Low-rank matrix factorization (LRMF) is a canonical problem in non-convex optimization, the objective function to be minimized is non-convex and even non-smooth, which makes the global convergence guarantee of gradient-based algorithm quite challenging.

The low-rank matrix recovery problem seeks to reconstruct an unknown $n_1 \times n_2$ rank-$r$ matrix from $m$ linear measurements, where $m\ll n_1n_2$. This problem has been extensively studied over the past few decades, leading to a variety of algorithms with solid theoretical guarantees. Among these, gradient descent based non-convex methods have become particularly popular due to their computational efficiency. However, these methods typically suffer from two key limitations: a sub-optimal sample complexity of $O((n_1 + n_2)r^2)$ and an iteration complexity of $O(κ\log(1/ε))$ to achieve $ε$-accuracy, resulting in slow convergence when the target matrix is ill-conditioned. Here, $κ$ denotes the condition number of the unknown matrix. Recent studies show that a preconditioned variant of GD, known as scaled gradient descent (ScaledGD), can significantly reduce the iteration complexity to $O(\log(1/ε))$. Nonetheless, its sample complexity remains sub-optimal at $O((n_1 + n_2)r^2)$. In contrast, a delicate virtual sequence technique demonstrates that the standard GD in the positive semidefinite (PSD) setting achieves the optimal sample complexity $O((n_1 + n_2)r)$, but converges more slowly with an iteration complexity $O(κ^2 \log(1/ε))$. In this paper, through a more refined analysis, we show that ScaledGD achieves both the optimal sample complexity $O((n_1 + n_2)r)$ and the improved iteration complexity $O(\log(1/ε))$. Notably, our results extend beyond the PSD setting to general low-rank matrix recovery problem. Numerical experiments further validate that ScaledGD accelerates convergence for ill-conditioned matrices with the optimal sampling complexity.

Q-linear convergence of AGN represents a substantial enhancement over the convergence of the existing methods for the overparameterized non-convex LRMS.

We consider the noisy matrix sensing problem in the over-parameterization setting, where the estimated rank $r$ is larger than the true rank $r_\star$. Specifically, our main objective is to recover a matrix $ X_\star \in \mathbb{R}^{n_1 \times n_2} $ with rank $ r_\star $ from noisy measurements using an over-parameterized factorized form $ LR^\top $, where $ L \in \mathbb{R}^{n_1 \times r}, \, R \in \mathbb{R}^{n_2 \times r} $ and $ \min\{n_1, n_2\} \ge r > r_\star $, with the true rank $ r_\star $ being unknown. Recently, preconditioning methods have been proposed to accelerate the convergence of matrix sensing problem compared to vanilla gradient descent, incorporating preconditioning terms $ (L^\top L + \lambda I)^{-1} $ and $ (R^\top R + \lambda I)^{-1} $ into the original gradient. However, these methods require careful tuning of the damping parameter $\lambda$ and are sensitive to initial points and step size. To address these limitations, we propose the alternating preconditioned gradient descent (APGD) algorithm, which alternately updates the two factor matrices, eliminating the need for the damping parameter and enabling faster convergence with larger step sizes. We theoretically prove that APGD achieves near-optimal error convergence at a linear rate, starting from arbitrary random initializations. Through extensive experiments, we validate our theoretical results and demonstrate that APGD outperforms other methods, achieving the fastest convergence rate. Notably, both our theoretical analysis and experimental results illustrate that APGD does not rely on the initialization procedure, making it more practical and versatile.

Tensors, which give a faithful and effective representation to deliver the intrinsic structure of multi-dimensional data, play a crucial role in an increasing number of signal processing and machine learning problems. However, tensor data are often accompanied by arbitrary signal corruptions, including missing entries and sparse noise. A fundamental challenge is to reliably extract the meaningful information from corrupted tensor data in a statistically and computationally efficient manner. This paper develops a scaled gradient descent (ScaledGD) algorithm to directly estimate the tensor factors with tailored spectral initializations under the tensor-tensor product (t-product) and tensor singular value decomposition (t-SVD) framework. In theory, ScaledGD achieves linear convergence at a constant rate that is independent of the condition number of the ground truth low-rank tensor for two canonical problems -- tensor robust principal component analysis and tensor completion -- as long as the level of corruptions is not too large and the sample size is sufficiently large, while maintaining the low per-iteration cost of gradient descent. To the best of our knowledge, ScaledGD is the first algorithm that provably has such properties for low-rank tensor estimation with the t-SVD decomposition. Finally, numerical examples are provided to demonstrate the efficacy of ScaledGD in accelerating the convergence rate of ill-conditioned low-rank tensor estimation in these two applications.

Robust matrix completion (RMC) is a widely used machine learning tool that simultaneously tackles two critical issues in low-rank data analysis: missing data entries and extreme outliers. This paper proposes a novel scalable and learnable non-convex approach, coined Learned Robust Matrix Completion (LRMC), for large-scale RMC problems. LRMC enjoys low computational complexity with linear convergence. Motivated by the proposed theorem, the free parameters of LRMC can be effectively learned via deep unfolding to achieve optimum performance. Furthermore, this paper proposes a flexible feedforward-recurrent-mixed neural network framework that extends deep unfolding from fix-number iterations to infinite iterations. The superior empirical performance of LRMC is verified with extensive experiments against state-of-the-art on synthetic datasets and real applications, including video background subtraction, ultrasound imaging, face modeling, and cloud removal from satellite imagery.

This work revisits the classical low-rank matrix factorization problem and unveils the critical role of initialization in shaping convergence rates for such nonconvex and nonsmooth optimization. We introduce Nystrom initialization, which significantly improves the global convergence of Scaled Gradient Descent (ScaledGD) in both symmetric and asymmetric matrix factorization tasks. Specifically, we prove that ScaledGD with Nystrom initialization achieves quadratic convergence in cases where only linear rates were previously known. Furthermore, we extend this initialization to low-rank adapters (LoRA) commonly used for finetuning foundation models. Our approach, NoRA, i.e., LoRA with Nystrom initialization, demonstrates superior performance across various downstream tasks and model scales, from 1B to 7B parameters, in large language and diffusion models.