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Anru R. Zhang

Publications and source records attributed to Anru R. Zhang.

At least 55 records · Page 3Linked to original sources
arXiv2024

Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model

This paper introduces a novel framework called Mode-wise Principal Subspace Pursuit (MOP-UP) to extract hidden variations in both the row and column dimensions for matrix data. To enhance the understanding of the framework, we introduce a class of matrix-variate spiked covariance models that serve as inspiration for the development of the MOP-UP algorithm. The MOP-UP algorithm consists of two steps: Average Subspace Capture (ASC) and Alternating Projection (AP). These steps are specifically designed to capture the row-wise and column-wise dimension-reduced subspaces which contain the most informative features of the data. ASC utilizes a novel average projection operator as initialization and achieves exact recovery in the noiseless setting. We analyze the convergence and non-asymptotic error bounds of MOP-UP, introducing a blockwise matrix eigenvalue perturbation bound that proves the desired bound, where classic perturbation bounds fail. The effectiveness and practical merits of the proposed framework are demonstrated through experiments on both simulated and real datasets. Lastly, we discuss generalizations of our approach to higher-order data.

stat.ME↗
arXiv2024

Enhancing convolutional neural network generalizability via low-rank weight approximation

Noise is ubiquitous during image acquisition. Sufficient denoising is often an important first step for image processing. In recent decades, deep neural networks (DNNs) have been widely used for image denoising. Most DNN-based image denoising methods require a large-scale dataset or focus on supervised settings, in which single/pairs of clean images or a set of noisy images are required. This poses a significant burden on the image acquisition process. Moreover, denoisers trained on datasets of limited scale may incur over-fitting. To mitigate these issues, we introduce a new self-supervised framework for image denoising based on the Tucker low-rank tensor approximation. With the proposed design, we are able to characterize our denoiser with fewer parameters and train it based on a single image, which considerably improves the model's generalizability and reduces the cost of data acquisition. Extensive experiments on both synthetic and real-world noisy images have been conducted. Empirical results show that our proposed method outperforms existing non-learning-based methods (e.g., low-pass filter, non-local mean), single-image unsupervised denoisers (e.g., DIP, NN+BM3D) evaluated on both in-sample and out-sample datasets. The proposed method even achieves comparable performances with some supervised methods (e.g., DnCNN).

cs.CV↗
arXiv2024

Soft Phenotyping for Sepsis via EHR Time-aware Soft Clustering

Objective: Sepsis is one of the most serious hospital conditions associated with high mortality. Sepsis is the result of a dysregulated immune response to infection that can lead to multiple organ dysfunction and death. Due to the wide variability in the causes of sepsis, clinical presentation, and the recovery trajectories, identifying sepsis sub-phenotypes is crucial to advance our understanding of sepsis characterization, to choose targeted treatments and optimal timing of interventions, and to improve prognostication. Prior studies have described different sub-phenotypes of sepsis using organ-specific characteristics. These studies applied clustering algorithms to electronic health records (EHRs) to identify disease sub-phenotypes. However, prior approaches did not capture temporal information and made uncertain assumptions about the relationships among the sub-phenotypes for clustering procedures. Methods: We developed a time-aware soft clustering algorithm guided by clinical variables to identify sepsis sub-phenotypes using data available in the EHR. Results: We identified six novel sepsis hybrid sub-phenotypes and evaluated them for medical plausibility. In addition, we built an early-warning sepsis prediction model using logistic regression. Conclusion: Our results suggest that these novel sepsis hybrid sub-phenotypes are promising to provide more accurate information on sepsis-related organ dysfunction and sepsis recovery trajectories which can be important to inform management decisions and sepsis prognosis.

stat.AP↗
arXiv2024

One-dimensional Tensor Network Recovery

We study the recovery of the underlying graphs or permutations for tensors in the tensor ring or tensor train format. Our proposed algorithms compare the matricization ranks after down-sampling, whose complexity is $O(d\log d)$ for $d$-th order tensors. We prove that our algorithms can almost surely recover the correct graph or permutation when tensor entries can be observed without noise. We further establish the robustness of our algorithms against observational noise. The theoretical results are validated by numerical experiments.

