# softMin

## PURPOSE

Calculates the softMin of a vector.

## SYNOPSIS

function M = softMin( D, sigma )

## DESCRIPTION

     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 Image&Video Toolbox      Version 2.0
     Copyright 2012 Piotr Dollar.  [pdollar-at-caltech.edu]
     Please email me if you find bugs, or have suggestions or questions!
     Licensed under the Simplified BSD License [see external/bsd.txt]