IA Scholar Query: Rapid, Robust, and Reliable Blind Deconvolution via Nonconvex Optimization.
https://scholar.archive.org/
Internet Archive Scholar query results feedeninfo@archive.orgThu, 15 Sep 2022 00:00:00 GMTfatcat-scholarhttps://scholar.archive.org/help1440A Unified Bregman Alternating Minimization Algorithm for Generalized DC Programming with Applications to Image Processing
https://scholar.archive.org/work/zomrj3gperb55a7xumsox6xkke
In this paper, we consider a class of nonconvex (not necessarily differentiable) optimization problems called generalized DC (Difference-of-Convex functions) programming, which is minimizing the sum of two separable DC parts and one two-block-variable coupled function. To circumvent the nonconvexity and nonseparability of the problem under consideration, we accordingly introduce a Unified Bregman Alternating Minimization Algorithm (UBAMA) by maximally exploiting the favorable DC structure of the objective. Specifically, we first follow the spirit of alternating minimization to update each block variable in a sequential order, which can efficiently tackle the nonseparablitity caused by the coupled function. Then, we employ the Fenchel-Young inequality to approximate the second DC components (i.e., concave parts) so that each subproblem becomes a convex optimization problem, thereby alleviating the computational burden of the nonconvex DC parts. Moreover, each subproblem absorbs a Bregman proximal regularization term, which is usually beneficial for inducing closed-form solutions of subproblems for many cases via choosing appropriate Bregman functions. It is remarkable that our algorithm not only covers some existing algorithms, but also enjoys implementable schemes with easier subproblems than some state-of-the-art first-order algorithms developed for generic nonconvex and nonsmooth optimization problems. Theoretically, we prove that the sequence generated by our algorithm converges to a critical point under the Kurdyka-Łojasiewicz condition. A series of numerical experiments on image data sets demonstrate the superiority of the proposed algorithmic framework over some existing methods.Hongjin He, Zhiyuan Zhangwork_zomrj3gperb55a7xumsox6xkkeThu, 15 Sep 2022 00:00:00 GMTInference for Low-rank Tensors – No Need to Debias
https://scholar.archive.org/work/4yb46coxlzf5poz6cfommdmtfa
In this paper, we consider the statistical inference for several low-rank tensor models. Specifically, in the Tucker low-rank tensor PCA or regression model, provided with any estimates achieving some attainable error rate, we develop the data-driven confidence regions for the singular subspace of the parameter tensor based on the asymptotic distribution of an updated estimate by two-iteration alternating minimization. The asymptotic distributions are established under some essential conditions on the signal-to-noise ratio (in PCA model) or sample size (in regression model). If the parameter tensor is further orthogonally decomposable, we develop the methods and non-asymptotic theory for inference on each individual singular vector. For the rank-one tensor PCA model, we establish the asymptotic distribution for general linear forms of principal components and confidence interval for each entry of the parameter tensor. Finally, numerical simulations are presented to corroborate our theoretical discoveries. In all these models, we observe that different from many matrix/vector settings in existing work, debiasing is not required to establish the asymptotic distribution of estimates or to make statistical inference on low-rank tensors. In fact, due to the widely observed statistical-computational-gap for low-rank tensor estimation, one usually requires stronger conditions than the statistical (or information-theoretic) limit to ensure the computationally feasible estimation is achievable. Surprisingly, such conditions "incidentally" render a feasible low-rank tensor inference without debiasing.Dong Xia and Anru R. Zhang and Yuchen Zhouwork_4yb46coxlzf5poz6cfommdmtfaFri, 29 Oct 2021 00:00:00 GMT