Calculates the softMin of a vector. Let D be a vector. Then the softMin of D is defined as: s = exp(-D/sigma^2) / sum( exp(-D/sigma^2) ) The softMin is a way of taking a dissimilarity (distance) vector D and converting it to a similarity vector s, such that sum(s)==1. If D is an NxK array, is is treated as N K-dimensional vectors, and the return is likewise an NxK array. This is useful if D is a distance matrix, generated by the likes of pdist2. Note that as sigma->0, softMin's behavior tends toward that of the standard min function. That is the softMin of a vector D has all zeros with a single 1 in the location of the smallest value of D. For example, "softMin([.2 .4 .1 .3],eps)" returns "[0 0 1 0]". As sigma->inf, then softMin(D,sigma) tends toward "ones(1,n)/n", where n==length(D). If D contains the squared euclidean distance between a point y and k points xi, then there is a probabilistic interpretation for softMin. If we think of the k points representing equal variant gaussians each with mean xi and std sigma, then the softMin returns the relative probability of y being generated by each gaussian. USAGE M = softMin( D, sigma ) INPUTS D - NxK dissimilarity matrix sigma - controls 'softness' of softMin OUTPUTS M - the softMin (indexes into D) EXAMPLE - 1 C = [0 0; 1 0; 0 1; 1 1]; x=[.7,.3; .1 .2]; D = pdist2( x, C ), M = softMin( D, .25 ) EXAMPLE - 2 fplot( 'softMin( [0.5 0.2 .4], x )', [0 5] ); xlabel('sigma'); ylabel('assignments') See also PDIST2, SOFTMAX Piotr's Computer Vision Matlab Toolbox Version 2.0 Copyright 2014 Piotr Dollar. [pdollar-at-gmail.com] Licensed under the Simplified BSD License [see external/bsd.txt]

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