The robust low-rank tensor completion problem addresses the challenge of recovering corrupted high-dimensional tensor data with missing entries, outliers, and sparse noise commonly found in real-world applications. Existing methodologies have encountered fundamental limitations due to their reliance on uniform regularization schemes, particularly the tensor nuclear norm and $\ell_1$ norm regularization approaches, which indiscriminately apply equal shrinkage to all singular values and sparse components, thereby compromising the preservation of critical tensor structures. The proposed tensor weighted correlated total variation (TWCTV) regularizer addresses these shortcomings through an $M$-product framework that combines a weighted Schatten-$p$ norm on gradient tensors for low-rankness with smoothness enforcement and weighted sparse components for noise suppression. The proposed weighting scheme adaptively reduces the thresholding level to preserve both dominant singular values and sparse components, thus improving the reconstruction of critical structural elements and nuanced details in the recovered signal. Through a systematic algorithmic approach, we introduce an enhanced alternating direction method of multipliers (ADMM) that offers both computational efficiency and theoretical substantiation, with convergence properties comprehensively analyzed within the $M$-product framework.Comprehensive numerical evaluations across image completion, denoising, and background subtraction tasks validate the superior performance of this approach relative to established benchmark methods.

We consider sequential decision making problem in the adversarial setting, where regret is measured with respect to the optimal sequence of actions and the feedback adheres the bandit setting. It is well-known that obtaining sublinear regret in this setting is impossible in general, which arises the question of when can we do better than linear regret? Previous works show that when the environment is guaranteed to vary slowly and furthermore we are given prior knowledge regarding its variation (i.e., a limit on the amount of changes suffered by the environment), then this task is feasible. The caveat however is that such prior knowledge is not likely to be available in practice, which causes the obtained regret bounds to be somewhat irrelevant. Our main result is a regret guarantee that scales with the variation parameter of the environment, without requiring any prior knowledge about it whatsoever. By that, we also resolve an open problem posted by Gur, Zeevi and Besbes [8]. An important key component in our result is a statistical test for identifying non-stationarity in a sequence of independent random variables. This test either identifies nonstationarity or upper-bounds the absolute deviation of the corresponding sequence of mean values in terms of its total variation. This test is interesting on its own right and has the potential to be found useful in additional settings.

We consider the problem of estimating a function defined over nlocations on a d-dimensional grid (having all side lengths equal to n1/d). When the function is constrained to have discrete total variation bounded by Cn, we derive the minimax optimal (squared) `2 estimation error rate, parametrized by n,Cn. Total variation denoising, also known as the fused lasso, is seen to be rate optimal. Several simpler estimators exist, such as Laplacian smoothing and Laplacian eigenmaps. A natural question is: can these simpler estimators perform just as well?

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The Job Shop Scheduling (JSS) problem can be viewed as an integer optimization program with linear objective function and linear, disjunctive constraints. Theconstraints(14c)enforceprecedencebetween tasks that must be scheduled in the specified order within their respective job. Themodel presented belowisusedtoconstruct solutions that are integral, and feasible tothe original problem constraints. However, the resolution frequency to solve OPFs is limited by their computational complexity. Additionally,the stochasticity introduced by renewable energy sources further increases the number of scenarios to consider. C.2 DatasetDetails Table 4 describes the power network benchmarks used, including the number of buses|N|, and transmission lines/transformers |E|.

Generative models at times produce "invalid" outputs, such as images with generation artifacts and unnatural sounds.

Sharp Inequalities between Total Variation and Hellinger Distances for Gaussian Mixtures Joonhyuk Jung 1 and Chao Gao 1 1 Department of Statistics, University of Chicago Abstract We study the relation between the total variation (TV) and Hellinger distances between two Gaussian location mixtures. Our first result establishes a general upper bound: for any two mixing distributions supported on a compact set, the Hellinger distance between the two mixtures is controlled by the TV distance raised to a power 1 o(1), where the o(1) term is of order 1/log log(1/TV). We also construct two sequences of mixing distributions that demonstrate the sharpness of this bound. Taken together, our results resolve an open problem raised in Jia et al. (2023) and thus lead to an entropic characterization of learning Gaussian mixtures in total variation. Our inequality also yields optimal robust estimation of Gaussian mixtures in Hellinger distance, which has a direct implication for bounding the minimax regret of empirical Bayes under Huber contamination. 1 Introduction The Gaussian location mixture is one of the most fundamental models used in nonparametric density estimation, Bayesian inference, and clustering (Lindsay, 1995; Dasgupta, 1999). Given a probability measure π supported on R d, the induced Gaussian mixture is defined by f π(x) = null R dϕ d(x θ)dπ(θ), where ϕ d(x) = (2π) d/2 exp( x 2 2/2) is the density function of the d-dimensional standard Gaussian distribution. In this paper, we study the relation between the total variation distance TV(p,q):= 1 2 null |p q| and the Hellinger distance H(p,q):= null 1 2 null ( p q) 2 of two Gaussian mixture densities.