math.NA↗
arXiv2024

Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay

We study the tensor-on-tensor regression, where the goal is to connect tensor responses to tensor covariates with a low Tucker rank parameter tensor/matrix without the prior knowledge of its intrinsic rank. We propose the Riemannian gradient descent (RGD) and Riemannian Gauss-Newton (RGN) methods and cope with the challenge of unknown rank by studying the effect of rank over-parameterization. We provide the first convergence guarantee for the general tensor-on-tensor regression by showing that RGD and RGN respectively converge linearly and quadratically to a statistically optimal estimate in both rank correctly-parameterized and over-parameterized settings. Our theory reveals an intriguing phenomenon: Riemannian optimization methods naturally adapt to over-parameterization without modifications to their implementation. We also prove the statistical-computational gap in scalar-on-tensor regression by a direct low-degree polynomial argument. Our theory demonstrates a "blessing of statistical-computational gap" phenomenon: in a wide range of scenarios in tensor-on-tensor regression for tensors of order three or higher, the computationally required sample size matches what is needed by moderate rank over-parameterization when considering computationally feasible estimators, while there are no such benefits in the matrix settings. This shows moderate rank over-parameterization is essentially "cost-free" in terms of sample size in tensor-on-tensor regression of order three or higher. Finally, we conduct simulation studies to show the advantages of our proposed methods and to corroborate our theoretical findings.

math.ST↗
arXiv2023

Computational and Statistical Thresholds in Multi-layer Stochastic Block Models

We study the problem of community recovery and detection in multi-layer stochastic block models, focusing on the critical network density threshold for consistent community structure inference. Using a prototypical two-block model, we reveal a computational barrier for such multi-layer stochastic block models that does not exist for its single-layer counterpart: When there are no computational constraints, the density threshold depends linearly on the number of layers. However, when restricted to polynomial-time algorithms, the density threshold scales with the square root of the number of layers, assuming correctness of a low-degree polynomial hardness conjecture. Our results provide a nearly complete picture of the optimal inference in multiple-layer stochastic block models and partially settle the open question in Lei and Lin (2022) regarding the optimality of the bias-adjusted spectral method.

math.ST↗
arXiv2023

Guaranteed Functional Tensor Singular Value Decomposition

This paper introduces the functional tensor singular value decomposition (FTSVD), a novel dimension reduction framework for tensors with one functional mode and several tabular modes. The problem is motivated by high-order longitudinal data analysis. Our model assumes the observed data to be a random realization of an approximate CP low-rank functional tensor measured on a discrete time grid. Incorporating tensor algebra and the theory of Reproducing Kernel Hilbert Space (RKHS), we propose a novel RKHS-based constrained power iteration with spectral initialization. Our method can successfully estimate both singular vectors and functions of the low-rank structure in the observed data. With mild assumptions, we establish the non-asymptotic contractive error bounds for the proposed algorithm. The superiority of the proposed framework is demonstrated via extensive experiments on both simulated and real data.

stat.ME↗
arXiv2023

Cocaine Use Prediction with Tensor-based Machine Learning on Multimodal MRI Connectome Data

This paper considers the use of machine learning algorithms for predicting cocaine use based on magnetic resonance imaging (MRI) connectomic data. The study utilized functional MRI (fMRI) and diffusion MRI (dMRI) data collected from 275 individuals, which was then parcellated into 246 regions of interest (ROIs) using the Brainnetome atlas. After data preprocessing, the datasets were transformed into tensor form. We developed a tensor-based unsupervised machine learning algorithm to reduce the size of the data tensor from $275$ (individuals) $\times 2$ (fMRI and dMRI) $\times 246$ (ROIs) $\times 246$ (ROIs) to $275$ (individuals) $\times 2$ (fMRI and dMRI) $\times 6$ (clusters) $\times 6$ (clusters). This was achieved by applying the high-order Lloyd algorithm to group the ROI data into 6 clusters. Features were extracted from the reduced tensor and combined with demographic features (age, gender, race, and HIV status). The resulting dataset was used to train a Catboost model using subsampling and nested cross-validation techniques, which achieved a prediction accuracy of 0.857 for identifying cocaine users. The model was also compared with other models, and the feature importance of the model was presented. Overall, this study highlights the potential for using tensor-based machine learning algorithms to predict cocaine use based on MRI connectomic data and presents a promising approach for identifying individuals at risk of substance abuse.

stat.AP↗
arXiv2023

Tensor Clustering with Planted Structures: Statistical Optimality and Computational Limits

