected derivatives of
xL = hL(x0, x1, . . . , xL−1) = fπL
(x0, x1, . . . , xL−1).
Here πl denotes the index of the layer fπl
that computes the value of the variable xl. There
is at most one such layer, or none if xl is an input or parameter of the original NN. In the
first case, the layer may depend on any of the variables prior to xl in the sequence, so that
general one has:
xl = fπl
(x0, . . . , xl−1).
At the beginning of backpropagation, since there are no intermediate variables between xL−1
and xL, the function hL is the same as the last layer fπL
. Thus the projected derivatives of
hL are the same as the projected derivatives of fπL
, resulting in the equation
∀t = 0, . . . , L − 1 : dxt ← dxt +
d pL, fπL
(x0, . . . , xt−1)
dxt
.
Here, for uniformity with the other iterations, we use the fact that dxl are initialized to zero
anaccumulate the values instead of storing them. In practice, the update operation needs to
be carried out only for the variables xl that are actual inputs to fπL
, which is often a tiny
fraction of all the variables in the DAG.
After the update, each dxt contains the projected derivative of function hL with respect
to the corresponding variable:
∀t = 0, . . . , L − 1 : dxt =
d pL, hL(x0, . . . , xl−1)
dxt
.
Given this information, the next iteration of backpropagation updates the variables to con-
tain the projected derivatives of hL−1 instead. In general, given the derivatives of hl+1,
backpropagation computes the derivatives of hl by using the relation
xL = hl(x0, x1, . . . , xl−1) = hl+1(x0, x1, . . . , xl−1, fπL
(x0, . . . , xl−1))
Applying the chain rule to this expression, for all 0 ≤ t ≤ l − 1:
d p, hl
d(vec xt)
=
d p, hl+1
d(vec xt)
+
d pL, hl+1
d(vec xl)
vec dxl
d vec fπl
d(vec xt)
.

This yields the update equation
∀t = 0, . . . , l − 1 : dxt ← dxt +
d pl, fπl
(x0, . . . , xl−1)
dxt
, where pl = dxl. (2.3)
Once more, the update needs to be explicitly carried out only for the variables xt that are
actual inputs of fπl
. In particular, if xl is a data input or a parameter of the original neural
network, then xl does not depend on any other variables or parameters and fπl
is a nullary
function (i.e. a function with no arguments). In this case, the update does not do anything.
After iteration L − l + 1 completes, backpropagation remains with:
∀t = 0, . . . , l − 1 : dxt =
d pL, hl(x0, . . . , xl−1)
dxt
.
Note that the derivatives for variables xt, l ≤ t ≤ L − 1 are not updated since hl does not
depend on any of those. Thus, after all L iterations are complete, backpropagation terminates
with
∀l = 1, . . . , L : dxl−1 =
d pL, hl(x0, . . . , xl−1)
dxl−1
.
As seen above, functions hl are obtained from the original network f by transforming variables
x0, . . . , xl−1 into to inputs. If xl−1 was already an input (data or parameter) of f, then the
derivative dxl−1 is applicable to f as well.
Backpropagation can be summarized as follows:
Given: a DAG neural network f with a single output xL, the values of all input variables
(including the parameters), and the value of the projection pL (usually xL is a scalar
and pL = pL = 1):
1. Sort all variables by computation order (x0, x1, . . . , xL) according to the DAG.
2. Perform a forward pass through the network to compute all the intermediate vari-
able values.
3. Initialize (dx0, . . . , dxL−1) to null tensors with the same size as the corresponding
variables.
4. For l = L, L − 1, . . . , 2, 1:
a) Find the index πl of the layer xl = fπl
(x0, . . . , xl−1) that evaluates variable xl.
If there is no such layer (because xl is an input or parameter of the network),
go to the next iteration.
b) Update the variables using the formula:
∀t = 0, . . . , l − 1 : dxt ← dxt +
d dxl, fπl
(x0, . . . , xl−1)
dxt
.
To do so eﬃciently, use the “backward mode” of the layer fπl
to compute its
derivative projected onto dxl as needed.

2.3.5 DAG backpropagation networks
Just like for sequences, backpropagation in DAGs can be implemented as a corresponding
BP-reversed DAG. To construct the reversed DAG:
1. For each layer fl, and variable/parameter xt and wl, create a corresponding layer dfl
and variable/parameter dxt and dwl.
2. If a variable xt (or parameter wl) is an input of fl, then it is an input of dfl as well.
3. If a variable xt (or parameter wl) is an input of fl, then the variable dxt (or the
parameter dwl) is an output dfl.
4. In the previous step, if a variable xt (or parameter wl) is input to two or more layers in
f, then dxt would be the output of two or more layers in the reversed network, which
creates a conﬂict. Resolve these conﬂicts by inserting a summation layer that adds
these contributions (this corresponds to the summation in the BP update equation
(2.3)).
The BP network corresponding to the DAG of Fig. 2.1 is given in Fig. 2.2.

f1 x1
x0 f3 x3
f2 x2 f5 x7
x5
x4 f4
x6
df1 dx1
dx0 df3 dx3
df2 dx2 df5 p7
dx5
dx4 df4
p6
w1
w2
w4
w5
Σ dw1
dw2
Σ
dw4
dw5
Figure 2.2: Backpropagation network for a DAG.

Chapter 3
Wrappers and pre-trained models
It is easy enough to combine the computational blocks of chapter 4 “manually”. However, it
is usually much more convenient to use them through a wrapper that can implement CNN
architectures given a model specification. The available wrappers are briefly summarised in
section 3.1.
MatConvNet also comes with many pre-trained models for image classification (most
of which are trained on the ImageNet ILSVRC challenge), image segmentation, text spotting,
and face recognition. These are very simple to use, as illustrated in section 3.2.
3.1 Wrappers
MatConvNet provides two wrappers: SimpleNN for basic chains of blocks (section 3.1.1)
and DagNN for blocks organized in more complex direct acyclic graphs (section 3.1.2).
3.1.1 SimpleNN
The SimpleNN wrapper is suitable for networks consisting of linear chains of computational
blocks. It is largely implemented by the vl_simplenn function (evaluation of the CNN and of
its derivatives), with a few other support functions such as vl_simplenn_move (moving the
CNN between CPU and GPU) and vl_simplenn_display (obtain and/or print information
about the CNN).
vl_simplenn takes as input a structure net representing the CNN as well as input x and
potentially output derivatives dzdy, depending on the mode of operation. Please refer to the
inline help of the vl_simplenn function for details on the input and output formats. In fact,
the implementation of vl_simplenn is a good example of how the basic neural net building
blocks can be used together and can serve as a basis for more complex implementations.
3.1.2 DagNN
The DagNN wrapper is more complex than SimpleNN as it has to support arbitrary graph
topologies. Its design is object oriented, with one class implementing each layer type. While
this adds complexity, and makes the wrapper slightly slower for tiny CNN architectures (e.g.
MNIST), it is in practice much more flexible and easier to extend.
21

22 CHAPTER 3. WRAPPERS AND PRE-TRAINED MODELS
DagNN is implemented by the dagnn.DagNN class (under the dagnn namespace).
3.2 Pre-trained models
vl_simplenn is easy to use with pre-trained models (see the homepage to download some).
For example, the following code downloads a model pre-trained on the ImageNet data and
applies it to one of MATLAB stock images:
% setup MatConvNet in MATLAB
run matlab/vl_setupnn
% download a pre−trained CNN from the web
urlwrite(...
'http://www.vlfeat.org/matconvnet/models/imagenet−vgg−f.mat', ...
'imagenet−vgg−f.mat') ;
net = load('imagenet−vgg−f.mat') ;
% obtain and preprocess an image
im = imread('peppers.png') ;
im_ = single(im) ; % note: 255 range
im_ = imresize(im_, net.meta.normalization.imageSize(1:2)) ;
im_ = im_ − net.meta.normalization.averageImage ;
Note that the image should be preprocessed before running the network. While preprocessing
specifics depend on the model, the pre-trained model contains a net.meta.normalization
field that describes the type of preprocessing that is expected. Note in particular that this
network takes images of a fixed size as input and requires removing the mean; also, image
intensities are normalized in the range [0,255].
The next step is running the CNN. This will return a res structure with the output of
the network layers:
% run the CNN
res = vl_simplenn(net, im_) ;
The output of the last layer can be used to classify the image. The class names are
contained in the net structure for convenience:
% show the classification result
scores = squeeze(gather(res(end).x)) ;
[bestScore, best] = max(scores) ;
figure(1) ; clf ; imagesc(im) ;
title(sprintf('%s (%d), score %.3f',...
net.meta.classes.description{best}, best, bestScore)) ;
Note that several extensions are possible. First, images can be cropped rather than
rescaled. Second, multiple crops can be fed to the network and results averaged, usually for
improved results. Third, the output of the network can be used as generic features for image
encoding.

