Guarantees of a Preconditioned Subgradient Algorithm for Overparameterized Asymmetric Low-rank Matrix Recovery

In this paper, we focus on a matrix factorization-based approach for robust low-rank and asymmetric matrix recovery from corrupted measurements. We address the challenging scenario where the rank of the sought matrix is unknown and employ an overparameterized approach using the variational form of the nuclear norm as a regularizer. We propose a subgradient algorithm that inherits the merits of preconditioned algorithms, whose rate of convergence does not depend on the condition number of the sought matrix, and addresses their current limitation, i.e., the lack of convergence guarantees in the case of asymmetric matrices with unknown rank. In this setting, we provide, for the first time in the literature, linear convergence guarantees for the derived overparameterized preconditioned subgradient algorithm in the presence of gross corruptions. Additionally, by applying our approach to matrix sensing, we highlight its merits when the measurement operator satisfies the mixed-norm restricted isometry properties. Lastly, we present numerical experiments that validate our theoretical results and demonstrate the effectiveness of our approach.

Riemannian Optimization for Non-convex Euclidean Distance Geometry with Global Recovery Guarantees

The problem of determining the configuration of points from partial distance information, known as the Euclidean Distance Geometry (EDG) problem, is fundamental to many tasks in the applied sciences. In this paper, we propose two algorithms grounded in the Riemannian optimization framework to address the EDG problem. Our approach formulates the problem as a low-rank matrix completion task over the Gram matrix, using partial measurements represented as expansion coefficients of the Gram matrix in a non-orthogonal basis. For the first algorithm, under a uniform sampling with replacement model for the observed distance entries, we demonstrate that, with high probability, a Riemannian gradient-like algorithm on the manifold of rank-$r$ matrices converges linearly to the true solution, given initialization via a one-step hard thresholding. This holds provided the number of samples, $m$, satisfies $m \geq \mathcal{O}(n^{7/4}r^2 \log(n))$. With a more refined initialization, achieved through resampled Riemannian gradient-like descent, we further improve this bound to $m \geq \mathcal{O}(nr^2 \log(n))$. Our analysis for the first algorithm leverages a non-self-adjoint operator and depends on deriving eigenvalue bounds for an inner product matrix of restricted basis matrices, leveraging sparsity properties for tighter guarantees than previously established. The second algorithm introduces a self-adjoint surrogate for the sampling operator. This algorithm demonstrates strong numerical performance on both synthetic and real data. Furthermore, we show that optimizing over manifolds of higher-than-rank-$r$ matrices yields superior numerical results, consistent with recent literature on overparameterization in the EDG problem.

Correlating Time Series with Interpretable Convolutional Kernels

This study addresses the problem of convolutional kernel learning in univariate, multivariate, and multidimensional time series data, which is crucial for interpreting temporal patterns in time series and supporting downstream machine learning tasks. First, we propose formulating convolutional kernel learning for univariate time series as a sparse regression problem with a non-negative constraint, leveraging the properties of circular convolution and circulant matrices. Second, to generalize this approach to multivariate and multidimensional time series data, we use tensor computations, reformulating the convolutional kernel learning problem in the form of tensors. This is further converted into a standard sparse regression problem through vectorization and tensor unfolding operations. In the proposed methodology, the optimization problem is addressed using the existing non-negative subspace pursuit method, enabling the convolutional kernel to capture temporal correlations and patterns. To evaluate the proposed model, we apply it to several real-world time series datasets. On the multidimensional rideshare and taxi trip data from New York City and Chicago, the convolutional kernels reveal interpretable local correlations and cyclical patterns, such as weekly seasonality. In the context of multidimensional fluid flow data, both local and nonlocal correlations captured by the convolutional kernels can reinforce tensor factorization, leading to performance improvements in fluid flow reconstruction tasks. Thus, this study lays an insightful foundation for automatically learning convolutional kernels from time series data, with an emphasis on interpretability through sparsity and non-negativity constraints.

