arXiv subjects

Jian-Feng Cai

Publications and source records attributed to Jian-Feng Cai.

At least 37 records · Page 2Linked to original sources

arXiv2024

A Fast and Provable Algorithm for Sparse Phase Retrieval

We study the sparse phase retrieval problem, which seeks to recover a sparse signal from a limited set of magnitude-only measurements. In contrast to prevalent sparse phase retrieval algorithms that primarily use first-order methods, we propose an innovative second-order algorithm that employs a Newton-type method with hard thresholding. This algorithm overcomes the linear convergence limitations of first-order methods while preserving their hallmark per-iteration computational efficiency. We provide theoretical guarantees that our algorithm converges to the $s$-sparse ground truth signal $\mathbf{x}^{\natural} \in \mathbb{R}^n$ (up to a global sign) at a quadratic convergence rate after at most $O(\log (\Vert\mathbf{x}^{\natural} \Vert /x_{\min}^{\natural}))$ iterations, using $Ω(s^2\log n)$ Gaussian random samples. Numerical experiments show that our algorithm achieves a significantly faster convergence rate than state-of-the-art methods.

cs.IT↗

arXiv2024

Interlacing Polynomial Method for the Column Subset Selection Problem

This paper investigates the spectral norm version of the column subset selection problem. Given a matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ and a positive integer $k\leq\text{rank}(\mathbf{A})$, the objective is to select exactly $k$ columns of $\mathbf{A}$ that minimize the spectral norm of the residual matrix after projecting $\mathbf{A}$ onto the space spanned by the selected columns. We use the method of interlacing polynomials introduced by Marcus-Spielman-Srivastava to derive a new upper bound on the minimal approximation error. This new bound is asymptotically sharp when the matrix $\mathbf{A}\in\mathbb{R}^{n\times d}$ obeys a spectral power-law decay. The relevant expected characteristic polynomials can be written as an extension of the expected polynomial for the restricted invertibility problem, incorporating two extra variable substitution operators. Finally, we propose a deterministic polynomial-time algorithm that achieves this error bound up to a computational error.

cs.DS↗

arXiv2023

Stochastic Three-Operator Splitting Algorithms for Nonconvex and Nonsmooth Optimization Arising from FLASH Radiotherapy

Radiation therapy (RT) aims to deliver tumoricidal doses with minimal radiation-induced normal-tissue toxicity. Compared to conventional RT (of conventional dose rate), FLASH-RT (of ultra-high dose rate) can provide additional normal tissue sparing, which however has created a new nonconvex and nonsmooth optimization problem that is highly challenging to solve. In this paper, we propose a stochastic three-operator splitting (STOS) algorithm to address the FLASH optimization problem. We establish the convergence and convergence rates of the STOS algorithm under the nonconvex framework for both unbiased gradient estimators and variance-reduced gradient estimators. These stochastic gradient estimators include the most popular ones, such as SGD, SAGA, SARAH, and SVRG, among others. The effectiveness of the STOS algorithm is validated using FLASH radiotherapy planning for patients.

math.OC↗

arXiv2023

Restoration Guarantee of Image Inpainting via Low Rank Patch Matrix Completion

In recent years, patch-based image restoration approaches have demonstrated superior performance compared to conventional variational methods. This paper delves into the mathematical foundations underlying patch-based image restoration methods, with a specific focus on establishing restoration guarantees for patch-based image inpainting, leveraging the assumption of self-similarity among patches. To accomplish this, we present a reformulation of the image inpainting problem as structured low-rank matrix completion, accomplished by grouping image patches with potential overlaps. By making certain incoherence assumptions, we establish a restoration guarantee, given that the number of samples exceeds the order of $rlog^2(N)$, where $N\times N$ denotes the size of the image and $r > 0$ represents the sum of ranks for each group of image patches. Through our rigorous mathematical analysis, we provide valuable insights into the theoretical foundations of patch-based image restoration methods, shedding light on their efficacy and offering guidelines for practical implementation.

cs.IT↗

arXiv2023

Fast Projected Newton-like Method for Precision Matrix Estimation under Total Positivity