3.3. LEARNING MODELS 23
3.3 Learning models
As MatConvNet can compute derivatives of the CNN using backpropagation, it is simple
to implement learning algorithms with it. A basic implementation of stochastic gradient
descent is therefore straightforward. Example code is provided in examples/cnn_train.
This code is flexible enough to allow training on NMINST, CIFAR, ImageNet, and probably
many other datasets. Corresponding examples are provided in the examples/ directory.
3.4 Running large scale experiments
For large scale experiments, such as learning a network for ImageNet, a NVIDIA GPU (at
least 6GB of memory) and adequate CPU and disk speeds are highly recommended. For
example, to train on ImageNet, we suggest the following:
ˆ Download the ImageNet data http://www.image-net.org/challenges/LSVRC. In-
stall it somewhere and link to it from data/imagenet12
ˆ Consider preprocessing the data to convert all images to have a height of 256 pixels.
This can be done with the supplied utils/preprocess-imagenet.sh script. In this
manner, training will not have to resize the images every time. Do not forget to point
the training code to the pre-processed data.
ˆ Consider copying the dataset into a RAM disk (provided that you have enough memory)
for faster access. Do not forget to point the training code to this copy.
ˆ Compile MatConvNet with GPU support. See the homepage for instructions.
Once your setup is ready, you should be able to run examples/cnn_imagenet (edit the
file and change any flag as needed to enable GPU support and image pre-fetching on multiple
threads).
If all goes well, you should expect to be able to train with 200-300 images/sec.
3.5 Reading images
MatConvNet provides the tool vl_imreadjpeg to quickly read images, transform them,
and move them to the GPU.
Image cropping and scaling. Several options in vl_imreadjpeg control how images are
cropped and rescaled. The procedure is as follows:
1. Given an input image of size H × W, first the size of the output image Ho × Wo
is determined. The output size is either equal the input size ((Ho, Wo) = (H, W)),
equal to a specified constant size, or obtained by setting the minimum side equal to
a specified constant and rescaling the other accordingly ((Ho, Wo) = s(H, W), s =
max{Ho/H, Wo/W}).

24 CHAPTER 3. WRAPPERS AND PRE-TRAINED MODELS
2. Next, the crop size Hc × Wc is determined, starting from the crop anisotropy a =
(Wo/Ho)/(Wc/Hc), i.e. the relative change of aspect ratio from the crop to the output:
(Hc, Wc) ∝ (Ho/a, aWo). One option is to choose a = (W/H)/(Wo/Ho) such that the
crop has the same aspect raio of the input image, which allows to squash a rectangular
input into a square output. Another option is to sample it as a ∼ U([a−, a+]) where
a−, a+ are, respectively, the minimum and maximum anisotropy.
3. The relative crop scale is determined by sampling a parameter ρ ∼ U([ρ−, ρ+]) where
ρ−, ρ+ are, respectively, the minimum and maximum relative crop sizes. The absolute
maximum size is determined by the size of the input image. Overall, the shape of the
crop is given by:
(Hc, Wc) = ρ(Ho/a, aWo) min{aH/Ho, W/(aWo)}.
4. Given the crop size (Hc, Wc), the crop is extracted relative to the input image either in
the middle (center crop) or randomly shifted.
5. Finally, it is also possible to flip a crop left-to-right with a 50% probability.
In the simples case, vl_imreadjpeg extract an image as is, without any processing. A a
standard center crop of 128 pixels can be obtained by setting Ho = Wo = 128, (resize
option), a− = a+ = 1 (CropAnisotropy option), and ρ− = ρ+ = 1 (CropSize option). In the
input image, this crop is isotropically stretched to fill either its width or height. If the input
image is rectangular, such a crop can either slide horizontally or vertically (CropLocation),
but not both. Setting ρ− = ρ+ = 0.9 makes the crop slightly smaller, allowing it to shift in
both directions. Setting ρ− = 0.9 and ρ+ = 1.0 allows picking differently-sized crops each
time. Setting a− = 0.9 and a+ = 1.2 allows the crops to be slightly elongated or widened.
Color post-processing. vl_imreadjpeg supports some basic colour postrpocessing. It
allows to subtract from all the pixels a constant shift µ ∈ R3
( µ can also be a Ho × Wo
image for fixed-sized crops). It also allows to add a random shift vector (sample independently
for each image in a batch), and to also perturb randomly the saturation and contrast of the
image. These transformations are discussed in detail next.
The brightness shift is a constant offset b added to all pixels in the image, similarly to the
vector µ, which is however subtracted and constant for all images in the batch. The shift is
randomly sampled from a Gaussian distribution with standard deviation B. Here, B ∈ R3×3
is the square root of the covariance matrix of the Gaussian, such that:
b ← Bω, ω ∼ N(0, I).
If x(u, v) ∈ R3
is an RGB triplet at location (u, v) in the image, average color subtraction
and brightness shift results in the transformation:
x(u, v) ← x(u, v) + b − µ.
After this shift is applied, the image contrast is changed as follow:
x(u, v) ← γx(u, v) + (1 − γ) avg
uv
[x(u, v)], γ ∼ U([1 − C, 1 + C])

3.5. READING IMAGES 25
where the coefficient γ is uniformly sampled in the interval [1 − C, 1 + C] where is C is the
contrast deviation coefficient. Note that, since γ can be larger than one, contrast can also
be increased.
The last transformation changes the saturation of the image. This is controlled by the
saturation deviation coefficient S:
x(u, v) ← σx(u, v) +
1 − σ
3
11 x(u, v), σ ∼ U([1 − S, 1 + S])
Overall, pixels are transformed as follows:
x(u, v) ← σI +
1 − σ
3
11 γx(u, v) + (1 − γ) avg
uv
[x(u, v)] + Bω − µ .
For grayscale images, changing the saturation does not do anything (unless ones applies first
a colored shift, which effectively transforms a grayscale image into a color one).

Chapter 4
Computational blocks
This chapters describes the individual computational blocks supported by MatConvNet.
The interface of a CNN computational block <block> is designed after the discussion in
chapter 2. The block is implemented as a MATLAB function y = vl_nn<block>(x,w) that
takes as input MATLAB arrays x and w representing the input data and parameters and
returns an array y as output. In general, x and y are 4D real arrays packing N maps or
images, as discussed above, whereas w may have an arbitrary shape.
The function implementing each block is capable of working in the backward direction
as well, in order to compute derivatives. This is done by passing a third optional argument
dzdy representing the derivative of the output of the network with respect to y; in this case,
the function returns the derivatives [dzdx,dzdw] = vl_nn<block>(x,w,dzdy) with respect to
the input data and parameters. The arrays dzdx, dzdy and dzdw have the same dimensions
of x, y and w respectively (see section 2.3).
Different functions may use a slightly different syntax, as needed: many functions can
take additional optional arguments, specified as property-value pairs; some do not have
parameters w (e.g. a rectified linear unit); others can take multiple inputs and parameters, in
which case there may be more than one x, w, dzdx, dzdy or dzdw. See the rest of the chapter
and MATLAB inline help for details on the syntax.1
The rest of the chapter describes the blocks implemented in MatConvNet, with a
particular focus on their analytical definition. Refer instead to MATLAB inline help for
further details on the syntax.
4.1 Convolution
The convolutional block is implemented by the function vl_nnconv. y=vl_nnconv(x,f,b) com-
putes the convolution of the input map x with a bank of K multi-dimensional filters f and
biases b. Here
x ∈ RH×W×D
, f ∈ RH ×W ×D×D
, y ∈ RH ×W ×D
.
1
Other parts of the library will wrap these functions into objects with a perfectly uniform interface;
however, the low-level functions aim at providing a straightforward and obvious interface even if this means
differing slightly from block to block.
27

28 CHAPTER 4. COMPUTATIONAL BLOCKS
P- = 2 P+ = 3
1 2 3 4
1 2 3 4
1 2 3 4
1 2 3 4
1 2
1 2 3 4
1 2 3 4
1 2 3 4
x
y
Figure 4.1: Convolution. The figure illustrates the process of filtering a 1D signal x by a
filter f to obtain a signal y. The filter has H = 4 elements and is applied with a stride of
Sh = 2 samples. The purple areas represented padding P− = 2 and P+ = 3 which is zero-
filled. Filters are applied in a sliding-window manner across the input signal. The samples of
x involved in the calculation of a sample of y are shown with arrow. Note that the rightmost
sample of x is never processed by any filter application due to the sampling step. While in
this case the sample is in the padded region, this can happen also without padding.
The process of convolving a signal is illustrated in fig. 4.1 for a 1D slice. Formally, the output
is given by
yi j d = bd +
H
i =1
W
j =1
D
d =1
fi j d × xi +i −1,j +j −1,d ,d .
The call vl_nnconv(x,f,[]) does not use the biases. Note that the function works with arbi-
trarily sized inputs and filters (as opposed to, for example, square images). See section 6.1
for technical details.
Padding and stride. vl_nnconv allows to specify top-bottom-left-right paddings
(P−
h , P+
h , P−
w , P+
w ) of the input array and subsampling strides (Sh, Sw) of the output array:
yi j d = bd +
H
i =1
W
j =1
D
d =1
fi j d × xSh(i −1)+i −P−
h ,Sw(j −1)+j −P−
w ,d ,d .
In this expression, the array x is implicitly extended with zeros as needed.
Output size. vl_nnconv computes only the “valid” part of the convolution; i.e. it requires
each application of a filter to be fully contained in the input support. The size of the output
is computed in section 5.2 and is given by:
H = 1 +
H − H + P−
h + P+
h
Sh
.