This paper studies the statistical and computational limits of high-order clustering with planted structures. We focus on two clustering models, constant high-order clustering (CHC) and rank-one higher-order clustering (ROHC), and study the methods and theory for testing whether a cluster exists (detection) and identifying the support of cluster (recovery). Specifically, we identify the sharp boundaries of signal-to-noise ratio for which CHC and ROHC detection/recovery are statistically possible. We also develop the tight computational thresholds: when the signal-to-noise ratio is below these thresholds, we prove that polynomial-time algorithms cannot solve these problems under the computational hardness conjectures of hypergraphic planted clique (HPC) detection and hypergraphic planted dense subgraph (HPDS) recovery. We also propose polynomial-time tensor algorithms that achieve reliable detection and recovery when the signal-to-noise ratio is above these thresholds. Both sparsity and tensor structures yield the computational barriers in high-order tensor clustering. The interplay between them results in significant differences between high-order tensor clustering and matrix clustering in literature in aspects of statistical and computational phase transition diagrams, algorithmic approaches, hardness conjecture, and proof techniques. To our best knowledge, we are the first to give a thorough characterization of the statistical and computational trade-off for such a double computational-barrier problem. Finally, we provide evidence for the computational hardness conjectures of HPC detection (via low-degree polynomial and Metropolis methods) and HPDS recovery (via low-degree polynomial method).

math.ST↗
arXiv2023

Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence

In this paper, we consider the estimation of a low Tucker rank tensor from a number of noisy linear measurements. The general problem covers many specific examples arising from applications, including tensor regression, tensor completion, and tensor PCA/SVD. We consider an efficient Riemannian Gauss-Newton (RGN) method for low Tucker rank tensor estimation. Different from the generic (super)linear convergence guarantee of RGN in the literature, we prove the first local quadratic convergence guarantee of RGN for low-rank tensor estimation in the noisy setting under some regularity conditions and provide the corresponding estimation error upper bounds. A deterministic estimation error lower bound, which matches the upper bound, is provided that demonstrates the statistical optimality of RGN. The merit of RGN is illustrated through two machine learning applications: tensor regression and tensor SVD. Finally, we provide the simulation results to corroborate our theoretical findings.

stat.ML↗
arXiv2023

Sampling is as easy as learning the score: theory for diffusion models with minimal data assumptions

We provide theoretical convergence guarantees for score-based generative models (SGMs) such as denoising diffusion probabilistic models (DDPMs), which constitute the backbone of large-scale real-world generative models such as DALL$\cdot$E 2. Our main result is that, assuming accurate score estimates, such SGMs can efficiently sample from essentially any realistic data distribution. In contrast to prior works, our results (1) hold for an $L^2$-accurate score estimate (rather than $L^\infty$-accurate); (2) do not require restrictive functional inequality conditions that preclude substantial non-log-concavity; (3) scale polynomially in all relevant problem parameters; and (4) match state-of-the-art complexity guarantees for discretization of the Langevin diffusion, provided that the score error is sufficiently small. We view this as strong theoretical justification for the empirical success of SGMs. We also examine SGMs based on the critically damped Langevin diffusion (CLD). Contrary to conventional wisdom, we provide evidence that the use of the CLD does not reduce the complexity of SGMs.

cs.LG↗
arXiv2023

On Geometric Connections of Embedded and Quotient Geometries in Riemannian Fixed-rank Matrix Optimization

In this paper, we propose a general procedure for establishing the geometric landscape connections of a Riemannian optimization problem under the embedded and quotient geometries. By applying the general procedure to the fixed-rank positive semidefinite (PSD) and general matrix optimization, we establish an exact Riemannian gradient connection under two geometries at every point on the manifold and sandwich inequalities between the spectra of Riemannian Hessians at Riemannian first-order stationary points (FOSPs). These results immediately imply an equivalence on the sets of Riemannian FOSPs, Riemannian second-order stationary points (SOSPs), and strict saddles of fixed-rank matrix optimization under the embedded and the quotient geometries. To the best of our knowledge, this is the first geometric landscape connection between the embedded and the quotient geometries for fixed-rank matrix optimization and it provides a concrete example of how these two geometries are connected in Riemannian optimization. In addition, the effects of the Riemannian metric and quotient structure on the landscape connection are discussed. We also observe an algorithmic connection between two geometries with some specific Riemannian metrics in fixed-rank matrix optimization: there is an equivalence between gradient flows under two geometries with shared spectra of Riemannian Hessians. A number of novel ideas and technical ingredients including a unified treatment for different Riemannian metrics, novel metrics for the Stiefel manifold, and new horizontal space representations under quotient geometries are developed to obtain our results. The results in this paper deepen our understanding of geometric and algorithmic connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature.

