Pose graph optimization (PGO) is fundamental to robot perception and navigation systems, serving as the mathematical backbone for solving simultaneous localization and mapping (SLAM). Existing solvers suffer from polynomial growth in computational complexity with graph size, hindering real-time deployment in large-scale scenarios. In this paper, by duplicating variables and introducing equality constraints, we reformulate the problem and propose a Parallelizable Riemannian Alternating Direction Method of Multipliers (PRADMM) to solve it efficiently. Compared with the state-of-the-art methods that usually exhibit polynomial time complexity growth with graph size, PRADMM enables efficient parallel computation across vertices regardless of graph size. Crucially, all subproblems admit closed-form solutions, ensuring PRADMM maintains exceptionally stable performance. Furthermore, by carefully exploiting the structures of the coefficient matrices in the constraints, we establish the global convergence of PRADMM under mild conditions, enabling larger relaxation step sizes within the interval (0,2). Extensive empirical validation on two synthetic datasets and multiple real-world 3D SLAM benchmarks confirms the superior computational performance of PRADMM.

Abstract. This paper focuses on the convergence of the alternating direction method of multipliers (ADMM) for solving linearly constrained optimization problems whose objective function is the sum of one weakly convex and two strongly convex functions. Many applications in image denoising and machine learning fields with a sparsity-driven regularization term can be approximated unbiasedly by weakly convex functions. Our new theoretical results provide convergence guarantees of the direct extension of ADMM (E-ADMM) without Lipschitz continuity of the gradient and Kurdyka–Łojasiewicz property, and establish the worst-case [Formula: see text] convergence rate in the nonergodic sense. Under further conditions such as Lipschitz continuity and nonsingularity, we derive the global linear rate of convergence. The numerical results on tensor robust principal component analysis and generalized elastic net regression illustrate that E-ADMM is efficient compared with some popular methods such as the difference-of-convex approach and accelerated proximal gradient descent.

In this paper, we explore a specific optimization problem that involves the combination of a differentiable nonconvex function and a nondifferentiable function. The differentiable component lacks a global Lipschitz continuous gradient, posing challenges for optimization. To address this issue and accelerate the convergence, we propose a Bregman proximal stochastic gradient method with extrapolation (BPSGE), which only requires smooth adaptivity of the differentiable part. Under variance reduction framework, we not only analyze the subsequential and global convergence of the proposed algorithm under certain conditions, but also analyze the sublinear convergence rate of the subsequence, and the complexity of the algorithm, revealing that the BPSGE algorithm requires at most O(epsilon\^\,(-2)) iterations in expectation to attain an epsilon-stationary point. To validate the effectiveness of our proposed algorithm, we conduct numerical experiments on three real-world applications: graph regularized nonnegative matrix factorization (NMF), matrix factorization with weakly-convex regularization, and NMF with nonconvex sparsity constraints. These experiments demonstrate that BPSGE is faster than the baselines without extrapolation. The code is available at: https://github.com/nothing2wang/BPSGE-Algorithm.

Recently, there has been a growing interest in the exploration of Nonlinear Matrix Decomposition (NMD) due to its close ties with neural networks. NMD aims to find a low-rank matrix from a sparse nonnegative matrix with a per-element nonlinear function. A typical choice is the Rectified Linear Unit (ReLU) activation function. To address over-fitting in the existing ReLU-based NMD model (ReLU-NMD), we propose a Tikhonov regularized ReLU-NMD model, referred to as ReLU-NMD-T. Subsequently, we introduce a momentum accelerated algorithm for handling the ReLU-NMD-T model. A distinctive feature, setting our work apart from most existing studies, is the incorporation of both positive and negative momentum parameters in our algorithm. Our numerical experiments on real-world datasets show the effectiveness of the proposed model and algorithm.

In diffusion-weighted magnetic resonance imaging, the estimation of the orientations of multiple nerve fibers in each voxel (the fiber orientation distribution (FOD)) is a critical issue for exploring the connection of cerebral tissue. In this paper, we establish a convex semidefinite programming (CSDP) model for the FOD estimation. One feature of the new model is that it can ensure the statistical meaning of FOD since as a probability density function, FOD must be nonnegative and have a unit mass. To construct such a statistically meaningful FOD, we consider its approximation by a sum of squares (SOS) polynomial and impose the unit-mass by a linear constraint. Another feature of the new model is that it introduces a new regularization based on the sparsity of nerve fibers. Due to the sparsity of the orientations of nerve fibers in cerebral white matter, a heuristic regularization is raised, which is inspired by the Z-eigenvalue of a symmetric tensor that closely relates to the SOS polynomial. To solve the CSDP efficiently, we propose a new Peaceman--Rachford splitting method and prove its global convergence. Numerical experiments on synthetic and real-world human brain data show that, when compared with some existing approaches for fiber estimations, the new method gives a sharp and smooth FOD. Further, the proposed Peaceman--Rachford splitting method is shown to have good numerical performances comparing several existing methods.

The positive definiteness of a diffusion tensor is important in magnetic resonance imaging because it reflects the phenomenon of water molecular diffusion in complicated biological tissue environments. To preserve this property, we represent it as an explicit positive semidefinite (PSD) matrix constraint and some linear matrix equalities. The objective function is the regularized linear least squares fitting for the log-linearized Stejskal--Tanner equation. The regularization term is the heuristic nuclear norm of the PSD matrix, since we expect it to be of low rank. In this way, we establish a convex quadratic semidefinite programming (SDP) model, whose global solution exists. The optimal solution could be solved by three efficient methods. While there are two state-of-the-art solvers---SDPT3 and QSDP---for the primal problem, we design a new augmented Lagrangian based alternating direction method (ADM) for the dual problem. Sensitivity analyses on the coefficients of the optimal diffusion tensor and the optimal objective function value with respect to noise-corrupted signals are presented. Experiments on synthetic data with multiple fibers show that the new method is robust to the Rician noise and outperforms several existing methods. Furthermore, when the fiber orientation distribution function is considered, the new method is competitive with the Q-ball imaging. Using the human brain data, we illustrate that the new method could capture the crossing of three nervous fiber bundles. Additionally, the new method generates positive definite generalized diffusion tensors in all voxels, while the unconstrained least squares fitting fails. Finally, we confirm that the ADM solver is more efficient than SDPT3 and QSDP for this special problem.