Recent work has generalized several results concerning the well-understood spiked Wigner matrix model of a low-rank signal matrix corrupted by additive i.i.d. Gaussian noise to the inhomogeneous case, where the noise has a variance profile. In particular, for the special case where the variance profile has a block structure, a series of results identified an effective spectral algorithm for detecting and estimating the signal, identified the threshold signal strength required for that algorithm to succeed, and proved information-theoretic lower bounds that, for some special signal distributions, match the above threshold. We complement these results by studying the computational optimality of this spectral algorithm. Namely, we show that, for a much broader range of signal distributions, whenever the spectral algorithm cannot detect a low-rank signal, then neither can any low-degree polynomial algorithm. This gives the first evidence for a computational hardness conjecture of Guionnet, Ko, Krzakala, and Zdeborová (2023). With similar techniques, we also prove sharp information-theoretic lower bounds for a class of signal distributions not treated by prior work. Unlike all of the above results on inhomogeneous models, our results do not assume that the variance profile has a block structure, and suggest that the same spectral algorithm might remain optimal for quite general profiles. We include a numerical study of this claim for an example of a smoothly-varying rather than piecewise-constant profile. Our proofs involve analyzing the graph sums of a matrix, which also appear in free and traffic probability, but we require new bounds on these quantities that are tighter than existing ones for non-negative matrices, which may be of independent interest.

In the context of a binary classification problem, the optimal linear combination of continuous predictors can be estimated by maximizing an empirical estimate of the area under the receiver operating characteristic (ROC) curve (AUC). For multi-category responses, the optimal predictor combination can similarly be obtained by maximization of the empirical hypervolume under the manifold (HUM). This problem is particularly relevant to medical research, where it may be of interest to diagnose a disease with various subtypes or predict a multi-category outcome. Since the empirical HUM is discontinuous, non-differentiable, and possibly multi-modal, solving this maximization problem requires a global optimization technique. Estimation of the optimal coefficient vector using existing global optimization techniques is computationally expensive, becoming prohibitive as the number of predictors and the number of outcome categories increases. We propose an efficient derivative-free black-box optimization technique based on pattern search to solve this problem. Through extensive simulation studies, we demonstrate that the proposed method achieves better performance compared to existing methods including the step-down algorithm. Finally, we illustrate the proposed method to predict swallowing difficulty after radiation therapy for oropharyngeal cancer based on radiation dose to various structures in the head and neck.