4.2. CONVOLUTION TRANSPOSE (DECONVOLUTION) 29
Note that the padded input must be at least as large as the filters: H + P−
h + P+
h ≥ H ,
otherwise an error is thrown.
Receptive field size and geometric transformations. Very often it is useful to geo-
metrically relate the indexes of the various array to the input data (usually images) in terms
of coordinate transformations and size of the receptive field (i.e. of the image region that
affects an output). This is derived in section 5.2.
Fully connected layers. In other libraries, fully connected blocks or layers are linear
functions where each output dimension depends on all the input dimensions. MatConvNet
does not distinguish between fully connected layers and convolutional blocks. Instead, the
former is a special case of the latter obtained when the output map y has dimensions W =
H = 1. Internally, vl_nnconv handles this case more efficiently when possible.
Filter groups. For additional flexibility, vl_nnconv allows to group channels of the input
array x and apply different subsets of filters to each group. To use this feature, specify
as input a bank of D filters f ∈ RH ×W ×D ×D
such that D divides the number of input
dimensions D. These are treated as g = D/D filter groups; the first group is applied to
dimensions d = 1, . . . , D of the input x; the second group to dimensions d = D + 1, . . . , 2D
and so on. Note that the output is still an array y ∈ RH ×W ×D
.
An application of grouping is implementing the Krizhevsky and Hinton network [8] which
uses two such streams. Another application is sum pooling; in the latter case, one can specify
D groups of D = 1 dimensional filters identical filters of value 1 (however, this is considerably
slower than calling the dedicated pooling function as given in section 4.3).
Dilation. vl_nnconv allows kernels to be spatially dilated on the fly by inserting zeros
between elements. For instance, a dilation factor d = 2 causes the 1D kernel [f1, f2] to be
implicitly transformed in the kernel [f1, 0, 0, f2]. Thus, with dilation factors dh, dw, a filter of
size (Hf , Wf ) is equivalent to a filter of size:
H = dh(Hf − 1) + 1, W = dw(Wf − 1) + 1.
With dilation, the convolution becomes:
yi j d = bd +
Hf
i =1
Wf
j =1
D
d =1
fi j d × xSh(i −1)+dh(i −1)−P−
h +1,Sw(j −1)+dw(j −1)−P−
w +1,d ,d .
4.2 Convolution transpose (deconvolution)
The convolution transpose block (sometimes referred to as “deconvolution”) is the transpose
of the convolution block described in section 4.1. In MatConvNet, convolution transpose
is implemented by the function vl_nnconvt.
In order to understand convolution transpose, let:
x ∈ RH×W×D
, f ∈ RH ×W ×D×D
, y ∈ RH ×W ×D
,

C- = 2 C+ = 3
1 2 3 4
1 2 3 4
1 2 3 4
1 2 4
2 3 4
1 2 3 4
1 2 3 4
y
x
1 2 3 4
Figure 4.2: Convolution transpose. The figure illustrates the process of filtering a 1D
signal x by a filter f to obtain a signal y. The filter is applied as a sliding-window, forming
a pattern which is the transpose of the one of fig. 4.1. The filter has H = 4 samples in
total, although each filter application uses two of them (blue squares) in a circulant manner.
The purple areas represent crops with C− = 2 and C+ = 3 which are discarded. The arrows
exemplify which samples of x are involved in the calculation of a particular sample of y. Note
that, differently from the forward convolution fig. 4.1, there is no need to add padding to
the input array; instead, the convolution transpose filters can be seen as being applied with
maximum input padding (more would result in zero output values), and the latter can be
reduced by cropping the output instead.
be the input tensor, filters, and output tensors. Imagine operating in the reverse direction
by using the filter bank f to convolve the output y to obtain the input x, using the defini-
tions given in section 4.1 for the convolution operator; since convolution is linear, it can be
expressed as a matrix M such that vec x = M vec y; convolution transpose computes instead
vec y = M vec x. This process is illustrated for a 1D slice in fig. 4.2.
There are two important applications of convolution transpose. The first one is the so
called deconvolutional networks [14] and other networks such as convolutional decoders that
use the transpose of a convolution. The second one is implementing data interpolation.
In fact, as the convolution block supports input padding and output downsampling, the
convolution transpose block supports input upsampling and output cropping.
Convolution transpose can be expressed in closed form in the following rather unwieldy
expression (derived in section 6.2):
yi j d =
D
d =1
q(H ,Sh)
i =0
q(W ,Sw)
j =0
f1+Shi +m(i +P−
h ,Sh), 1+Swj +m(j +P−
w ,Sw), d ,d ×
x1−i +q(i +P−
h ,Sh), 1−j +q(j +P−
w ,Sw), d (4.1)
where
m(k, S) = (k − 1) mod S, q(k, n) =
k − 1
S
,

4.3. SPATIAL POOLING 31
(Sh, Sw) are the vertical and horizontal input upsampling factors, (P−
h , P+
h , P−
h , P+
h ) the output
crops, and x and f are zero-padded as needed in the calculation. Note also that filter k is
stored as a slice f:,:,k,: of the 4D tensor f.
The height of the output array y is given by
H = Sh(H − 1) + H − P−
h − P+
h .
A similar formula holds true for the width. These formulas are derived in section 5.3 along
with an expression for the receptive field of the operator.
We now illustrate the action of convolution transpose in an example (see also fig. 4.2).
Consider a 1D slice in the vertical direction, assume that the crop parameters are zero,
and that Sh > 1. Consider the output sample yi where the index i is chosen such that
Sh divides i − 1; according to (4.1), this sample is obtained as a weighted summation of
xi /Sh
, xi /Sh−1, ... (note that the order is reversed). The weights are the filter elements f1,
fSh
,f2Sh
, . . . subsampled with a step of Sh. Now consider computing the element yi +1; due to
the rounding in the quotient operation q(i , Sh), this output sample is obtained as a weighted
combination of the same elements of the input x that were used to compute yi ; however,
the filter weights are now shifted by one place to the right: f2, fSh+1,f2Sh+1, . . . . The same
is true for i + 2, i + 3, . . . until we hit i + Sh. Here the cycle restarts after shifting x to
the right by one place. Effectively, convolution transpose works as an interpolating filter.
4.3 Spatial pooling
vl_nnpool implements max and sum pooling. The max pooling operator computes the max-
imum response of each feature channel in a H × W patch
yi j d = max
1≤i ≤H ,1≤j ≤W
xi +i −1,j +j −1,d.
resulting in an output of size y ∈ RH ×W ×D
, similar to the convolution operator of sec-
tion 4.1. Sum-pooling computes the average of the values instead:
yi j d =
1
W H 1≤i ≤H ,1≤j ≤W
xi +i −1,j +j −1,d.
Detailed calculation of the derivatives is provided in section 6.3.
Padding and stride. Similar to the convolution operator of section 4.1, vl_nnpool sup-
ports padding the input; however, the effect is different from padding in the convolutional
block as pooling regions straddling the image boundaries are cropped. For max pooling,
this is equivalent to extending the input data with −∞; for sum pooling, this is similar to
padding with zeros, but the normalization factor at the boundaries is smaller to account for
the smaller integration area.

4.4 Activation functions
MatConvNet supports the following activation functions:
ˆ ReLU. vl_nnrelu computes the Rectified Linear Unit (ReLU):
yijd = max{0, xijd}.
ˆ Sigmoid. vl_nnsigmoid computes the sigmoid:
yijd = σ(xijd) =
1
1 + e−xijd
.
See section 6.4 for implementation details.
4.5 Spatial bilinear resampling
vl_nnbilinearsampler uses bilinear interpolation to spatially warp the image according to
an input transformation grid. This operator works with an input image x, a grid g, and an
output image y as follows:
x ∈ RH×W×C
, g ∈ [−1, 1]2×H ×W
, y ∈ RH ×W ×C
.
The same transformation is applied to all the features channels in the input, as follows:
yi j c =
H
i=1
W
j=1
xijc max{0, 1 − |αvg1i j + βv − i|} max{0, 1 − |αug2i j + βu − j|}, (4.2)
where, for each feature channel c, the output yi j c at the location (i , j ), is a weighted sum
of the input values xijc in the neighborhood of location (g1i j , g2i j ). The weights, as given
in (4.2), correspond to performing bilinear interpolation. Furthermore, the grid coordinates
are expressed not in pixels, but relative to a reference frame that extends from −1 to 1 for
all spatial dimensions of the input image; this is given by choosing the coefficients as:
αv =
H − 1
2
, βv = −
H + 1
2
, αu =
W − 1
2
, βu = −
W + 1
2
.
See section 6.5 for implementation details.
4.6 Region of interest pooling
The region of interest (ROI) pooling block applies max or average pooling to specified sub-
windows of a tensor. A region is a rectangular region R = (u−, v−, u+, v+). The region itself
is partitioned into (H , W ) tiles along the vertical and horizontal directions. The edges of
the tiles have coordinates
vi = v− + (v+ − v− + 1)(i − 1), i = 1, . . . , H ,
uj = u− + (u+ − u− + 1)(j − 1), j = 1, . . . , W .

