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Although robust statistical estimators are less affected by outlying observations, their computation is usually more challenging. This is particularly the case in high-dimensional sparse settings. The availability of new optimization procedures, mainly developed in the computer science domain, offers new possibilities for the field of robust statistics. This paper investigates how such procedures can be used for robust sparse association estimators. The problem can be split into a robust estimation step followed by an optimization for the remaining decoupled, (bi-)convex problem. A combination of the augmented Lagrangian algorithm and adaptive gradient descent is implemented to also include suitable constraints for inducing sparsity. We provide results concerning the precision of the algorithm and show the advantages over existing algorithms in this context. High-dimensional empirical examples underline the usefulness of this procedure. Extensions to other robust sparse estimators are possible.

We present a novel hybrid (model- and learning-based) architecture for fusing the most significant features from conventional aerial images and integral aerial images that result from synthetic aperture sensing for removing occlusion caused by dense vegetation. It combines the environment's spatial references with features of unoccluded targets. Our method out-beats the state-of-the-art, does not require manually tuned parameters, can be extended to an arbitrary number and combinations of spectral channels, and is reconfigurable to address different use-cases.

By leveraging the no-cloning principle of quantum mechanics, unclonable cryptography enables us to achieve novel cryptographic protocols that are otherwise impossible classically. Two most notable examples of unclonable cryptography are quantum copy-protection and unclonable encryption. Despite receiving a lot of attention in recent years, two important open questions still remain: copy-protection for point functions in the plain model, which is usually considered as feasibility demonstration, and unclonable encryption with unclonable indistinguishability security in the plain model. In this work, by relying on previous works of Coladangelo, Liu, Liu, and Zhandry (Crypto'21) and Culf and Vidick (Quantum'22), we establish a new monogamy-of-entanglement property for subspace coset states, which allows us to obtain the following new results: - We show that copy-protection of point functions exists in the plain model, with different challenge distributions (including arguably the most natural ones). - We show, for the first time, that unclonable encryption with unclonable indistinguishability security exists in the plain model.

Optimization problems involving minimization of a rank-one convex function over constraints modeling restrictions on the support of the decision variables emerge in various machine learning applications. These problems are often modeled with indicator variables for identifying the support of the continuous variables. In this paper we investigate compact extended formulations for such problems through perspective reformulation techniques. In contrast to the majority of previous work that relies on support function arguments and disjunctive programming techniques to provide convex hull results, we propose a constructive approach that exploits a hidden conic structure induced by perspective functions. To this end, we first establish a convex hull result for a general conic mixed-binary set in which each conic constraint involves a linear function of independent continuous variables and a set of binary variables. We then demonstrate that extended representations of sets associated with epigraphs of rank-one convex functions over constraints modeling indicator relations naturally admit such a conic representation. This enables us to systematically give perspective formulations for the convex hull descriptions of these sets with nonlinear separable or non-separable objective functions, sign constraints on continuous variables, and combinatorial constraints on indicator variables. We illustrate the efficacy of our results on sparse nonnegative logistic regression problems.

Motivated by limitations on the depth of near-term quantum devices, we study the depth-computation trade-off in the query model, where the depth corresponds to the number of adaptive query rounds and the computation per layer corresponds to the number of parallel queries per round. We achieve the strongest known separation between quantum algorithms with $r$ versus $r-1$ rounds of adaptivity. We do so by using the $k$-fold Forrelation problem introduced by Aaronson and Ambainis (SICOMP'18). For $k=2r$, this problem can be solved using an $r$ round quantum algorithm with only one query per round, yet we show that any $r-1$ round quantum algorithm needs an exponential (in the number of qubits) number of parallel queries per round. Our results are proven following the Fourier analytic machinery developed in recent works on quantum-classical separations. The key new component in our result are bounds on the Fourier weights of quantum query algorithms with bounded number of rounds of adaptivity. These may be of independent interest as they distinguish the polynomials that arise from such algorithms from arbitrary bounded polynomials of the same degree.

Selecting the best regularization parameter in inverse problems is a classical and yet challenging problem. Recently, data-driven approaches have become popular to tackle this challenge. These approaches are appealing since they do require less a priori knowledge, but their theoretical analysis is limited. In this paper, we propose and study a statistical machine learning approach, based on empirical risk minimization. Our main contribution is a theoretical analysis, showing that, provided with enough data, this approach can reach sharp rates while being essentially adaptive to the noise and smoothness of the problem. Numerical simulations corroborate and illustrate the theoretical findings. Our results are a step towards grounding theoretically data-driven approaches to inverse problems.

We propose a theory for matrix completion that goes beyond the low-rank structure commonly considered in the literature and applies to general matrices of low description complexity. Specifically, complexity of the sets of matrices encompassed by the theory is measured in terms of Hausdorff and upper Minkowski dimensions. Our goal is the characterization of the number of linear measurements, with an emphasis on rank-$1$ measurements, needed for the existence of an algorithm that yields reconstruction, either perfect, with probability 1, or with arbitrarily small probability of error, depending on the setup. Concretely, we show that matrices taken from a set $\mathcal{U}$ such that $\mathcal{U}-\mathcal{U}$ has Hausdorff dimension $s$ can be recovered from $k>s$ measurements, and random matrices supported on a set $\mathcal{U}$ of Hausdorff dimension $s$ can be recovered with probability 1 from $k>s$ measurements. What is more, we establish the existence of recovery mappings that are robust against additive perturbations or noise in the measurements. Concretely, we show that there are $\beta$-H\"older continuous mappings recovering matrices taken from a set of upper Minkowski dimension $s$ from $k>2s/(1-\beta)$ measurements and, with arbitrarily small probability of error, random matrices supported on a set of upper Minkowski dimension $s$ from $k>s/(1-\beta)$ measurements. The numerous concrete examples we consider include low-rank matrices, sparse matrices, QR decompositions with sparse R-components, and matrices of fractal nature.

Despite the advancements in high-performance computing and modern numerical algorithms, the cost remains prohibitive for multi-query kinetic plasma simulations. In this work, we develop data-driven reduced-order models (ROM) for collisionless electrostatic plasma dynamics, based on the kinetic Vlasov-Poisson equation. Our ROM approach projects the equation onto a linear subspace defined by principal proper orthogonal decomposition (POD) modes. We introduce an efficient tensorial method to update the nonlinear term using a precomputed third-order tensor. We capture multiscale behavior with a minimal number of POD modes by decomposing the solution into multiple time windows using a physical-time indicator and creating a temporally-local ROM. Applied to 1D-1V simulations, specifically the benchmark two-stream instability case, our time-windowed reduced-order model (TW-ROM) with the tensorial approach solves the equation approximately 280 times faster than Eulerian simulations while maintaining a maximum relative error of 4% for the training data and 13% for the testing data.

In the aim of reducing the computational cost of the resolution of parameter-dependent eigenvalue problems, a model order reduction (MOR) procedure is proposed. We focus on the case of non-self-adjoint generalized eigenvalue problems, such as the stationary multigroup neutron diffusion equations. The method lies in an approximation of the manifold of solutions using a Proper Orthogonal Decomposition approach. The numerical method is composed of two stages. In the offline stage, we build a reduced space which approximates the manifold. In the online stage, for any given new set of parameters, we solve a reduced problem on the reduced space within a much smaller computational time than the required time to solve the high-fidelity problem. This method is applied to core computations in the APOLLO3 code.