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Proximal methods for structured nonsmooth optimization over Riemannian submanifolds

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arXiv 2024年

作者： Li, Qia Zhang, Na Yan, Hanwei Feng, Junyu School of Computer Science and Engineering Guangdong Province Key Laboratory of Computational Science Sun Yat-sen University Guangzhou China Department of Applied Mathematics College of Mathematics and Informatics South China Agricultural University Guangzhou China School of Computer Science and Engineering Sun Yat-sen University Guangzhou China

In this paper, we consider a class of structured nonsmooth optimization problems over an embedded submanifold of a Euclidean space, where the first part of the objective is the sum of a difference-of-convex (DC) function and a smooth function, while the remaining part is the square of a weakly convex function over a smooth function. This model problem has many important applications in machine learning and scientific computing, for example, the sparse generalized eigenvalue problem. We propose a manifold proximal-gradient-subgradient algorithm (MPGSA) and show that under mild conditions any accumulation point of the solution sequence generated by it is a critical point of the underlying problem. By assuming the Kurdyka-Lojasiewicz property of an auxiliary function, we further establish the convergence of the full sequence generated by MPGSA under some suitable conditions. When the second component of the DC function involved is the maximum of finite continuously differentiable convex functions, we also propose an enhanced MPGSA with guaranteed subsequential convergence to a lifted B-stationary points of the optimization problem. Finally, some preliminary numerical experiments are conducted to illustrate the efficiency of the proposed *** Codes 90C26, 90C30, 65K05 Copyright © 2024, The Authors. All rights reserved.

关键词： Discriminant analysis

Unbiased Parameter Estimation for Partially Observed Diffusions

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arXiv 2023年

作者： Awadelkarim, Elsiddig Jasra, Ajay Ruzayqat, Hamza Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia

In this article we consider the estimation of static parameters for partially observed diffusion process with discrete-time observations over a fixed time interval. In particular, we assume that one must time-discretize the partially observed diffusion process and work with the model with bias and consider maximizing the resulting log-likelihood. Using a novel double randomization scheme, based upon Markovian stochastic approximation we develop a new method to unbiasedly estimate the static parameters, that is, to obtain the maximum likelihood estimator with no time discretization bias. Under assumptions we prove that our estimator is unbiased and investigate the method in several numerical examples, showing that it can empirically out-perform existing unbiased *** Codes 60J22, 62M05, 65C40, 62M20 © 2023, CC BY.

关键词： Parameter estimation

Decentralized and Equitable Optimal Transport

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Decentralized and Equitable Optimal Transport

American Control Conference (ACC)

作者： Ivan Lau Shiqian Ma César A. Uribe The Department of Computer Science National University of Singapore Singapore The Department of Electrical and Computer Engineering Rice University Houston TX USA The Department of Computational Applied Mathematics and Operations Research Rice University Houston TX USA

ISBN: (数字)9798350382655

ISBN: (纸本)9798350382662

This paper considers the decentralized (discrete) optimal transport (D-OT) problem. In this setting, a network of agents seeks to design a transportation plan jointly, where the cost function is the sum of privately held costs for each agent. We reformulate the D-OT problem as a constraint-coupled optimization problem and propose a single-loop decentralized algorithm with an iteration complexity of $O(1/\epsilon)$ that matches existing centralized first-order approaches. Moreover, we propose the decentralized equitable optimal transport (DE-OT) problem. In DE-OT, in addition to cooperatively designing a transportation plan that minimizes transportation costs, agents seek to ensure equity in their individual costs. The iteration complexity of the proposed method to solve DE-OT is also $O(1/\epsilon)$ . This rate improves existing centralized algorithms, where the best iteration complexity obtained is $O(1/\epsilon^{2})$ .

关键词： Costs Transportation Cost function Complexity theory

A Class of Fast and Accurate Multi-layer Block Summation and Dot Product Algorithms 18th

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A Class of Fast and Accurate Multi-layer Block Summation a...