4.7. NORMALIZATION 33
Following the implementation of [3], the H × W pooling tiles are given by
Ωi j = { vi + 1, . . . , vi +1 } × { ui + 1, . . . , ui +1 }.
Then the input and output tensors are as follows:
x ∈ RH×W×C
, y ∈ RH ×W ×C
,
where
yi j c = max
(i,j)∈Ωi j
xijc.
Alternatively, max can be replaced by the averaging operator.
The extent of each region is defined by four coordinates as specified above; however,
differently from tensor indexes, these use (0, 0) as the coordinate of the top-left pixel. In
fact, if there is a single tile (H = W = 1), then the region (0, 0, H − 1, W − 1) covers the
whole input image:
Ω11 = {1, . . . , W} × {1, . . . , H}.
In more details, the input of the block is a sequence of K regions. Each region pools one
of the T images in the batch stored in x ∈ RH×W×C×T
. Regions are therefore specified as a
tensor R ∈ R5×K
, where the first coordinate is the index of the pooled image in the batch.
The output is a y ∈ RH ×W ×C×K
tensor.
For compatibility with [3], furthermore, the region coordinates are rounded to the nearest
integer before the definitions above are used. Note also that, due to the discretization details,
1) tiles always contain at least one pixel, 2) there can be a pixel of overlap between them
and 3) the discretization has a slight bias towards left-top pixels.
4.7 Normalization
4.7.1 Local response normalization (LRN)
vl_nnnormalize implements the Local Response Normalization (LRN) operator. This oper-
ator is applied independently at each spatial location and to groups of feature channels as
follows:
yijk = xijk

κ + α
t∈G(k)
x2
ijt


−β
,
where, for each output channel k, G(k) ⊂ {1, 2, . . . , D} is a corresponding subset of input
channels. Note that input x and output y have the same dimensions. Note also that the
operator is applied uniformly at all spatial locations.
See section 6.6.1 for implementation details.
4.7.2 Batch normalization
vl_nnbnorm implements batch normalization [5]. Batch normalization is somewhat different
from other neural network blocks in that it performs computation across images/feature

maps in a batch (whereas most blocks process different images/feature maps individually).
y = vl_nnbnorm(x, w, b) normalizes each channel of the feature map x averaging over spatial
locations and batch instances. Let T be the batch size; then
x, y ∈ RH×W×K×T
, w ∈ RK
, b ∈ RK
.
Note that in this case the input and output arrays are explicitly treated as 4D tensors in
order to work with a batch of feature maps. The tensors w and b define component-wise
multiplicative and additive constants. The output feature map is given by
yijkt = wk
xijkt − µk
σ2
k +
+bk, µk =
1
HWT
H
i=1
W
j=1
T
t=1
xijkt, σ2
k =
1
HWT
H
i=1
W
j=1
T
t=1
(xijkt−µk)2
.
4.7.3 Spatial normalization
vl_nnspnorm implements spatial normalization. The spatial normalization operator acts on
different feature channels independently and rescales each input feature by the energy of the
features in a local neighbourhood . First, the energy of the features in a neighbourhood
W × H is evaluated
n2
i j d =
1
W H 1≤i ≤H ,1≤j ≤W
x2
i +i −1− H −1
2
,j +j −1− W −1
2
,d
.
In practice, the factor 1/W H is adjusted at the boundaries to account for the fact that
neighbors must be cropped. Then this is used to normalize the input:
yi j d =
1
(1 + αn2
i j d)β
xi j d.
4.7.4 Softmax
vl_nnsoftmax computes the softmax operator:
yijk =
exijk
D
t=1 exijt
.
Note that the operator is applied across feature channels and in a convolutional manner
at all spatial locations. Softmax can be seen as the combination of an activation function
(exponential) and a normalization operator. See section 6.6.4 for implementation details.

4.8. CATEGORICAL LOSSES 35
4.8 Categorical losses
The purpose of a categorical loss function (x, c) is to compare a prediction x to a ground
truth class label c. As in the rest of MatConvNet, the loss is treated as a convolutional
operator, in the sense that the loss is evaluated independently at each spatial location. How-
ever, the contribution of different samples are summed together (possibly after weighting)
and the output of the loss is a scalar. Section 4.8.1 losses useful for multi-class classification
and the section 4.8.2 losses useful for binary attribute prediction. Further technical details
are in section 6.7. vl_nnloss implements the following all of these.
4.8.1 Classification losses
Classification losses decompose additively as follows:
(x, c) =
ijn
wij1n (xij:n, cij:n). (4.3)
Here x ∈ RH×W×C×N
and c ∈ {1, . . . , C}H×W×1×N
, such that the slice xij:n represent a vector
of C class scores and and cij1n is the ground truth class label. The ìnstanceWeights` option
can be used to specify the tensor w of weights, which are otherwise set to all ones; w has
the same dimension as c.
Unless otherwise noted, we drop the other indices and denote by x and c the slice xij:n
and the scalar cij1n. vl_nnloss automatically skips all samples such that c = 0, which can
be used as an “ignore” label.
Classification error. The classification error is zero if class c is assigned the largest score
and zero otherwise:
(x, c) = 1 c = argmax
k
xc . (4.4)
Ties are broken randomly.
Top-K classification error. The top-K classification error is zero if class c is within the
top K ranked scores:
(x, c) = 1 [|{k : xk ≥ xc}| ≤ K] . (4.5)
The classification error is the same as the top-1 classification error.
Log loss or negative posterior log-probability. In this case, x is interpreted as a vector
of posterior probabilities p(k) = xk, k = 1, . . . , C over the C classes. The loss is the negative
log-probability of the ground truth class:
(x, c) = − log xc. (4.6)
Note that this makes the implicit assumption x ≥ 0, k xk = 1. Note also that, unless
xc > 0, the loss is undefined. For these reasons, x is usually the output of a block such as

softmax that can guarantee these conditions. However, the composition of the naive log loss
and softmax is numerically unstable. Thus this is implemented as a special case below.
Generally, for such a loss to make sense, the score xc should be somehow in competition
with the other scores xk, k = c. If this is not the case, minimizing (4.6) can trivially be
achieved by maxing all xk large, whereas the intended effect is that xc should be large com-
pared to the xk, k = c. The softmax block makes the score compete through the normalization
factor.
Softmax log-loss or multinomial logistic loss. This loss combines the softmax block
and the log-loss block into a single block:
(x, c) = − log
exc
C
k=1 exk
= −xc + log
C
k=1
exk
. (4.7)
Combining the two blocks explicitly is required for numerical stability. Note that, by combin-
ing the log-loss with softmax, this loss automatically makes the score compete: (bx, c) ≈ 0
when xc k=c xk.
This loss is implemented also in the deprecated function vl_softmaxloss.
Multi-class hinge loss. The multi-class logistic loss is given by
(x, c) = max{0, 1 − xc}. (4.8)
Note that (x, c) = 0 ⇔ xc ≥ 1. This, just as for the log-loss above, this loss does not
automatically make the score competes. In order to do that, the loss is usually preceded by
the block:
yc = xc − max
k=c
xk.
Hence yc represent the confidence margin between class c and the other classes k = c. Just
like softmax log-loss combines softmax and loss, the next loss combines margin computation
and hinge loss.
Structured multi-class hinge loss. The structured multi-class logistic loss, also know as
Crammer-Singer loss, combines the multi-class hinge loss with a block computing the score
margin:
(x, c) = max 0, 1 − xc + max
k=c
xk . (4.9)
4.8.2 Attribute losses
Attribute losses are similar to classification losses, but in this case classes are not mutually
exclusive; they are, instead, binary attributes. Attribute losses decompose additively as
follows:
(x, c) =
ijkn
wijkn (xijkn, cijkn). (4.10)

