18. A storage medium storing instructions defining a function for optimizing a sum-of-exponentials function including a summation of exponentials of the form ∑ k = 1 K ⁢ ⅇ β k T ⁢ x by optimization operations comprising: upper bounding the sum of exponentials ∑ k = 1 K ⁢ ⅇ β k T ⁢ x by a product of sigmoids; upper bounding terms of the form log(1+e x ) in the product of sigmoids to generate a double majorization upper bound; minimizing the double majorization upper bound respective to the parameters β to generate optimized parameters β; and outputting the optimized sum-of-exponentials function represented at least by the optimized parameters β.
19. The storage medium as set forth in claim 18 , wherein the optimization function is configured to optimize a softmax function of the form ⅇ β k T ⁢ x ∑ k ′ = 1 K ⁢ ⅇ β k ′ T ⁢ x .