softmin

 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.  

 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 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 dist_euclidean or dist_chisquared.

 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.

 INPUTS
   D       - NxK dissimilarity matrix 
   sigma   - controls 'softness' of softmin

 OUTPUTS
   M       - the softmin

 EXAMPLE
   % example 1
   C = [0 0; 1 0; 0 1; 1 1]; x=[.7,.3; .1 .2];
   D = dist_euclidean( x, C );
   M = softmin( D, 1 )
   % example 2
   fplot( 'softmin( [0.5 0.2 .4], x )', [eps 10] );

 DATESTAMP
   29-Sep-2005  2:00pm

 See also DIST_EUCLIDEAN, DIST_CHISQUARED