math.OC↗
arXiv2023

Phase transition for detecting a small community in a large network

How to detect a small community in a large network is an interesting problem, including clique detection as a special case, where a naive degree-based $χ^2$-test was shown to be powerful in the presence of an Erdős-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $χ^2$-test may be a modeling artifact, and it may disappear once we replace the Erdős-Renyi model by a broader network model. We show that the recent SgnQ test is more appropriate for such a setting. The test is optimal in detecting communities with sizes comparable to the whole network, but has never been studied for our setting, which is substantially different and more challenging. Using a degree-corrected block model (DCBM), we establish phase transitions of this testing problem concerning the size of the small community and the edge densities in small and large communities. When the size of the small community is larger than $\sqrt{n}$, the SgnQ test is optimal for it attains the computational lower bound (CLB), the information lower bound for methods allowing polynomial computation time. When the size of the small community is smaller than $\sqrt{n}$, we establish the parameter regime where the SgnQ test has full power and make some conjectures of the CLB. We also study the classical information lower bound (LB) and show that there is always a gap between the CLB and LB in our range of interest.

math.ST↗
arXiv2022

Recursive Importance Sketching for Rank Constrained Least Squares: Algorithms and High-order Convergence

In this paper, we propose {\it \underline{R}ecursive} {\it \underline{I}mportance} {\it \underline{S}ketching} algorithm for {\it \underline{R}ank} constrained least squares {\it \underline{O}ptimization} (RISRO). The key step of RISRO is recursive importance sketching, a new sketching framework based on deterministically designed recursive projections, which significantly differs from the randomized sketching in the literature \citep{mahoney2011randomized,woodruff2014sketching}. Several existing algorithms in the literature can be reinterpreted under this new sketching framework and RISRO offers clear advantages over them. RISRO is easy to implement and computationally efficient, where the core procedure in each iteration is to solve a dimension-reduced least squares problem. We establish the local quadratic-linear and quadratic rate of convergence for RISRO under some mild conditions. We also discover a deep connection of RISRO to the Riemannian Gauss-Newton algorithm on fixed rank matrices. The effectiveness of RISRO is demonstrated in two applications in machine learning and statistics: low-rank matrix trace regression and phase retrieval. Simulation studies demonstrate the superior numerical performance of RISRO.

math.OC↗
arXiv2022

Exact Clustering in Tensor Block Model: Statistical Optimality and Computational Limit

High-order clustering aims to identify heterogeneous substructures in multiway datasets that arise commonly in neuroimaging, genomics, social network studies, etc. The non-convex and discontinuous nature of this problem pose significant challenges in both statistics and computation. In this paper, we propose a tensor block model and the computationally efficient methods, \emph{high-order Lloyd algorithm} (HLloyd), and high-order spectral clustering (HSC), for high-order clustering. The convergence guarantees and statistical optimality are established for the proposed procedure under a mild sub-Gaussian noise assumption. Under the Gaussian tensor block model, we completely characterize the statistical-computational trade-off for achieving high-order exact clustering based on three different signal-to-noise ratio regimes. The analysis relies on new techniques of high-order spectral perturbation analysis and a ``singular-value-gap-free'' error bound in tensor estimation, which are substantially different from the matrix spectral analyses in the literature. Finally, we show the merits of the proposed procedures via extensive experiments on both synthetic and real datasets.

stat.ME↗
arXiv2022

Core Shrinkage Covariance Estimation for Matrix-variate Data

A separable covariance model for a random matrix provides a parsimonious description of the covariances among the rows and among the columns of the matrix, and permits likelihood-based inference with a very small sample size. However, in many applications the assumption of exact separability is unlikely to be met, and data analysis with a separable model may overlook or misrepresent important dependence patterns in the data. In this article, we propose a compromise between separable and unstructured covariance estimation. We show how the set of covariance matrices may be uniquely parametrized in terms of the set of separable covariance matrices and a complementary set of "core" covariance matrices, where the core of a separable covariance matrix is the identity matrix. This parametrization defines a Kronecker-core decomposition of a covariance matrix. By shrinking the core of the sample covariance matrix with an empirical Bayes procedure, we obtain an estimator that can adapt to the degree of separability of the population covariance matrix.

stat.ME↗
arXiv2022

Sparse Group Lasso: Optimal Sample Complexity, Convergence Rate, and Statistical Inference

We study sparse group Lasso for high-dimensional double sparse linear regression, where the parameter of interest is simultaneously element-wise and group-wise sparse. This problem is an important instance of the simultaneously structured model -- an actively studied topic in statistics and machine learning. In the noiseless case, matching upper and lower bounds on sample complexity are established for the exact recovery of sparse vectors and for stable estimation of approximately sparse vectors, respectively. In the noisy case, upper and matching minimax lower bounds for estimation error are obtained. We also consider the debiased sparse group Lasso and investigate its asymptotic property for the purpose of statistical inference. Finally, numerical studies are provided to support the theoretical results.

math.ST↗