We study the problem of estimating precision matrices in Gaussian distributions that are multivariate totally positive of order two ($\mathrm{MTP}_2$). The precision matrix in such a distribution is an M-matrix. This problem can be formulated as a sign-constrained log-determinant program. Current algorithms are designed using the block coordinate descent method or the proximal point algorithm, which becomes computationally challenging in high-dimensional cases due to the requirement to solve numerous nonnegative quadratic programs or large-scale linear systems. To address this issue, we propose a novel algorithm based on the two-metric projection method, incorporating a carefully designed search direction and variable partitioning scheme. Our algorithm substantially reduces computational complexity, and its theoretical convergence is established. Experimental results on synthetic and real-world datasets demonstrate that our proposed algorithm provides a significant improvement in computational efficiency compared to the state-of-the-art methods.

cs.LG↗

arXiv2023

Computationally Efficient and Statistically Optimal Robust Low-rank Matrix and Tensor Estimation

Low-rank matrix estimation under heavy-tailed noise is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a novel Riemannian sub-gradient (RsGrad) algorithm which is not only computationally efficient with linear convergence but also is statistically optimal, be the noise Gaussian or heavy-tailed. Convergence theory is established for a general framework and specific applications to absolute loss, Huber loss, and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of dual-phase convergence. In phase one, RsGrad behaves as in a typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator which is already observed in the existing literature. Interestingly, during phase two, RsGrad converges linearly as if minimizing a smooth and strongly convex objective function and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Lastly, RsGrad is applicable for low-rank tensor estimation under heavy-tailed noise where a statistically optimal rate is attainable with the same phenomenon of dual-phase convergence, and a novel shrinkage-based second-order moment method is guaranteed to deliver a warm initialization. Numerical simulations confirm our theoretical discovery and showcase the superiority of RsGrad over prior methods.

math.ST↗

arXiv2023

Computationally Efficient and Statistically Optimal Robust High-Dimensional Linear Regression

High-dimensional linear regression under heavy-tailed noise or outlier corruption is challenging, both computationally and statistically. Convex approaches have been proven statistically optimal but suffer from high computational costs, especially since the robust loss functions are usually non-smooth. More recently, computationally fast non-convex approaches via sub-gradient descent are proposed, which, unfortunately, fail to deliver a statistically consistent estimator even under sub-Gaussian noise. In this paper, we introduce a projected sub-gradient descent algorithm for both the sparse linear regression and low-rank linear regression problems. The algorithm is not only computationally efficient with linear convergence but also statistically optimal, be the noise Gaussian or heavy-tailed with a finite 1 + epsilon moment. The convergence theory is established for a general framework and its specific applications to absolute loss, Huber loss and quantile loss are investigated. Compared with existing non-convex methods, ours reveals a surprising phenomenon of two-phase convergence. In phase one, the algorithm behaves as in typical non-smooth optimization that requires gradually decaying stepsizes. However, phase one only delivers a statistically sub-optimal estimator, which is already observed in the existing literature. Interestingly, during phase two, the algorithm converges linearly as if minimizing a smooth and strongly convex objective function, and thus a constant stepsize suffices. Underlying the phase-two convergence is the smoothing effect of random noise to the non-smooth robust losses in an area close but not too close to the truth. Numerical simulations confirm our theoretical discovery and showcase the superiority of our algorithm over prior methods.

math.ST↗

arXiv2023

A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recovery

The low-rank matrix recovery problem often arises in various fields, including signal processing, machine learning, and imaging science. The Riemannian gradient descent (RGD) algorithm has proven to be an efficient algorithm for solving this problem. In this paper, we present a preconditioned Riemannian gradient descent (PRGD) for low-rank matrix recovery. The preconditioner, noted for its simplicity and computational efficiency, is constructed by weighting the (i,j)-th entry of the gradient matrix according to the norms of the i-th row and the j-th column. We establish the theoretical recovery guarantee for PRGD under the restricted isometry property assumption. Experimental results indicate that PRGD can accelerate RGD by up to tenfold in solving low-rank matrix recovery problems such as matrix completion.

math.OC↗

arXiv2023

Provable Sample-Efficient Sparse Phase Retrieval Initialized by Truncated Power Method

