Advanced Techniques for Mobile Robotics
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This document discusses Gaussian mixture models (GMMs) as an advanced clustering technique for mobile robotics. GMMs assume data points are generated from a mixture of Gaussian distributions rather than fixed clusters. The expectation-maximization (EM) algorithm is used to estimate the parameters of each Gaussian component to maximize the likelihood of the data. EM iterates between assigning data points to components (E-step) and re-estimating the component parameters (M-step) until convergence. GMMs can represent any continuous distribution and are more flexible than k-means clustering, but are also more computationally expensive.
Advanced Techniques for Mobile Robotics
- 1. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Gaussian Mixture Models Advanced Techniques for Mobile Robotics
- 2. Recap K-Means § Can be applied for clustering data § Computes new centroids for the clusters in an iterative manner § Converges to a local optimum § Uses a fixed variance § But the shapes of the clusters can be different in reality! 2
- 3. Motivation 3image source: wikipedia k-means Clustering based on a mixture of Gaussians
- 4. Mixtures of Gaussians § Assume that the data points are generated by sampling from a continuous function § A mixture of Gaussians is such a generative model § K Gaussians with means and covariance matrices § Each point is generated from one mixture component (but we don’t know from which one) § Use mixing coefficients (probability that a data point is generated from component k) 4
- 5. EM for Gaussian Mixtures Models (GMMs) § E-step: Softly assign data points to mixture components § Similar to k-means but considers mixing coefficients 5 data point mixture component
- 6. EM for Gaussian Mixtures Models (GMMs) § E-step: Softly assign data points to mixture components § M-step: Re-estimate the parameters for each mixture component based on the soft assignments 6 where “soft” assignments to k (as in k-means)
- 7. EM with GMMs 7image source: C. M. Bishop
- 8. 8 Properties of Gaussian Mixture Models § Can represent any continuous distribution § Number of mixture components must be estimated separately (as with k-means) § EM for GMMs is computationally more expensive than for k-means § EM converges slower than for k-means § Results depend on the initialization § K-means can be used for initialization (to speed up convergence and to find a “better” local optimum)
- 9. Initialization with K-Means § Run k-means N times § Take best result (highest likelihood) § Use this result to initialize EM for the GMM § Set to the mean of cluster j from k-means § Set to the covariance of the data points associated with cluster j 9
- 10. 10 Further Reading E. Alpaydin Introduction to Machine Learning C.M. Bishop Pattern Recognition and Machine Learning J. A. Bilmes A Gentle Tutorial of the EM algorithm and its Applications to Parameter Estimation (Technical report)