This paper studies decentralized optimization over a compact submanifold within a communication network of n nodes, where each node possesses a smooth non-convex local cost function, and the goal is to jointly minimize the sum of these local costs. We focus particularly on the online setting, where local data is processed in real-time as it streams in, without the need for full data storage. We propose a decentralized projected Riemannian stochastic recursive momentum (DPRSRM) method that employs local hybrid stochastic gradient estimators and uses the network to track the global gradient. DPRSRM achieves an oracle complexity of O(epsilon^(-3/2)), outperforming existing methods that have at most O(epsilon^(-2)) complexity. Our method requires only O(1) gradient evaluations per iteration for each local node and does not require restarting with a large batch gradient. Furthermore, we demonstrate the effectiveness of our proposed methods compared to state-of-the-art ones through numerical experiments on principal component analysis problems and low-rank matrix completion.

We study the problem of minimizing the sum of a smooth function and a nonsmooth convex regularizer over a compact Riemannian submanifold embedded in Euclidean space. By introducing an auxiliary splitting variable, we propose an adaptive Riemannian alternating direction method of multipliers (ARADMM), which, for the first time, achieves convergence without requiring smoothing of the nonsmooth term. In contrast to conventional Riemannian ADMM methods that require exactly solving a nested subproblem at each iteration, our approach involves only one Riemannian gradient evaluation and one proximal update per iteration. Through careful and adaptive coordination of the stepsizes and penalty parameters, we establish an optimal iteration complexity of order $\mathcal{O}(\epsilon^{-3})$ for finding an $\epsilon$-approximate KKT point, matching the complexity of existing smoothing technique-based Riemannian ADMM methods. Extensive numerical experiments on sparse PCA and robust subspace recovery demonstrate that our ARADMM consistently outperforms state-of-the-art Riemannian ADMM variants in convergence speed and solution quality.

Learning pseudo-contractive denoisers is a fundamental challenge in the theoretical analysis of Plug-and-Play (PnP) methods and the Regularization by Denoising (RED) framework. While spectral methods attempt to address this challenge using the power iteration method, they fail to guarantee the truly pseudo-contractive property and suffer from high computational complexity. In this work, we rethink gradient step (GS) denoisers and establish a theoretical connection between GS denoisers and pseudo-contractive operators. We show that GS denoisers, with the gradients of convex potential functions parameterized by input convex neural networks (ICNNs), can achieve truly pseudo-contractive properties. Furthermore, we integrate the learned truly pseudo-contractive denoiser into the RED-PRO (RED via fixed-point projection) model, definitely ensuring convergence in terms of both iterative sequences and objective functions. Extensive numerical experiments confirm that the learned GS denoiser satisfies the truly pseudo-contractive property and, when integrated into RED-PRO, provides a favorable trade-off between interpretability and empirical performance on inverse problems.

This paper aims to develop a Newton-type method to solve a class of nonconvex composite programs. In particular, the nonsmooth part is possibly nonconvex. To tackle the nonconvexity, we develop a notion of strong prox-regularity which is related to the singleton property and Lipschitz continuity of the associated proximal operator, and we verify it in various classes of functions, including weakly convex functions, indicator functions of proximally smooth sets, and two specific sphere-related nonconvex nonsmooth functions. In this case, the problem class we are concerned with covers smooth optimization problems on manifold and certain composite optimization problems on manifold. For the latter, the proposed algorithm is the first second-order type method. Combining with the semismoothness of the proximal operator, we design a projected semismooth Newton method to find a root of the natural residual induced by the proximal gradient method. Due to the possible nonconvexity of the feasible domain, an extra projection is added to the usual semismooth Newton step and new criteria are proposed for the switching between the projected semismooth Newton step and the proximal step. The global convergence is then established under the strong prox-regularity. Based on the BD regularity condition, we establish local superlinear convergence. Numerical experiments demonstrate the effectiveness of our proposed method compared with state-of-the-art ones.

As field-programmable gate array (FPGA) architectures continue to evolve and become more complex, the heterogeneity and clock constraints imposed by modern FPGAs have posed significant challenges to FPGA placement. This article proposes a high-performance placement engine for modern large-scale FPGAs with heterogeneity and clock constraints. To improve efficiency and scalability, we develop a clustering method considering both internal/external connectivity and the balance of block types to build the hierarchy. In each hierarchy level, we propose a hybrid penalty and augmented Lagrangian method (HPALM) to convert the FPGA global placement with heterogeneity and clock constraints into a series of unconstrained optimization subproblems, then use the Adam method to solve each subproblem. In particular, we prove that the HPALM is globally convergent for global placement. Besides, a matching-based IP block legalization is developed to legalize the DSPs and RAMs, and a multistage packing is presented to cluster LUTs and FFs into HCLBs. Finally, we propose a history-based legalization to legalize CLBs in an FPGA, and a simulated-annealing-based detailed placement is presented to reduce the wirelength while maintaining legality. Compared with the state-of-the-art works, experimental results based on the ISPD 2017 contest benchmarks show that the proposed algorithm can achieve the shortest routed wirelength in a reasonable runtime.

The sparse representation-based classifier (SRC) has been developed and verified as having great potential for real-world face recognition. In this paper, we propose a discriminative projection and representation-based classification (DPRC) method to enhance the discriminant ability of the SRC. The proposed method first obtains a discriminative projection matrix not only maximizing the ratio of the distance within interclass over the distance within intraclass, but also minimizing the linear approximation error within intraclass. Then it maps the original data onto the discriminative space, and adopts an SRC method to obtain the final solution. An inexact augmented Lagrangian method of multiplier is proposed for finding the optimal representation vector in our framework, and a proximal alternating minimization method is adopted to the iteration subproblems of the proposed method. The proposed method is proven to have the subsequence convergence property. Experimental results on Yale, ORL, and AR face image databases demonstrate that, compared with some existing feature extraction methods based on the SRC, the proposed DPRC method is more efficient.