We study the sparse phase retrieval problem, recovering an $s$-sparse length-$n$ signal from $m$ magnitude-only measurements. Two-stage non-convex approaches have drawn much attention in recent studies for this problem. Despite non-convexity, many two-stage algorithms provably converge to the underlying solution linearly when appropriately initialized. However, in terms of sample complexity, the bottleneck of those algorithms often comes from the initialization stage. Although the refinement stage usually needs only $m=Ω(s\log n)$ measurements, the widely used spectral initialization in the initialization stage requires $m=Ω(s^2\log n)$ measurements to produce a desired initial guess, which causes the total sample complexity order-wisely more than necessary. To reduce the number of measurements, we propose a truncated power method to replace the spectral initialization for non-convex sparse phase retrieval algorithms. We prove that $m=Ω(\bar{s} s\log n)$ measurements, where $\bar{s}$ is the stable sparsity of the underlying signal, are sufficient to produce a desired initial guess. When the underlying signal contains only very few significant components, the sample complexity of the proposed algorithm is $m=Ω(s\log n)$ and optimal. Numerical experiments illustrate that the proposed method is more sample-efficient than state-of-the-art algorithms.

cs.IT↗

arXiv2023

Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric

This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.

math.OC↗

arXiv2023

Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion

We study the robust matrix completion problem for the low-rank Hankel matrix, which detects the sparse corruptions caused by extreme outliers while we try to recover the original Hankel matrix from the partial observation. In this paper, we explore the convenient Hankel structure and propose a novel non-convex algorithm, coined Hankel Structured Gradient Descent (HSGD), for large-scale robust Hankel matrix completion problems. HSGD is highly computing- and sample-efficient compared to the state-of-the-arts. The recovery guarantee with a linear convergence rate has been established for HSGD under some mild assumptions. The empirical advantages of HSGD are verified on both synthetic datasets and real-world nuclear magnetic resonance signals.

cs.IT↗

arXiv2022

Quantum state tomography via non-convex Riemannian gradient descent

The recovery of an unknown density matrix of large size requires huge computational resources. The recent Factored Gradient Descent (FGD) algorithm and its variants achieved state-of-the-art performance since they could mitigate the dimensionality barrier by utilizing some of the underlying structures of the density matrix. Despite their theoretical guarantee of a linear convergence rate, the convergence in practical scenarios is still slow because the contracting factor of the FGD algorithms depends on the condition number $κ$ of the ground truth state. Consequently, the total number of iterations can be as large as $O(\sqrtκ\ln(\frac{1}{\varepsilon}))$ to achieve the estimation error $\varepsilon$. In this work, we derive a quantum state tomography scheme that improves the dependence on $κ$ to the logarithmic scale; namely, our algorithm could achieve the approximation error $\varepsilon$ in $O(\ln(\frac{1}{κ\varepsilon}))$ steps. The improvement comes from the application of the non-convex Riemannian gradient descent (RGD). The contracting factor in our approach is thus a universal constant that is independent of the given state. Our theoretical results of extremely fast convergence and nearly optimal error bounds are corroborated by numerical results.

quant-ph↗

arXiv2022

Approximation Theory of Wavelet Frame Based Image Restoration

In this paper, we analyze the error estimate of a wavelet frame based image restoration method from degraded and incomplete measurements. We present the error between the underlying original discrete image and the approximate solution which has the minimal $\ell_1$-norm of the canonical wavelet frame coefficients among all possible solutions. Then we further connect the error estimate for the discrete model to the approximation to the underlying function from which the underlying image comes.

math.AP↗

arXiv2022

Two Stage Continuous Domain Regularization for Piecewise Constant Image Restoration

The finite-rate-of-innovation (FRI) framework which corresponds a signal/image to a structured low-rank matrix is emerging as an alternative to the traditional sparse regularization. This is because such an off-the-grid approach is able to alleviate the basis mismatch between the true support in the continuous domain and the discrete grid. In this paper, we propose a two-stage off-the-grid regularization model for the image restoration. Given that the discontinuities/edges of the image lie in the zero level set of a band-limited periodic function, we can derive that the Fourier samples of the gradient of the image satisfy an annihilation relation, resulting in a low-rank two-fold Hankel matrix. In addition, since the singular value decomposition of a low-rank Hankel matrix corresponds to an adaptive tight frame system which can represent the image with sparse canonical coefficients, our approach consists of the following two stages. The first stage learns the tight wavelet frame system from a given measurement, and the second stage restores the image via the analysis approach based sparse regularization. The numerical results are presented to demonstrate that the proposed approach is compared favorably against several popular discrete regularization approaches and structured low-rank matrix approaches.