Here x ∈ RH×W×C×N
and c ∈ {−1, +1}H×W×C×N
, such that the scalar xijkn represent
a confidence that attribute k is on and cij1n is the ground truth attribute label. The
ìnstanceWeights` option can be used to specify the tensor w of weights, which are oth-
erwise set to all ones; w has the same dimension as c.
Unless otherwise noted, we drop the other indices and denote by x and c the scalars xijkn
and cijkn. As before, samples with c = 0 are skipped.
Binary error. This loss is zero only if the sign of x − τ agrees with the ground truth label
c:
(x, c|τ) = 1[sign(x − τ) = c]. (4.11)
Here τ is a configurable threshold, often set to zero.
Binary log-loss. This is the same as the multi-class log-loss but for binary attributes.
Namely, this time xk ∈ [0, 1] is interpreted as the probability that attribute k is on:
(x, c) =
− log x, c = +1,
− log(1 − x), c = −1,
(4.12)
= − log c x −
1
2
+
1
2
. (4.13)
Similarly to the multi-class log loss, the assumption x ∈ [0, 1] must be enforced by the block
computing x.
Binary logistic loss. This is the same as the multi-class logistic loss, but this time x/2
represents the confidence that the attribute is on and −x/2 that it is off. This is obtained
by using the logistic function σ(x)
(x, c) = − log σ(cx) = − log
1
1 + e−cx
= − log
e
cx
2
e
cx
2 + e−cx
2
. (4.14)
Binary hinge loss. This is the same as the structured multi-class hinge loss but for binary
attributes:
(x, c) = max{0, 1 − cx}. (4.15)
There is a relationship between the hinge loss and the structured multi-class hinge loss which
is analogous to the relationship between binary logistic loss and multi-class logistic loss.
Namely, the hinge loss can be rewritten as:
(x, c) = max 0, 1 −
cx
2
+ max
k=c
kx
2
Hence the hinge loss is the same as the structure multi-class hinge loss for C = 2 classes,
where x/2 is the score associated to class c = 1 and −x/2 the score associated to class c = −1.

4.9 Comparisons
4.9.1 p-distance
The vl_nnpdist function computes the p-distance between the vectors in the input data x
and a target ¯x:
yij =
d
|xijd − ¯xijd|p
1
p
Note that this operator is applied convolutionally, i.e. at each spatial location ij one extracts
and compares vectors xij:. By specifying the option 'noRoot', true it is possible to compute
a variant omitting the root:
yij =
d
|xijd − ¯xijd|p
, p > 0.

Chapter 5
Geometry
This chapter looks at the geometry of the CNN input-output mapping.
5.1 Preliminaries
In this section we are interested in understanding how components in a CNN depend on
components in the layers before it, and in particular on components of the input. Since
CNNs can incorporate blocks that perform complex operations, such as for example cropping
their inputs based on data-dependent terms (e.g. Fast R-CNN), this information is generally
available only at “run time” and cannot be uniquely determined given only the structure
of the network. Furthermore, blocks can implement complex operations that are difficult to
characterise in simple terms. Therefore, the analysis will be necessarily limited in scope.
We consider blocks such as convolutions for which one can deterministically establish
dependency chains between network components. We also assume that all the inputs x and
outputs y are in the usual form of spatial maps, and therefore indexed as xi,j,d,k where i, j
are spatial coordinates.
Consider a layer y = f(x). We are interested in establishing which components of x
influence which components of y. We also assume that this relation can be expressed in
terms of a sliding rectangular window field, called receptive field. This means that the output
component yi ,j depends only on the input components xi,j where (i, j) ∈ Ω(i , j ) (note that
feature channels are implicitly coalesced in this discussion). The set Ω(i , j ) is a rectangle
defined as follows:
i ∈ αh(i − 1) + βh + −
∆h − 1
2
,
∆h − 1
2
(5.1)
j ∈ αv(j − 1) + βv + −
∆v − 1
2
,
∆v − 1
2
(5.2)
where (αh, αv) is the stride, (βh, βv) the offset, and (∆h, ∆v) the receptive field size.
39

40 CHAPTER 5. GEOMETRY
5.2 Simple filters
We now compute the receptive field geometry (αh, αv, βh, βv, ∆h, ∆v) for the most common
operators, namely filters. We consider in particular simple filters that are characterised by
an integer size, stride, and padding.
It suffices to reason in 1D. Let H bet the vertical filter dimension, Sh the subampling
stride, and P−
h and P+
h the amount of zero padding applied to the top and the bottom of the
input x. Here the value yi depends on the samples:
xi : i ∈ [1, H ] + Sh(i − 1) − P−
h = −
H − 1
2
,
H − 1
2
+ Sh(i − 1) − P−
h +
H + 1
2
.
Hence
αh = Sh, βh =
H + 1
2
− P−
h , ∆h = H .
A similar relation holds for the horizontal direction.
Note that many blocks (e.g. max pooling, LNR, ReLU, most loss functions etc.) have a
filter-like receptive field geometry. For example, ReLU can be considered a 1 × 1 filter, such
that H = Sh = 1 and P−
h = P+
h = 0. Note that in this case αh = 1, βh = 1 and ∆h = 1.
In addition to computing the receptive field geometry, we are often interested in determin-
ing the sizes of the arrays x and y throughout the architecture. In the case of filters, and once
more reasoning for a 1D slice, we notice that yi can be obtained for i = 1, 2, . . . , H where
H is the largest value of i before the receptive fields falls outside x (including padding). If
H is the height of the input array x, we get the condition
H + Sh(H − 1) − P−
h ≤ H + P+
h .
Hence
H =
H − H + P−
h + P+
h
Sh
+ 1. (5.3)
5.2.1 Pooling in Caffe
MatConvNet treats pooling operators like filters, using the rules above. In the library Caffe,
this is done slightly differently, creating some incompatibilities. In their case, the pooling
window is allowed to shift enough such that the last application always includes the last pixel
of the input. If the stride is greater than one, this means that the last application of the
pooling window can be partially outside the input boundaries even if padding is “officially”
zero.
More formally, if H is the pool size and H the size of the signal, the last application of
the pooling window has index i = H such that
Sh(i − 1) + H i =H
≥ H ⇔ H =
H − H
Sh
+ 1.
If there is padding, the same logic applies after padding the input image, such that the output
has height:
H =
H − H + P−
h + P+
h
Sh
+ 1.

5.2. SIMPLE FILTERS 41
This is the same formula as for above filters, but with the ceil instead of floor operator. Note
that in practice P−
h = P+
h = Ph since Caffe does not support asymmetric padding.
Unfortunately, it gets more complicated. Using the formula above, it can happen that
the last padding application is completely outside the input image and Caffe tries to avoid
it. This requires
Sh(i − 1) − P−
h + 1 i =H
≤ H ⇔ H ≤
H − 1 + P−
h
Sh
+ 1. (5.4)
Using the fact that for integers a, b, one has a/b = (a + b − 1)/b , we can rewrite the
expression for H as follows
H =
H − H + P−
h + P+
h
Sh
+ 1 =
H − 1 + P−
h
Sh
+
P+
h + Sh − H
Sh
+ 1.
Hence if P+
h + Sh ≤ H then the second term is less than zero and (5.4) is satisfied. In
practice, Caffe assumes that P+
h , P−
h ≤ H − 1, as otherwise the first filter application falls
entirely in the padded region. Hence, we can upper bound the second term:
P+
h + Sh − H
Sh
≤
Sh − 1
Sh
≤ 1.
We conclude that, for any choices of P+
h and Sh allowed by Caffe, the formula above may
violate constraint (5.4) by at most one unit. Caffe has a special provision for that and lowers
H by one when needed. Furthermore, we see that if P+
h = 0 and Sh ≤ H (which is often
the case and may be assumed by Caffe), then the equation is also satisfied and Caffe skips
the check.
Next, we find MatConvNet equivalents for these parameters. Assume that Caffe applies
a symmetric padding Ph. Then in MatConvNet P−
h = Ph to align the top part of the output
signal. To match Caffe, the last sample of the last filter application has to be on or to the
right of the last Caffe-padded pixel:
Sh





H − H + P−
h + P+
h
Sh
+ 1
MatConvNet rightmost pooling index
−1





+ H
MatConvNet rightmost pooled input sample
≥ H + 2P−
h
Caffe rightmost input sample with padding
.
Rearranging
H − H + P−
h + P+
h
Sh
≥
H − H + 2P−
h
Sh
Using a/b = (a − b + 1)/b we get the equivalent condition:
H − H + 2P−
h
Sh
+
P+
h − P−
h − Sh + 1
Sh
≥
H − H + 2P−
h
Sh