S$^3$Attention: Improving Long Sequence Attention with Smoothed Skeleton Sketching

Attention based models have achieved many remarkable breakthroughs in numerous applications. However, the quadratic complexity of Attention makes the vanilla Attention based models hard to apply to long sequence tasks. Various improved Attention structures are proposed to reduce the computation cost by inducing low rankness and approximating the whole sequence by sub-sequences. The most challenging part of those approaches is maintaining the proper balance between information preservation and computation reduction: the longer sub-sequences used, the better information is preserved, but at the price of introducing more noise and computational costs. In this paper, we propose a smoothed skeleton sketching based Attention structure, coined S$^3$Attention, which significantly improves upon the previous attempts to negotiate this trade-off. S$^3$Attention has two mechanisms to effectively minimize the impact of noise while keeping the linear complexity to the sequence length: a smoothing block to mix information over long sequences and a matrix sketching method that simultaneously selects columns and rows from the input matrix. We verify the effectiveness of S$^3$Attention both theoretically and empirically. Extensive studies over Long Range Arena (LRA) datasets and six time-series forecasting show that S$^3$Attention significantly outperforms both vanilla Attention and other state-of-the-art variants of Attention structures.

Guaranteed Sampling Flexibility for Low-tubal-rank Tensor Completion

While Bernoulli sampling is extensively studied in tensor completion, t-CUR sampling approximates low-tubal-rank tensors via lateral and horizontal subtensors. However, both methods lack sufficient flexibility for diverse practical applications. To address this, we introduce Tensor Cross-Concentrated Sampling (t-CCS), a novel and straightforward sampling model that advances the matrix cross-concentrated sampling concept within a tensor framework. t-CCS effectively bridges the gap between Bernoulli and t-CUR sampling, offering additional flexibility that can lead to computational savings in various contexts. A key aspect of our work is the comprehensive theoretical analysis provided. We establish a sufficient condition for the successful recovery of a low-rank tensor from its t-CCS samples. In support of this, we also develop a theoretical framework validating the feasibility of t-CUR via uniform random sampling and conduct a detailed theoretical sampling complexity analysis for tensor completion problems utilizing the general Bernoulli sampling model. Moreover, we introduce an efficient non-convex algorithm, the Iterative t-CUR Tensor Completion (ITCURTC) algorithm, specifically designed to tackle the t-CCS-based tensor completion. We have intensively tested and validated the effectiveness of the t-CCS model and the ITCURTC algorithm across both synthetic and real-world datasets.

Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent

This paper studies the robust Hankel recovery problem, which simultaneously removes the sparse outliers and fulfills missing entries from the partial observation. We propose a novel non-convex algorithm, coined Hankel Structured Newton-Like Descent (HSNLD), to tackle the robust Hankel recovery problem. HSNLD is highly efficient with linear convergence, and its convergence rate is independent of the condition number of the underlying Hankel matrix. The recovery guarantee has been established under some mild conditions. Numerical experiments on both synthetic and real datasets show the superior performance of HSNLD against state-of-the-art algorithms.

On the Robustness of Cross-Concentrated Sampling for Matrix Completion

Matrix completion is one of the crucial tools in modern data science research. Recently, a novel sampling model for matrix completion coined cross-concentrated sampling (CCS) has caught much attention. However, the robustness of the CCS model against sparse outliers remains unclear in the existing studies. In this paper, we aim to answer this question by exploring a novel Robust CCS Completion problem. A highly efficient non-convex iterative algorithm, dubbed Robust CUR Completion (RCURC), is proposed. The empirical performance of the proposed algorithm, in terms of both efficiency and robustness, is verified in synthetic and real datasets.

Towards Constituting Mathematical Structures for Learning to Optimize

Learning to Optimize (L2O), a technique that utilizes machine learning to learn an optimization algorithm automatically from data, has gained arising attention in recent years. A generic L2O approach parameterizes the iterative update rule and learns the update direction as a black-box network. While the generic approach is widely applicable, the learned model can overfit and may not generalize well to out-of-distribution test sets. In this paper, we derive the basic mathematical conditions that successful update rules commonly satisfy. Consequently, we propose a novel L2O model with a mathematics-inspired structure that is broadly applicable and generalized well to out-of-distribution problems. Numerical simulations validate our theoretical findings and demonstrate the superior empirical performance of the proposed L2O model.

Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption

We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis (RPCA), that aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called Robust Tensor CUR Decompositions (RTCUR), for large-scale non-convex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.

Non-convex approaches for low-rank tensor completion under tubal sampling

Tensor completion is an important problem in modern data analysis. In this work, we investigate a specific sampling strategy, referred to as tubal sampling. We propose two novel non-convex tensor completion frameworks that are easy to implement, named tensor $L_1$-$L_2$ (TL12) and tensor completion via CUR (TCCUR). We test the efficiency of both methods on synthetic data and a color image inpainting problem. Empirical results reveal a trade-off between the accuracy and time efficiency of these two methods in a low sampling ratio. Each of them outperforms some classical completion methods in at least one aspect.