Regularization is a central tool for addressing ill-posedness in inverse problems and statistical estimation, with the choice of a suitable penalty often determining the reliability and interpretability of downstream solutions. While recent work has characterized optimal regularizers for well-specified data distributions, practical deployments are often complicated by distributional uncertainty and the need to enforce structural constraints such as convexity. In this paper, we introduce a framework for distributionally robust optimal regularization, which identifies regularizers that remain effective under perturbations of the data distribution. Our approach leverages convex duality to reformulate the underlying distributionally robust optimization problem, eliminating the inner maximization and yielding formulations that are amenable to numerical computation. We show how the resulting robust regularizers interpolate between memorization of the training distribution and uniform priors, providing insights into their behavior as robustness parameters vary. For example, we show how certain ambiguity sets, such as those based on the Wasserstein-1 distance, naturally induce regularity in the optimal regularizer by promoting regularizers with smaller Lipschitz constants. We further investigate the setting where regularizers are required to be convex, formulating a convex program for their computation and illustrating their stability with respect to distributional shifts. Taken together, our results provide both theoretical and computational foundations for designing regularizers that are reliable under model uncertainty and structurally constrained for robust deployment.

Conditional risk minimization arises in high-stakes decisions where risk must be assessed in light of side information, such as stressed economic conditions, specific customer profiles, or other contextual covariates. Constructing reliable conditional distributions from limited data is notoriously difficult, motivating a series of optimal-transport-based proposals that address this uncertainty in a distributionally robust manner. Yet these approaches remain fragmented, each constrained by its own limitations: some rely on point estimates or restrictive structural assumptions, others apply only to narrow classes of risk measures, and their structural connections are unclear. We introduce a universal framework for distributionally robust conditional risk minimization, built on a novel union-ball formulation in optimal transport. This framework offers three key advantages: interpretability, by subsuming existing methods as special cases and revealing their deep structural links; tractability, by yielding convex reformulations for virtually all major risk functionals studied in the literature; and scalability, by supporting cutting-plane algorithms for large-scale conditional risk problems. Applications to portfolio optimization with rank-dependent expected utility highlight the practical effectiveness of the framework, with conditional models converging to optimal solutions where unconditional ones clearly do not.

We consider a point cloud $X_n := \{ x_1, \dots, x_n \}$ uniformly distributed on the flat torus $\mathbb{T}^d : = \mathbb{R}^d / \mathbb{Z}^d $, and construct a geometric graph on the cloud by connecting points that are within distance $\epsilon$ of each other. We let $\mathcal{P}(X_n)$ be the space of probability measures on $X_n$ and endow it with a discrete Wasserstein distance $W_n$ as introduced independently by Maas and Zhou et al. for general finite Markov chains. We show that as long as $\epsilon= \epsilon_n$ decays towards zero slower than an explicit rate depending on the level of uniformity of $X_n$, then the space $(\mathcal{P}(X_n), W_n)$ converges in the Gromov-Hausdorff sense towards the space of probability measures on $\mathbb{T}^d$ endowed with the Wasserstein distance.

In this note we prove that reconstruction from magnitudes of frame coefficients (the so called "phase retrieval problem") can be performed using Lipschitz continuous maps. Specifically we show that when the nonlinear analysis map $\alpha:{\mathcal H}\rightarrow\mathbb{R}^m$ is injective, with $(\alpha(x))_k=||^2$, where $\{f_1,\ldots,f_m\}$ is a frame for the Hilbert space ${\mathcal H}$, then there exists a left inverse map $\omega:\mathbb{R}^m\rightarrow {\mathcal H}$ that is Lipschitz continuous. Additionally we obtain the Lipschitz constant of this inverse map in terms of the lower Lipschitz constant of $\alpha$. Surprisingly the increase in Lipschitz constant is independent of the space dimension or frame redundancy.

In this paper we consider the problem of learning an n x n Kernel matrix from m similarity matrices under general convex loss. Past research have extensively studied the m =1 case and have derived several algorithms which require sophisticated techniques like ACCP, SOCP, etc. The existing algorithms do not apply if one uses arbitrary losses and often can not handle m > 1 case. We present several provably convergent iterative algorithms, where each iteration requires either an SVM or a Multiple Kernel Learning (MKL) solver for m > 1 case. One of the major contributions of the paper is to extend the well known Mirror Descent(MD) framework to handle Cartesian product of psd matrices. This novel extension leads to an algorithm, called EMKL, which solves the problem in O(m^2 log n) iterations; in each iteration one solves an MKL involving m kernels and m eigen-decomposition of n x n matrices. By suitably defining a restriction on the objective function, a faster version of EMKL is proposed, called REKL, which avoids the eigen-decomposition. An alternative to both EMKL and REKL is also suggested which requires only an SVM solver. Experimental results on real world protein data set involving several similarity matrices illustrate the efficacy of the proposed algorithms.