18th IFIP WG 10.3 International Conference on Network and Parallel Computing, NPC 2021

作者： He, Kang Barrio, Roberto Chen, Lin Jiang, Hao Liu, Jie Gu, Tongxiang Qi, Jin Science and Technology on Parallel and Distributed Processing Laboratory National University of Defense Technology Changsha410073 China Department of Applied Mathematics University of Zaragoza ZaragozaE50009 Spain College of Computer National University of Defense Technology Changsha410073 China Institute of Applied Physics and Computational Mathematics Beijing100000 China

ISBN: (纸本)9783030935702

Basic recursive summation and common dot product algorithm have a backward error bound that grows linearly with the vector dimension. Blanchard [1] proposed a class of fast and accurate summation and dot product algorithms respectively called FABsum and FABdot, which trades off the calculation accuracy and speed by the block size. Castaldo [2] proposed a multi-layer block summation and dot product algorithm called SuperBlocksum and SuperBlockdot that can increase the accuracy while adding almost no additional calculations. We combine the idea of [1] with the multi-layer block structure to propose SuperFABsum (for "super fast and accurate block summation") and SuperFABdot (for "super fast and accurate block dot product"). Our algorithms have two variants, one is SuperFAB(within), the other is SuperFAB(outside). Our algorithms further improve accuracy and speed compared with FAB and SuperBlock. We conducted accuracy and speed tests on the high-performance FT2000+ processor. Experimental results show that SuperFABdot(within) algorithm is more accurate than FABdot and SuperBlockdot. Compared with FABdot, SuperFABdot(outside) algorithm can achieve up to 1.2 × performance speedup while ensuring similar accuracy. © 2022, IFIP International Federation for Information Processing.

关键词： Artificial intelligence

Quasi-Monte Carlo for partial differential equations with generalized Gaussian input uncertainty

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arXiv 2024年

作者： Guth, Philipp A. Kaarnioja, Vesa Johann Radon Institute for Computational and Applied Mathematics Austrian Academy of Sciences Altenbergerstraße 69 LinzAT-4040 Austria Department of Mathematics and Computer Science Free University of Berlin Arnimallee 6 BerlinDE-14195 Germany

There has been a surge of interest in uncertainty quantification for parametric partial differential equations (PDEs) with Gevrey regular inputs. The Gevrey class contains functions that are infinitely smooth with a growth condition on the higher-order partial derivatives, but which are nonetheless not analytic in general. Recent studies by Chernov and Lê (Comput. Math. Appl., 2024, and SIAM J. Numer. Anal., 2024) as well as Harbrecht, Schmidlin, and Schwab (Math. Models Methods Appl. Sci., 2024) analyze the setting wherein the input random field is assumed to be uniformly bounded with respect to the uncertain parameters. In this paper, we relax this assumption and allow for parameter-dependent bounds. The parametric inputs are modeled as generalized Gaussian random variables, and we analyze the application of quasi-Monte Carlo (QMC) integration to assess the PDE response statistics using randomly shifted rank-1 lattice rules. In addition to the QMC error analysis, we also consider the dimension truncation and finite element errors in this *** Codes 65D30, 65D32, 35R60 Copyright © 2024, The Authors. All rights reserved.

关键词： Monte Carlo methods

On Convergence of MRQI and IMRQI Methods for Hermitian Eigenvalue Problems

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Communications on applied mathematics and Computation 2021年第1期3卷 189-197页

作者： Fang Chen Cun-Qiang Miao Galina V.Muratova School of Applied Science Beijing Information Science and Technology UniversityBeijing 100192China School of Mathematics and Statistics Central South UniversityChangsha 410083China Laboratory of Computational Mechanics I.I.Vorovich Institute of MathematicsMechanics and Computer ScienceSouthern Federal UniversityRostov-on-Don 344090Russia

Bai et *** the multistep Rayleigh quotient iteration(MRQI)as well as its inexact variant(IMRQI)in a recent work(***.77:2396–2406,2019).These methods can be used to effectively compute an eigenpair of a Hermitian *** convergence theorems of these methods were established under two conditions imposed on the initial guesses for the target eigenvalue and *** this paper,we show that these two conditions can be merged into a relaxed one,so the convergence conditions in these theorems can be weakened,and the resulting convergence theorems are applicable to a broad class of *** addition,we give detailed discussions about the new convergence condition and the corresponding estimates of the convergence errors,leading to rigorous convergence theories for both the MRQI and the IMRQI.

