LogSpecT: Feasible Graph Learning Model from Stationary Signals with Recovery Guarantees

Graph learning from signals is a core task in Graph Signal Processing (GSP). One of the most commonly used models to learn graphs from stationary signals is SpecT. However, its practical formulation rSpecT is known to be sensitive to hyperparameter selection and, even worse, to suffer from infeasibility. In this paper, we give the first condition that guarantees the infeasibility of rSpecT and design a novel model (LogSpecT) and its practical formulation (rLogSpecT) to overcome this issue. Contrary to rSpecT, the novel practical model rLogSpecT is always feasible. Furthermore, we provide recovery guarantees of rLogSpecT, which are derived from modern optimization tools related to epi-convergence. These tools could be of independent interest and significant for various learning problems. To demonstrate the advantages of rLogSpecT in practice, a highly efficient algorithm based on the linearized alternating direction method of multipliers (L-ADMM) is proposed. The subproblems of L-ADMM admit closed-form solutions and the convergence is guaranteed. Extensive numerical results on both synthetic and real networks corroborate the stability and superiority of our proposed methods, underscoring their potential for various graph learning applications

Rotation Group Synchronization via Quotient Manifold

Rotation group $\mathcal{SO}(d)$ synchronization is an important inverse problem and has attracted intense attention from numerous application fields such as graph realization, computer vision, and robotics. In this paper, we focus on the least-squares estimator of rotation group synchronization with general additive noise models, which is a nonconvex optimization problem with manifold constraints. Unlike the phase/orthogonal group synchronization, there are limited provable approaches for solving rotation group synchronization. First, we derive improved estimation results of the least-squares/spectral estimator, illustrating the tightness and validating the existing relaxation methods of solving rotation group synchronization through the optimum of relaxed orthogonal group version under near-optimal noise level for exact recovery. Moreover, departing from the standard approach of utilizing the geometry of the ambient Euclidean space, we adopt an intrinsic Riemannian approach to study orthogonal/rotation group synchronization. Benefiting from a quotient geometric view, we prove the positive definite condition of quotient Riemannian Hessian around the optimum of orthogonal group synchronization problem, and consequently the Riemannian local error bound property is established to analyze the convergence rate properties of various Riemannian algorithms. As a simple and feasible method, the sequential convergence guarantee of the (quotient) Riemannian gradient method for solving orthogonal/rotation group synchronization problem is studied, and we derive its global linear convergence rate to the optimum with the spectral initialization. All results are deterministic without any probabilistic model

Nonsmooth Composite Nonconvex-Concave Minimax Optimization

Nonconvex-concave minimax optimization has received intense interest in machine learning, including learning with robustness to data distribution, learning with non-decomposable loss, adversarial learning, to name a few. Nevertheless, most existing works focus on the gradient-descent-ascent (GDA) variants that can only be applied in smooth settings. In this paper, we consider a family of minimax problems whose objective function enjoys the nonsmooth composite structure in the variable of minimization and is concave in the variables of maximization. By fully exploiting the composite structure, we propose a smoothed proximal linear descent ascent (\textit{smoothed} PLDA) algorithm and further establish its $\mathcal{O}(\epsilon^{-4})$ iteration complexity, which matches that of smoothed GDA~\cite{zhang2020single} under smooth settings. Moreover, under the mild assumption that the objective function satisfies the one-sided Kurdyka-\L{}ojasiewicz condition with exponent $\theta \in (0,1)$, we can further improve the iteration complexity to $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$. To the best of our knowledge, this is the first provably efficient algorithm for nonsmooth nonconvex-concave problems that can achieve the optimal iteration complexity $\mathcal{O}(\epsilon^{-2})$ if $\theta \in (0,1/2]$. As a byproduct, we discuss different stationarity concepts and clarify their relationships quantitatively, which could be of independent interest. Empirically, we illustrate the effectiveness of the proposed smoothed PLDA in variation regularized Wasserstein distributionally robust optimization problems

Doubly Smoothed GDA: Global Convergent Algorithm for Constrained Nonconvex-Nonconcave Minimax Optimization

Nonconvex-nonconcave minimax optimization has received intense attention over the last decade due to its broad applications in machine learning. Unfortunately, most existing algorithms cannot be guaranteed to converge globally and even suffer from limit cycles. To address this issue, we propose a novel single-loop algorithm called doubly smoothed gradient descent ascent method (DSGDA), which naturally balances the primal and dual updates. The proposed DSGDA can get rid of limit cycles in various challenging nonconvex-nonconcave examples in the literature, including Forsaken, Bilinearly-coupled minimax, Sixth-order polynomial, and PolarGame. We further show that under an one-sided Kurdyka-\L{}ojasiewicz condition with exponent $\theta\in(0,1)$ (resp. convex primal/concave dual function), DSGDA can find a game-stationary point with an iteration complexity of $\mathcal{O}(\epsilon^{-2\max\{2\theta,1\}})$ (resp. $\mathcal{O}(\epsilon^{-4})$). These match the best results for single-loop algorithms that solve nonconvex-concave or convex-nonconcave minimax problems, or problems satisfying the rather restrictive one-sided Polyak-\L{}ojasiewicz condition. Our work demonstrates, for the first time, the possibility of having a simple and unified single-loop algorithm for solving nonconvex-nonconcave, nonconvex-concave, and convex-nonconcave minimax problems