Fast algorithm for convex optimization with application to density estimation and clustering

A non-transitory storage medium storing instructions executable by an electronic processing device to perform a method comprising:

• selecting a pair of parameters wi, wj from a set of K adjustable parameters of a log-likelihood function having the form
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• maximizing a value of the log-likelihood function incorporating a change δ to the selected pair of parameters wi, wj of the form
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• respective to the parameter −wi≦δ≦wj to generate an optimal value δopt for the change δ;
• replacing (wi)new ←(wi)old+δopt and (wj)new ←(wj)old−δopt;
• and
• repeating the selecting, maximizing, and replacing for different pairs of parameters wi, wj of the set of K adjustable parameters to identify optimized parameter values for the set of K adjustable parameters.

A system comprising:

• one or more electronic processors configured to perform a concave log-likelihood function maximization process defined by the following operations:
• selecting a pair of parameters wi, wj from a set of K adjustable parameters of a concave log-likelihood function,
• maximizing a value of the concave log-likelihood function respective to an adjustment value δ to generate an optimal adjustment value δopt, wherein the value of one member wi of the selected pair of parameters wi, wj is increased by the adjustment value δ and the value of the other member wj of the selected pair of parameters wi, wj is decreased by the adjustment value δ and the adjustment value δ satisfies the condition −Wi≦δ≦wj,
• updating values of the plurality of adjustable parameters by increasing the value of the one member wi, of the selected pair of parameters wi, wj the optimized adjustment value δopt and decreasing the value of the other member wj of the selected pair of parameters wi, wj by the optimized adjustment value δopt, and
• repeating the selecting, maximizing, and updating for different pairs of parameters wi, wj of the set of K adjustable parameters to identify optimized values of the set of K adjustable parameters.