Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have focused on solving problem instances in isolation, ignoring the fact that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks (GNNs), as a key building block for combinatorial tasks, either as solvers or as helper functions. GNNs are an inductive bias that e ectively encodes combinatorial and relational input due to their permutation-invariance and sparsity awareness. This paper presents a conceptual review of recent key advancements in this emerging eld, aiming at both the optimization and machine learning researcher.

We consider the problem of estimating the mean vector of a pvariate normal (θ, Σ) distribution under invariant quadratic loss, (δ − θ) ′ Σ −1 (δ − θ), when the covariance is unknown. We propose a new class of estimators that dominate the usual estimator δ 0 (X) = X. The proposed estimators of θ depend upon X and an independent Wishart matrix S with n degrees of freedom, however, S is singular almost surely when p > n. The proof of domination involves the development of some new unbiased estimators of risk for the p > n setting. We also find some relationships between the amount of domination and the magnitudes of n and p.

Combinatorial optimization is a well-established area in operations research and computer science. Until recently, its methods have mostly focused on solving problem instances in isolation, ignoring the fact that they often stem from related data distributions in practice. However, recent years have seen a surge of interest in using machine learning, especially graph neural networks, as a key building block for combinatorial tasks, either directly as solvers or by enhancing the former. This paper presents a conceptual review of recent key advancements in this emerging field, aiming at researchers in both optimization and machine learning.

High-dimensional prediction typically comprises two steps: variable selection and subsequent least-squares refitting on the selected variables. However, the standard variable selection procedures, such as the lasso, hinge on tuning parameters that need to be calibrated. Cross-validation, the most popular calibration scheme, is computationally costly and lacks finite sample guarantees. In this paper, we introduce an alternative scheme, easy to implement and both computationally and theoretically efficient.MSC 2010 subject classifications: Primary 62G08; secondary 62J07.

We study the behavior of a real p-dimensional Wishart random matrix with n degrees of freedom when n, p Ñ 8 but p{n Ñ 0. We establish the existence of phase transitions when p grows at the order n pK`1q{pK`3q for every k P N, and derive expressions for approximating densities between every two phase transitions. To do this, we make use of a novel tool we call the G-transform of a distribution, which is closely related to the characteristic function. We also derive an extension of the t-distribution to the real symmetric matrices, which naturally appears as the conjugate distribution to the Wishart under a G-transformation, and show its empirical spectral distribution obeys a semicircle law when p{n Ñ 0. Finally, we discuss how the phase transitions of the Wishart distribution might originate from changes in rates of convergence of symmetric t statistics.MSC 2010 subject classifications: Primary 60B20, 60B10; secondary 60E10.