Clustering and classification employing softmax function including efficient bounds

摘要

A function optimization method includes the operations of: constructing an upper bound using a double majorization bounding process to a sum-of-exponentials function including a summation of exponentials of the form; $$ \sum _{k=1}^K{e}^{{\beta }_k^{T}x};$$optimizing the constructed upper bound respective to parameters β to generate optimized parameters β; and outputting the optimized sum-of-exponentials function represented at least by the optimized parameters β. An inference process includes the operations of: invoking the function optimization method respective to a softmax function constrained by discrete observations y defining categorization observation conditioned by continuous variables x representing at least one input object; and applying the optimized softmax function output by the invocation of the softmax function optimization method to the continuous variables x representing at least one input object to generate classification probabilities.