Removing the ceil operator lower bounds the left-hand side of the equation and produces the
sufficient condition
P+
h ≥ P−
h + Sh − 1.
As before, this may still be too much padding, causing the last pool window application to
be entirely in the rightmost padded area. MatConvNet places the restriction P+
h ≤ H − 1,
so that
P+
h = min{P−
h + Sh − 1, H − 1}.
For example, a pooling region of width H = 3 samples with a stride of Sh = 1 samples and
null Caffe padding P−
h = 0, would result in a right MatConvNet padding of P+
h = 1.
5.3 Convolution transpose
The convolution transpose block is similar to a simple filter, but somewhat more complex.
Recall that convolution transpose (section 6.2) is the transpose of the convolution operator,
which in turn is a filter. Reasoning for a 1D slice, let xi be the input to the convolution
transpose block and yi its output. Furthermore let Uh, C−
h , C+
h and H be the upsampling
factor, top and bottom crops, and filter height, respectively.
If we look at the convolution transpose backward, from the output to the input (see also
fig. 4.2), the data dependencies are the same as for the convolution operator, studied in
section 5.2. Hence there is an interaction between xi and yi only if
1 + Uh(i − 1) − C−
h ≤ i ≤ H + Uh(i − 1) − C−
h (5.5)
where cropping becomes padding and upsampling becomes downsampling. Turning this
relation around, we find that
i + C−
h − H
Uh
+ 1 ≤ i ≤
i + C−
h − 1
Uh
+ 1.
Note that, due to rounding, it is not possible to express this set tightly in the form outlined
above. We can however relax these two relations (hence obtaining a slightly larger receptive
field) and conclude that
αh =
1
Uh
, βh =
2C−
h − H + 1
2Uh
+ 1, ∆h =
H − 1
Uh
+ 1.
Next, we want to determine the height H of the output y of convolution transpose as
a function of the heigh H of the input x and the other parameters. Swapping input and
output in (5.3) results in the constraint:
H = 1 +
H − H + C−
h + C+
h
Uh
.
If H is now given as input, it is not possible to recover H uniquely from this expression;
instead, all the following values are possible
Uh(H − 1) + H − C−
h − C+
h ≤ H < UhH + H − C−
h − C+
h .

5.4. TRANSPOSING RECEPTIVE FIELDS 43
This is due to the fact that Uh acts as a downsampling factor in the standard convolution
direction and some of the samples to the right of the convolution input y may be ignored by
the filter (see also fig. 4.1 and fig. 4.2).
Since the height of y is then determined up to Sh samples, and since the extra samples
would be ignored by the computation and stay zero, we choose the tighter definition and set
H = Uh(H − 1) + H − C−
h − C+
h .
5.4 Transposing receptive fields
Suppose we have determined that a later y = f(x) has a receptive field transformation
(αh, βh, ∆h) (along one spatial slice). Now suppose we are given a block x = g(y) which
is the “transpose” of f, just like the convolution transpose layer is the transpose of the
convolution layer. By this, we mean that, if yi depends on xi due to f, then xi depends on
yi due to g.
Note that, by definition of receptive fields, f relates the inputs and outputs index pairs
(i, i ) given by (5.1), which can be rewritten as
−
∆h − 1
2
≤ i − αh(i − 1) − βh ≤
∆h − 1
2
.
A simple manipulation of this expression results in the equivalent expression:
−
(∆h + αh − 1)/αh − 1
2
≤ i −
1
αh
(i − 1) −
1 + αh − βh
αh
≤
(∆h + αh − 1)/αh − 1
2αh
.
Hence, in the reverse direction, this corresponds to a RF transformation
ˆαh =
1
αh
, ˆβh =
1 + αh − βh
αh
, ˆ∆h =
∆h + αh − 1
αh
.
Example 1. For convolution, we have found the parameters:
αh = Sh, βh =
H + 1
2
− P−
h , ∆h = H .
Using the formulas just found, we can obtain the RF transformation for convolution transpose:
ˆαh =
1
αh
=
1
Sh
,
ˆβh =
1 + Sh − (H + 1)/2 + P−
h
Sh
=
P−
h − H /2 + 1/2
Sh
+ 1 =
2P−
h − H + 1
Sh
+ 1,
ˆ∆h =
H + Sh − 1
Sh
=
H − 1
Sh
+ 1.
Hence we find again the formulas obtained in section 5.3.

5.5 Composing receptive fields
Consider now the composition of two layers h = g ◦ f with receptive fields (αf , βf , ∆f ) and
(αg, βg, ∆g) (once again we consider only a 1D slice in the vertical direction, the horizontal
one being the same). The goal is to compute the receptive field of h.
To do so, pick a sample ig in the domain of g. The first and last sample if in the domain
of f to affect ig are given by:
if = αf (ig − 1) + βf ±
∆f − 1
2
.
Likewise, the first and last sample ig to affect a given output sample ih are given by
ig = αg(ih − 1) + βg ±
∆g − 1
2
.
Substituting one relation into the other, we see that the first and last sample if in the domain
of g ◦ f to affect ih are:
if = αf αg(ih − 1) + βg ±
∆g − 1
2
− 1 + βf ±
∆f − 1
2
= αf αg(ih − 1) + αf (βg − 1) + βf ±
αf (∆g − 1) + ∆f − 1
2
.
We conclude that
αh = αf αg, βh = αf (βg − 1) + βf , ∆h = αf (∆g − 1) + ∆f .
5.6 Overlaying receptive fields
Consider now the combination h(f(x1), g(x2)) where the domains of f and g are the same.
Given the rule above, it is possible to compute how each output sample ih depends on each
input sample if through f and on each input sample ig through g. Suppose that this gives
receptive fields (αhf , βhf , ∆hf ) and (αhg, βhg, ∆hg) respectively. Now assume that the domain
of f and g coincide, i.e. x = x1 = x2. The goal is to determine the combined receptive field.
This is only possible if, and only if, α = αhg = αhf . Only in this case, in fact, it is
possible to find a sliding window receptive field that tightly encloses the receptive field due
to g and f at all points according to formulas (5.1). We say that these two receptive fields
are compatible. The range of input samples i = if = ig that affect any output sample ih is
then given by
imax = α(ih − 1) + a, a = min βhf −
∆hf − 1
2
, βhg −
∆hg − 1
2
,
imin = α(ih − 1) + b, b = max βhf +
∆hf − 1
2
, βhg +
∆hg − 1
2
.
We conclude that the combined receptive field is
α = αhg = αhf , β =
a + b
2
, ∆ = b − a + 1.

Chapter 6
Implementation details
This chapter contains calculations and details.
6.1 Convolution
It is often convenient to express the convolution operation in matrix form. To this end, let
φ(x) be the im2row operator, extracting all W × H patches from the map x and storing
them as rows of a (H W ) × (H W D) matrix. Formally, this operator is given by:
[φ(x)]pq =
(i,j,d)=t(p,q)
xijd
where the correspondence between indexes (i, j, d) and (p, q) is given by the map (i, j, d) =
t(p, q) where:
i = i + i − 1, j = j + j − 1, p = i + H (j − 1), q = i + H (j − 1) + H W (d − 1).
In practice, this map is slightly modiﬁed to account for the padding, stride, and dilation
factors. It is also useful to deﬁne the “transposed” operator row2im:
[φ∗
(M)]ijd =
(p,q)∈t−1(i,j,d)
Mpq.
Note that φ and φ∗
are linear operators. Both can be expressed by a matrix H ∈
R(H W H W D)×(HWD)
such that
vec(φ(x)) = H vec(x), vec(φ∗
(M)) = H vec(M).
Hence we obtain the following expression for the vectorized output (see [7]):
vec y = vec (φ(x)F) =
(I ⊗ φ(x)) vec F, or, equivalently,
(F ⊗ I) vec φ(x),
where F ∈ R(H W D)×K
is the matrix obtained by reshaping the array f and I is an identity
matrix of suitable dimensions. This allows obtaining the following formulas for the deriva-
tives:
dz
d(vec F)
=
dz
d(vec y)
(I ⊗ φ(x)) = vec φ(x)
dz
dY
45

46 CHAPTER 6. IMPLEMENTATION DETAILS
where Y ∈ R(H W )×K
is the matrix obtained by reshaping the array y. Likewise:
dz
d(vec x)
=
dz
d(vec y)
(F ⊗ I)
d vec φ(x)
d(vec x)
= vec
dz
dY
F H
In summary, after reshaping these terms we obtain the formulas:
vec y = vec (φ(x)F) ,
dz
dF
= φ(x)
dz
dY
,
dz
dX
= φ∗ dz
dY
F
where X ∈ R(HW)×D
is the matrix obtained by reshaping x. Notably, these expressions are
used to implement the convolutional operator; while this may seem inefficient, it is instead
a fast approach when the number of filters is large and it allows leveraging fast BLAS and
GPU BLAS implementations.
6.2 Convolution transpose
In order to understand the definition of convolution transpose, let y to be obtained from x
by the convolution operator as defined in section 4.1 (including padding and downsampling).
Since this is a linear operation, it can be rewritten as vec y = M vec x for a suitable matrix M;
convolution transpose computes instead vec x = M vec y. While this is simple to describe
in term of matrices, what happens in term of indexes is tricky. In order to derive a formula
for the convolution transpose, start from standard convolution (for a 1D signal):
yi =
H
i =1
fi xS(i −1)+i −P−
h
, 1 ≤ i ≤ 1 +
H − H + P−
h + P+
h
S
,
where S is the downsampling factor, P−
h and P+
h the padding, H the length of the input
signal x and H the length of the filter f. Due to padding, the index of the input data x
may exceed the range [1, H]; we implicitly assume that the signal is zero padded outside this
range.
In order to derive an expression of the convolution transpose, we make use of the identity
vec y (M vec x) = (vec y M) vec x = vec x (M vec y). Expanding this in formulas:
b
i =1
yi
W
i =1
fi xS(i −1)+i −P−
h
=
+∞
i =−∞
+∞
i =−∞
yi fi xS(i −1)+i −P−
h
=
+∞
i =−∞
+∞
k=−∞
yi fk−S(i −1)+P−
h
xk
=
+∞
i =−∞
+∞
k=−∞
yi f
(k−1+P−
h ) mod S+S 1−i +
k−1+P −
h
S
+1
xk
=
+∞
k=−∞
xk
+∞
q=−∞
y k−1+P −
h
S
+2−q
f(k−1+P−
h ) mod S+S(q−1)+1.