关键词： Hermitian eigenvalue problem MRQI IMRQI Convergence

Towards Fair and Robust Face Parsing for Generative AI: A Multi-Objective Approach

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arXiv 2025年

作者： Abraham, Sophia J. Hauenstein, Jonathan D. Scheirer, Walter J. Department of Computer Science and Engineering University of Notre Dame Notre Dame IN46556 United States Department of Applied and Computational Mathematics and Statistics University of Notre Dame Notre Dame IN46556 United States

Face parsing is a fundamental task in computer vision, enabling applications such as identity verification, facial editing, and controllable image synthesis. However, existing face parsing models often lack fairness and robustness, leading to biased segmentation across demographic groups and errors under occlusions, noise, and domain shifts. These limitations affect downstream face synthesis, where segmentation biases can degrade generative model outputs. We propose a multi-objective learning framework that optimizes accuracy, fairness, and robustness in face parsing. Our approach introduces a homotopy-based loss function that dynamically adjusts the importance of these objectives during training. To evaluate its impact, we compare multi-objective and single-objective U-Net models in a GAN-based face synthesis pipeline (Pix2PixHD). Our results show that fairness-aware and robust segmentation improves photorealism and consistency in face generation. Additionally, we conduct preliminary experiments using ControlNet, a structured conditioning model for diffusion-based synthesis, to explore how segmentation quality influences guided image generation. Our findings demonstrate that multi-objective face parsing improves demographic consistency and robustness, leading to higher-quality GAN-based synthesis.1 © 2025, CC BY-NC-ND.

关键词：

STEERABLE TRANSFORMERS FOR VOLUMETRIC DATA

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arXiv 2024年

作者： Kundu, Soumyabrata Kondor, Risi Department of Statistics University of Chicago ChicagoIL60637 United States Department of Computer Science Department of Statistics Computational and Applied Mathematics University of Chicago ChicagoIL60637 United States

In this work we introduce Steerable Transformers, an extension of the Vision Transformer mechanism that maintains equivariance to the special Euclidean group SE(d). We propose an equivariant attention mechanism that operates on features extracted by steerable convolutions. Operating in Fourier space, our network utilizes Fourier space non-linearities. Our experiments in both two and three dimensions show that adding steerable transformer layers to steerable convolutional networks enhances performance. © 2024, CC BY.

关键词： Convolution

Globally injective ReLU networks

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The Journal of Machine Learning Research

The Journal of Machine Learning Research 2022年第1期23卷 4544-4598页

作者： Michael Puthawala Konik Kothari Matti Lassas Ivan Dokmanić Maarten De Hoop Department of Computational and Applied Mathematics Rice University Houston TX Department of Computer Science University of Illinois at Urbana-Champaign Champaign IL Department of Mathematics and Statistics University of Helsinki Finland Department of Mathematics and Computer Science University of Basel Basel Switzerland

Injectivity plays an important role in generative models where it enables inference; in inverse problems and compressed sensing with generative priors it is a precursor to well posedness. We establish sharp characterizations of injectivity of fully-connected and convolutional ReLU layers and networks. First, through a layerwise analysis, we show that an expansivity factor of two is necessary and sufficient for injectivity by constructing appropriate weight matrices. We show that global injectivity with iid Gaussian matrices, a commonly used tractable model, requires larger expansivity between 3.4 and 10.5. We also characterize the stability of inverting an injective network via worst-case Lipschitz constants of the inverse. We then use arguments from differential topology to study injectivity of deep networks and prove that any Lipschitz map can be approximated by an injective ReLU network. Finally, using an argument based on random projections, we show that an end-to-end--rather than layerwise--doubling of the dimension suffices for injectivity. Our results establish a theoretical basis for the study of nonlinear inverse and inference problems using neural networks.

关键词： bilipschitz networks injective generative models random neural networks deep learning universal approximation