math.NA↗

arXiv2022

Approximation Theory of Total Variation Minimization for Data Completion

Total variation (TV) minimization is one of the most important techniques in modern signal/image processing, and has wide range of applications. While there are numerous recent works on the restoration guarantee of the TV minimization in the framework of compressed sensing, there are few works on the restoration guarantee of the restoration from partial observations. This paper is to analyze the error of TV based restoration from random entrywise samples. In particular, we estimate the error between the underlying original data and the approximate solution that interpolates (or approximates with an error bound depending on the noise level) the given data that has the minimal TV seminorm among all possible solutions. Finally, we further connect the error estimate for the discrete model to the sparse gradient restoration problem and to the approximation to the underlying function from which the underlying true data comes.

math.AP↗

arXiv2022

Nearly optimal bounds for the global geometric landscape of phase retrieval

The phase retrieval problem is concerned with recovering an unknown signal $\bf{x} \in \mathbb{R}^n$ from a set of magnitude-only measurements $y_j=|\langle \bf{a}_j,\bf{x} \rangle|, \; j=1,\ldots,m$. A natural least squares formulation can be used to solve this problem efficiently even with random initialization, despite its non-convexity of the loss function. One way to explain this surprising phenomenon is the benign geometric landscape: (1) all local minimizers are global; and (2) the objective function has a negative curvature around each saddle point and local maximizer. In this paper, we show that $m=O(n \log n)$ Gaussian random measurements are sufficient to guarantee the loss function of a commonly used estimator has such benign geometric landscape with high probability. This is a step toward answering the open problem given by Sun-Qu-Wright, in which the authors suggest that $O(n \log n)$ or even $O(n)$ is enough to guarantee the favorable geometric property.

cs.IT↗

arXiv2022

Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization

We investigate a generalized framework to estimate a latent low-rank plus sparse tensor, where the low-rank tensor often captures the multi-way principal components and the sparse tensor accounts for potential model mis-specifications or heterogeneous signals that are unexplainable by the low-rank part. The framework is flexible covering both linear and non-linear models, and can easily handle continuous or categorical variables. We propose a fast algorithm by integrating the Riemannian gradient descent and a novel gradient pruning procedure. Under suitable conditions, the algorithm converges linearly and can simultaneously estimate both the low-rank and sparse tensors. The statistical error bounds of final estimates are established in terms of the gradient of loss function. The error bounds are generally sharp under specific statistical models, e.g., the robust tensor PCA and the community detection in hypergraph networks with outlier vertices. Moreover, our method achieves non-trivial error bounds for heavy-tailed tensor PCA whenever the noise has a finite $2+\varepsilon$ moment. We apply our method to analyze the international trade flow dataset and the statistician hypergraph co-authorship network, both yielding new and interesting findings.

stat.ME↗

arXiv2022

Sample-Efficient Sparse Phase Retrieval via Stochastic Alternating Minimization

In this work we propose a nonconvex two-stage \underline{s}tochastic \underline{a}lternating \underline{m}inimizing (SAM) method for sparse phase retrieval. The proposed algorithm is guaranteed to have an exact recovery from $O(s\log n)$ samples if provided the initial guess is in a local neighbour of the ground truth. Thus, the proposed algorithm is two-stage, first we estimate a desired initial guess (e.g. via a spectral method), and then we introduce a randomized alternating minimization strategy for local refinement. Also, the hard-thresholding pursuit algorithm is employed to solve the sparse constraint least square subproblems. We give the theoretical justifications that SAM find the underlying signal exactly in a finite number of iterations (no more than $O(\log m)$ steps) with high probability. Further, numerical experiments illustrates that SAM requires less measurements than state-of-the-art algorithms for sparse phase retrieval problem.

math.NA↗