6.3. SPATIAL POOLING 47
Summation ranges have been extended to infinity by assuming that all signals are zero padded
as needed. In order to recover such ranges, note that k ∈ [1, H] (since this is the range of
elements of x involved in the original convolution). Furthermore, q ≥ 1 is the minimum value
of q for which the filter f is non zero; likewise, q ≤ (H − 1)/S + 1 is a fairly tight upper
bound on the maximum value (although, depending on k, there could be an element less).
Hence
xk =
1+ H −1
S
q=1
y k−1+P −
h
S
+2−q
f(k−1+P−
h ) mod S+S(q−1)+1, k = 1, . . . , H. (6.1)
Note that the summation extrema in (6.1) can be refined slightly to account for the finite
size of y and w:
max 1,
k − 1 + P−
h
S
+ 2 − H ≤ q
≤ 1 + min
H − 1 − (k − 1 + P−
h ) mod S
S
,
k − 1 + P−
h
S
.
The size H of the output of convolution transpose is obtained in section 5.3.
6.3 Spatial pooling
Since max pooling simply selects for each output element an input element, the relation can
be expressed in matrix form as vec y = S(x) vec x for a suitable selector matrix S(x) ∈
{0, 1}(H W D)×(HWD)
. The derivatives can be written as: dz
d(vec x)
= dz
d(vec y)
S(x), for all
but a null set of points, where the operator is not differentiable (this usually does not pose
problems in optimization by stochastic gradient). For max-pooling, similar relations exists
with two differences: S does not depend on the input x and it is not binary, in order to
account for the normalization factors. In summary, we have the expressions:
vec y = S(x) vec x,
dz
d vec x
= S(x)
dz
d vec y
. (6.2)
6.4 Activation functions
6.4.1 ReLU
The ReLU operator can be expressed in matrix notation as
vec y = diag s vec x,
dz
d vec x
= diag s
dz
d vec y
where s = [vec x > 0] ∈ {0, 1}HWD
is an indicator vector.

6.4.2 Sigmoid
The derivative of the sigmoid function is given by
dz
dxijd
=
dz
dyijd
dyijd
dxijd
=
dz
dyijd
−1
(1 + e−xijd )2
(−e−xijd
)
=
dz
dyijd
yijd(1 − yijd).
In matrix notation:
dz
dx
=
dz
dy
y (11 − y).
6.5 Spatial bilinear resampling
The projected derivative d p, φ(x, g) /dx of the spatial bilinaer resampler operator with
respect to the input image x can be found as follows:
∂
∂xijc
i j c
pi k c
H
i =1
W
j =1
xi j c max{0, 1 − |αvg1i j + βv − i |} max{0, 1 − |αug2i j + βu − j |}
=
i j
pi k c max{0, 1 − |αvg1i j + βv − i|} max{0, 1 − |αug2i j + βu − j|}. (6.3)
Note that the formula is similar to Eq. 4.2, with the diﬀerence that summation is on i rather
than i.
The projected derivative d p, φ(x, g) /dg with respect to the grid is similar:
∂
∂g1i j
i j c
pi k c
H
i=1
W
j=1
xijc max{0, 1 − |αvg1i j + βv − i|} max{0, 1 − |αug2i j + βu − j|}
= −
c
pi j c
H
i=1
W
j=1
αvxijc max{0, 1−|αvg2i j +βv−j|} sign(αvg1i j +βv−j)1{−1<αug2i j +βu<1}.
(6.4)
A similar expression holds for ∂g2i j
6.6 Normalization
6.6.1 Local response normalization (LRN)
The derivative is easily computed as:
dz
dxijd
=
dz
dyijd
L(i, j, d|x)−β
− 2αβxijd
k:d∈G(k)
dz
dyijk
L(i, j, k|x)−β−1
xijk
where
L(i, j, k|x) = κ + α
t∈G(k)
x2
ijt.

6.6. NORMALIZATION 49
6.6.2 Batch normalization
The derivative of the network output z with respect to the multipliers wk and biases bk is
given by
dz
dwk
=
i j k t
dz
dyi j k t
dyi j k t
dwk
=
i j t
dz
dyi j kt
xi j kt − µk
σ2
k +
,
dz
dbk
=
i j k t
dz
dyi j k t
dyi j k t
dwk
=
i j t
dz
dyi j kt
.
The derivative of the network output z with respect to the block input x is computed as
follows:
dz
dxijkt
=
i j k t
dz
dyi j k t
dyi j k t
dxijkt
.
Since feature channels are processed independently, all terms with k = k are zero. Hence
dz
dxijkt
=
i j t
dz
dyi j kt
dyi j kt
dxijkt
,
where
dyi j kt
dxijkt
= wk δi=i ,j=j ,t=t −
dµk
dxijkt
1
σ2
k +
−
wk
2
(xi j kt − µk) σ2
k +
−3
2
dσ2
k
dxijkt
,
the derivatives with respect to the mean and variance are computed as follows:
dµk
dxijkt
=
1
HWT
,
dσ2
k
dxi j kt
=
2
HWT ijt
(xijkt − µk) δi=i ,j=j ,t=t −
1
HWT
=
2
HWT
(xi j kt − µk) ,
and δE is the indicator function of the event E. Hence
dz
dxijkt
=
wk
σ2
k +
dz
dyijkt
−
1
HWT i j kt
dz
dyi j kt
−
wk
2(σ2
k + )
3
2
i j kt
dz
dyi j kt
(xi j kt − µk)
2
HWT
(xijkt − µk)
i.e.
dz
dxijkt
=
wk
σ2
k +
dz
dyijkt
−
1
HWT i j kt
dz
dyi j kt
−
wk
σ2
k +
xijkt − µk
σ2
k +
1
HWT i j kt
dz
dyi j kt
xi j kt − µk
σ2
k +
.

We can identify some of these terms with the ones computed as derivatives of bnorm with
respect to wk and µk:
dz
dxijkt
=
wk
σ2
k +
dz
dyijkt
−
1
HWT
dz
dbk
−
xijkt − µk
σ2
k +
1
HWT
dz
dwk
.
6.6.3 Spatial normalization
The neighbourhood norm n2
i j d can be computed by applying average pooling to x2
ijd using
vl_nnpool with a W ×H pooling region, top padding H −1
2
, bottom padding H − H−1
2
−1,
and similarly for the horizontal padding.
The derivative of spatial normalization can be obtained as follows:
dz
dxijd
=
i j d
dz
dyi j d
dyi j d
dxijd
=
i j d
dz
dyi j d
(1 + αn2
i j d)−β dxi j d
dxijd
− αβ
dz
dyi j d
(1 + αn2
i j d)−β−1
xi j d
dn2
i j d
d(x2
ijd)
dx2
ijd
dxijd
=
dz
dyijd
(1 + αn2
ijd)−β
− 2αβxijd
i j d
dz
dyi j d
(1 + αn2
i j d)−β−1
xi j d
dn2
i j d
d(x2
ijd)
=
dz
dyijd
(1 + αn2
ijd)−β
− 2αβxijd
i j d
ηi j d
dn2
i j d
d(x2
ijd)
, ηi j d =
dz
dyi j d
(1 + αn2
i j d)−β−1
xi j d
Note that the summation can be computed as the derivative of the vl_nnpool block.
6.6.4 Softmax
Care must be taken in evaluating the exponential in order to avoid underﬂow or overﬂow.
The simplest way to do so is to divide the numerator and denominator by the exponential of
the maximum value:
yijk =
exijk−maxd xijd
D
t=1 exijt−maxd xijd
.
The derivative is given by:
dz
dxijd
=
k
dz
dyijk
exijd
L(x)−1
δ{k=d} − exijd
exijk
L(x)−2
, L(x) =
D
t=1
exijt
.
Simplifying:
dz
dxijd
= yijd
dz
dyijd
−
K
k=1
dz
dyijk
yijk .
In matrix form:
dz
dX
= Y
dz
dY
−
dz
dY
Y 11

