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This paper extends the classic theory of convex optimization to theminimization of functions that are equal to the negated logarithm of what weterm as a sum-log-concave function, i.e., a sum of log-concave functions. Inparticular, we show that such functions are in general not convex but stillsatisfy generalized convexity inequalities. These inequalities unveil the keyimportance of a certain vector that we call the cross-gradient and that is, ingeneral, distinct from the usual gradient. Thus, we propose the Cross GradientDescent (XGD) algorithm moving in the opposite direction of the cross-gradientand derive a convergence analysis. As an application of our sum-log-concaveframework, we introduce the so-called checkered regression method relying on asum-log-concave function. This classifier extends (multiclass) logisticregression to non-linearly separable problems since it is capable oftessellating the feature space by using any given number of hyperplanes,creating a checkerboard-like pattern of decision regions.

Beyond Log-Concavity: Theory and Algorithm for Sum-Log-Concave Optimization