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Muhang Tian, Bernie Chen, Allan Guo, Shiyi Jiang, Anru R. Zhang

Electronic Health Records (EHRs) are rich sources of patient-level data, including laboratory tests, medications, and diagnoses, offering valuable resources for medical data analysis. However, concerns about privacy often restrict access to EHRs, hindering downstream analysis. Researchers have explored various methods for generating privacy-preserving EHR data. In this study, we introduce a new method for generating diverse and realistic synthetic EHR time series data using Denoising Diffusion Probabilistic Models (DDPM). We conducted experiments on six datasets, comparing our proposed method with seven existing methods. Our results demonstrate that our approach significantly outperforms all existing methods in terms of data utility while requiring less training effort. Our approach also enhances downstream medical data analysis by providing diverse and realistic synthetic EHR data.

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Runshi Tang, Ming Yuan, Anru R. Zhang

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.

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Jiashun Jin, Zheng Tracy Ke, Paxton Turner, Anru R. Zhang

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 $\chi^2$-test was shown to be powerful in the presence of an Erd\H{o}s-Renyi background. Using Sinkhorn's theorem, we show that the signal captured by the $\chi^2$-test may be a modeling artifact, and it may disappear once we replace the Erd\H{o}s-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.

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Sitan Chen, Sinho Chewi, Jerry Li, Yuanzhi Li, Adil Salim, Anru R. Zhang

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.

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Chenyin Gao, Shu Yang, Anru R. Zhang

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 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).

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Yuetian Luo, Anru R. Zhang

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 give the first rigorous evidence for the statistical-computational gap in scalar-on-tensor regression under the low-degree polynomials framework. 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.

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Sitan Chen, Jerry Li, Yuanzhi Li, Anru R. Zhang

We consider the problem of learning high dimensional polynomial transformations of Gaussians. Given samples of the form $p(x)$, where $x\sim N(0, \mathrm{Id}_r)$ is hidden and $p: \mathbb{R}^r \to \mathbb{R}^d$ is a function where every output coordinate is a low-degree polynomial, the goal is to learn the distribution over $p(x)$. This problem is natural in its own right, but is also an important special case of learning deep generative models, namely pushforwards of Gaussians under two-layer neural networks with polynomial activations. Understanding the learnability of such generative models is crucial to understanding why they perform so well in practice. Our first main result is a polynomial-time algorithm for learning quadratic transformations of Gaussians in a smoothed setting. Our second main result is a polynomial-time algorithm for learning constant-degree polynomial transformations of Gaussian in a smoothed setting, when the rank of the associated tensors is small. In fact our results extend to any rotation-invariant input distribution, not just Gaussian. These are the first end-to-end guarantees for learning a pushforward under a neural network with more than one layer. Along the way, we also give the first polynomial-time algorithms with provable guarantees for tensor ring decomposition, a popular generalization of tensor decomposition that is used in practice to implicitly store large tensors.

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Yuetian Luo, Xudong Li, Anru R. Zhang

In this paper, we propose a general procedure for establishing the 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 on 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 for fixed-rank matrix optimization under two geometries with some specific Riemannian metrics. A number of novel ideas and technical ingredients including a unified treatment for different Riemannian metrics and new horizontal space representations under quotient geometries are developed to obtain our results. The results in this paper deepen our understanding of geometric connections of Riemannian optimization under different Riemannian geometries and provide a few new theoretical insights to unanswered questions in the literature.

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Yuetian Luo, Xudong Li, Anru R. Zhang

In this paper, we consider the geometric landscape connection of the widely studied manifold and factorization formulations in low-rank positive semidefinite (PSD) and general matrix optimization. We establish an equivalence on the set of first-order stationary points (FOSPs) and second-order stationary points (SOSPs) between the manifold and the factorization formulations. We further give a sandwich inequality on the spectrum of Riemannian and Euclidean Hessians at FOSPs, which can be used to transfer more geometric properties from one formulation to another. Similarities and differences on the landscape connection under the PSD case and the general case are discussed. To the best of our knowledge, this is the first geometric landscape connection between the manifold and the factorization formulations for handling rank constraints. In the general low-rank matrix optimization, the landscape connection of two factorization formulations (unregularized and regularized ones) is also provided. By applying these geometric landscape connections, we are able to solve unanswered questions in literature and establish stronger results in the applications on geometric analysis of phase retrieval, well-conditioned low-rank matrix optimization, and the role of regularization in factorization arising from machine learning and signal processing.

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