where X, Y ∈ RHW×D
are the matrices obtained by reshaping the arrays x and y. Note that
the numerical implementation of this expression is straightforward once the output Y has
been computed with the caveats above.
6.7 Categorical losses
This section obtains the projected derivatives of the categorical losses in section 4.8. Recall
that all losses give a scalar output, so the projection tensor p is trivial (a scalar).
6.7.1 Classiﬁcation losses
Top-K classiﬁcation error. The derivative is zero a.e.
Log-loss. The projected derivative is:
∂p (x, c)
∂xk
= −p
∂ log(xc)
∂xk
= −pxcδk=c.
Softmax log-loss. The projected derivative is given by:
∂p (x, c)
∂xk
= −p
∂
∂xk
xc − log
C
t=1
ext
= −p δk=c −
exc
C
t=1 ext
.
In brackets, we can recognize the output of the loss itself:
y = (x, c) =
exc
C
t=1 ext
.
Hence the loss derivatives rewrites:
∂p (x, c)
∂xk
= −p (δk=c − y) .
Multi-class hinge loss. The projected derivative is:
∂p (x, c)
∂xk
= −p 1[xc < 1] δk=c.
Structured multi-class hinge loss. The projected derivative is:
∂p (x, c)
∂xk
= −p 1[xc < 1 + max
t=c
xt] (δk=c − δk=t∗ ), t∗
= argmax
t=1,2,...,C
xt.
6.7.2 Attribute losses
Binary error. The derivative of the binary error is 0 a.e.

Binary log-loss. The projected derivative is:
∂p (x, c)
∂x
= −p
c
c x − 1
2
+ 1
2
.
Binary logistic loss. The projected derivative is:
∂p (x, c)
∂x
= −p
∂
∂x
log
1
1 + e−cx
= −p
ce−cx
1 + e−cx
= −p
c
ecx + 1
= −pc σ(−cx).
Binary hinge loss. The projected derivative is
∂p (x, c)
∂x
= −pc 1[cx < 1].
6.8 Comparisons
6.8.1 p-distance
The derivative of the operator without root is given by:
dz
dxijd
=
dz
dyij
p|xijd − ¯xijd|p−1
sign(xijd − ¯xijd).
The derivative of the operator with root is given by:
dz
dxijd
=
dz
dyij
1
p d
|xijd − ¯xijd |p
1
p
−1
p|xijd − ¯xijd|p−1
sign(xijd − ¯xijd)
=
dz
dyij
|xijd − ¯xijd|p−1
sign(xijd − ¯xijd)
yp−1
ij
,
dz
d¯xijd
= −
dz
dxijd
.
The formulas simplify a little for p = 1, 2 which are therefore implemented as special cases.
6.9 Other implementation details
6.9.1 Normal sampler
The function vl::randn() uses the Ziggurah method [10] to sample from a Normally-
distributed random variable. Let f(x) = 1√
2π
exp −1
2
x2
the standard Normal distribution.
The sampler encloses f(x) in a simple shape made of K − 1 horizontal rectangles and a base
composed of a rectangle tapering oﬀ in an exponential distribution. These are deﬁned by
points x1 > x2 > x3 > · · · > xK = 0 such that (for the right half of f(x)) the layers of the
Ziggurat are given by
∀k = 1, . . . , K − 1 : Rk = [f(xk), f(xk+1)] × [0, xk].

6.9. OTHER IMPLEMENTATION DETAILS 53
and such that its basis is given by
R0 = ([0, f(x1)] × [0, x1]) ∪ {(x, y) : x ≥ x1, y ≤ f(x1) exp(−x1(x − x1))}
Note that, since the last point xK = 0, (half of) the distribution is enclosed by the Ziggurat,
i.e. ∀x ≥ 0 : (x, f(x)) ∈ ∪K
k=0Rk.
The first point x1 in the sequence determines the area of the Ziggurat base:
A = |R0| = f(x1)x1 + f(x1)/x1.
The other points are defined recursively such that the area is the same for all rectangles:
A = |Rk| = (f(xk+1) − f(xk))xk ⇒ xk+1 = f−1
(A/xk + f(xk)).
There are two degrees of freedom: the number of subdivisions K and the point x1. Given
K, the goal is to choose x1 such that the K-th points xK = 0 lands on zero, enclosing
tightly f(x). The required value of x1 is easily found using bisection and, for K = 256, is
x1 = 3.655420419026953. Given x1, A and all other points in the sequence can be derived
easily using the formulas above.
The Ziggurath can be used to quickly sample from the Normal distribution. In order to
do so, one first samples a point (x, y) uniformly at random from the Ziggurat ∪K
k=0Rk and
then rejects pairs (x, y) that do not belong to the graph of f(x), i.e. y > f(x). Specifically:
1. Sample a point (x, y) uniformly from the Ziggurat. To do so, sample uniformly at
random an index k ∈ {0, 1, . . . , K − 1} and two scalars u, v in the interval [0, 1). Then,
for k ≥ 1, set x = uxk and y = vf(xk+1) + (1 − v)f(xk) (for k = 0 see below). Since all
regions Rk have the same area and (x, y) are then drawn uniformly form the selected
rectangle, this samples a point (x, y) from the Ziggurat uniformly at random.
2. If y ≤ f(x), accept x as a sample; otherwise, sample again. Note that, when x ≤ xk+1,
the test y ≤ f(xk+1) < f(x) is always successful, and the variable y and test can be
skipped in the step above.
Next, we complete the procedure for k = 0, when R0 is not just a rectangle but rather the
union of a rectangle and an exponential distribution. To sample from R0 uniformly, we either
choose the rectangle or the exponential distribution with a probability proportional to their
area. Reusing the notation (and corresponding code) above, we can express this as sampling
x = ux0 and accepting the latter as a sample from the rectangle component if ux0 ≤ x1; here
the pseudo-point x0 is defined such that x1/x0 = f(x1)x1/A, i.e. x0 = A/f(x1). If the test
fails, we sample instead from the exponential distribution x ∼ x1 exp(−x1(x − x1)), x ≥ x1.
To do so, let z = x1 exp(−x1(x − x1)); then x = x1 − (1/x1) ln z/x1 and dx = | − (x1/z)|dz,
where z ∈ (0, x1]. Since x1 exp(−x1(x − x1))dx = (1/x1)dz is uniform, we can implement
this by sampling u uniformly in (0, 1] and setting x = x1 − (1/x1) ln u. Finally, recall that
the goal is to sample from the Normal distribution, not the exponential, so the latter sample
must be refined by rejection sampling. As before, this requires sampling a pair (x, y) under
the exponential distribution graph. Given x sampled from the exponential distribution, we
sample the corresponding y uniformly at random in the interval [0, f(x1) exp(−x1(x − x1))],

and write the latter as y = vf(x1) exp(−x1(x − x1)), where v is uniform in [0, 1]. The
latter is then accepted provided that y is below the Normal distribution graph f(x), i.e.
vf(x1) exp(−x1(x − x1)) ≤ f(x). A short calculation yields the test:
−2 ln v ≥ x2
1 + x2
− 2x1x = (x1 − x)2
= ((1/x1) ln u)2
.
6.9.2 Euclid’s algorithm
Euclid’s algorithm finds the greatest common divisor (GCD) of two non-negative integers a
and b. Recall that the GCD is the largest integer that divides both a and b:
gcd(a, b) = max{d ∈ N : d|a ∧ d|b}.
Lemma 1 (Euclid’s algorithm). Let a, b ∈ N and let q ∈ Z such that a − qb ≥ 0. Then
gcd(a, b) = gcd(a − qb, b).
Proof. Let d be a divisor of both a and b. Then d divides a − qb as well because:
a − qb
d
=
a
d
∈Z
−q
b
d
∈Z
⇒
a − qb
d
∈ Z.
Hence gcd(a, b) ≤ gcd(q − qb, b). In the same way, we can show that, if d divides a − qb as
well as b, then it must divide a too, hence gcd(a − qb, b) ≤ gcd(a, b).
Euclid’s algorithm starts with a > b ≥ 1 and sets q to the quotient of the integer division
a/b. Due to the lemma above, the GCD of a and b is the same as the GCD of the remainder
r = a − qb = (a mod b) and b:
gcd(a, b) = gcd(a, a mod b).
Since the remainder (a mod b) < b is strictly smaller than b, now GCD is called with smaller
arguments. The recursion terminates when a zero reminder is generated, because
gcd(a, 0) = a.
We can modify the algorithms to also find two integers u, v, the Bézout’s coefficients, such
that:
au + bv = gcd(a, b).
To do so, we replace a = b(a/b) + r as above:
ru + bv = gcd(a, b) = gcd(r, b), v =
a
b
u + v.
The recursion terminates when r = 0, in which case
bv = gcd(0, b) = b ⇒ v = b.

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