arXiv subjects

Yue M. Lu

Publications and source records attributed to Yue M. Lu.

At least 55 records · Page 3Linked to original sources

arXiv2020

The role of regularization in classification of high-dimensional noisy Gaussian mixture

We consider a high-dimensional mixture of two Gaussians in the noisy regime where even an oracle knowing the centers of the clusters misclassifies a small but finite fraction of the points. We provide a rigorous analysis of the generalization error of regularized convex classifiers, including ridge, hinge and logistic regression, in the high-dimensional limit where the number $n$ of samples and their dimension $d$ go to infinity while their ratio is fixed to $α= n/d$. We discuss surprising effects of the regularization that in some cases allows to reach the Bayes-optimal performances. We also illustrate the interpolation peak at low regularization, and analyze the role of the respective sizes of the two clusters.

stat.ML↗

arXiv2019

A Solvable High-Dimensional Model of GAN

We present a theoretical analysis of the training process for a single-layer GAN fed by high-dimensional input data. The training dynamics of the proposed model at both microscopic and macroscopic scales can be exactly analyzed in the high-dimensional limit. In particular, we prove that the macroscopic quantities measuring the quality of the training process converge to a deterministic process characterized by an ordinary differential equation (ODE), whereas the microscopic states containing all the detailed weights remain stochastic, whose dynamics can be described by a stochastic differential equation (SDE). This analysis provides a new perspective different from recent analyses in the limit of small learning rate, where the microscopic state is always considered deterministic, and the contribution of noise is ignored. From our analysis, we show that the level of the background noise is essential to the convergence of the training process: setting the noise level too strong leads to failure of feature recovery, whereas setting the noise too weak causes oscillation. Although this work focuses on a simple copy model of GAN, we believe the analysis methods and insights developed here would prove useful in the theoretical understanding of other variants of GANs with more advanced training algorithms.

cs.LG↗

arXiv2019

Nonconvex Optimization Meets Low-Rank Matrix Factorization: An Overview

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

cs.LG↗

arXiv2019

Phase Transitions of Spectral Initialization for High-Dimensional Nonconvex Estimation

We study a spectral initialization method that serves a key role in recent work on estimating signals in nonconvex settings. Previous analysis of this method focuses on the phase retrieval problem and provides only performance bounds. In this paper, we consider arbitrary generalized linear sensing models and present a precise asymptotic characterization of the performance of the method in the high-dimensional limit. Our analysis also reveals a phase transition phenomenon that depends on the ratio between the number of samples and the signal dimension. When the ratio is below a minimum threshold, the estimates given by the spectral method are no better than random guesses drawn from a uniform distribution on the hypersphere, thus carrying no information; above a maximum threshold, the estimates become increasingly aligned with the target signal. The computational complexity of the method, as measured by the spectral gap, is also markedly different in the two phases. Worked examples and numerical results are provided to illustrate and verify the analytical predictions. In particular, simulations show that our asymptotic formulas provide accurate predictions for the actual performance of the spectral method even at moderate signal dimensions.

cs.IT↗

arXiv2019

Generalized Approximate Survey Propagation for High-Dimensional Estimation

In Generalized Linear Estimation (GLE) problems, we seek to estimate a signal that is observed through a linear transform followed by a component-wise, possibly nonlinear and noisy, channel. In the Bayesian optimal setting, Generalized Approximate Message Passing (GAMP) is known to achieve optimal performance for GLE. However, its performance can significantly degrade whenever there is a mismatch between the assumed and the true generative model, a situation frequently encountered in practice. In this paper, we propose a new algorithm, named Generalized Approximate Survey Propagation (GASP), for solving GLE in the presence of prior or model mis-specifications. As a prototypical example, we consider the phase retrieval problem, where we show that GASP outperforms the corresponding GAMP, reducing the reconstruction threshold and, for certain choices of its parameters, approaching Bayesian optimal performance. Furthermore, we present a set of State Evolution equations that exactly characterize the dynamics of GASP in the high-dimensional limit.

cond-mat.dis-nn↗

arXiv2019

MMSE Approximation For Sparse Coding Algorithms Using Stochastic Resonance

Sparse coding refers to the pursuit of the sparsest representation of a signal in a typically overcomplete dictionary. From a Bayesian perspective, sparse coding provides a Maximum a Posteriori (MAP) estimate of the unknown vector under a sparse prior. In this work, we suggest enhancing the performance of sparse coding algorithms by a deliberate and controlled contamination of the input with random noise, a phenomenon known as stochastic resonance. The proposed method adds controlled noise to the input and estimates a sparse representation from the perturbed signal. A set of such solutions is then obtained by projecting the original input signal onto the recovered set of supports. We present two variants of the described method, which differ in their final step. The first is a provably convergent approximation to the Minimum Mean Square Error (MMSE) estimator, relying on the generative model and applying a weighted average over the recovered solutions. The second is a relaxed variant of the former that simply applies an empirical mean. We show that both methods provide a computationally efficient approximation to the MMSE estimator, which is typically intractable to compute. We demonstrate our findings empirically and provide a theoretical analysis of our method under several different cases.

eess.SP↗

arXiv2018

Optimal Spectral Initialization for Signal Recovery with Applications to Phase Retrieval

We present the optimal design of a spectral method widely used to initialize nonconvex optimization algorithms for solving phase retrieval and other signal recovery problems. Our work leverages recent results that provide an exact characterization of the performance of the spectral method in the high-dimensional limit. This characterization allows us to map the task of optimal design to a constrained optimization problem in a weighted $L^2$ function space. The latter has a closed-form solution. Interestingly, under a mild technical condition, our results show that there exists a fixed design that is uniformly optimal over all sampling ratios. Numerical simulations demonstrate the performance improvement brought by the proposed optimal design over existing constructions in the literature. In a recent work, Mondelli and Montanari have shown the existence of a weak reconstruction threshold below which the spectral method cannot provide useful estimates. Our results serve to complement that work by deriving the fundamental limit of the spectral method beyond the aforementioned threshold.

cs.IT↗

arXiv2018

Subspace Estimation from Incomplete Observations: A High-Dimensional Analysis

We present a high-dimensional analysis of three popular algorithms, namely, Oja's method, GROUSE and PETRELS, for subspace estimation from streaming and highly incomplete observations. We show that, with proper time scaling, the time-varying principal angles between the true subspace and its estimates given by the algorithms converge weakly to deterministic processes when the ambient dimension $n$ tends to infinity. Moreover, the limiting processes can be exactly characterized as the unique solutions of certain ordinary differential equations (ODEs). A finite sample bound is also given, showing that the rate of convergence towards such limits is $\mathcal{O}(1/\sqrt{n})$. In addition to providing asymptotically exact predictions of the dynamic performance of the algorithms, our high-dimensional analysis yields several insights, including an asymptotic equivalence between Oja's method and GROUSE, and a precise scaling relationship linking the amount of missing data to the signal-to-noise ratio. By analyzing the solutions of the limiting ODEs, we also establish phase transition phenomena associated with the steady-state performance of these techniques.

cs.LG↗

arXiv2018

Super Resolution Phase Retrieval for Sparse Signals

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts to recover the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. Solving the phase retrieval problem is equivalent to recovering a signal from its auto-correlation function. In this paper, we assume the original signal to be sparse; this is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. We propose an algorithm that resolves the phase retrieval problem in three stages: i) we leverage the finite rate of innovation sampling theory to super-resolve the auto-correlation function from a limited number of samples, ii) we design a greedy algorithm that identifies the locations of a sparse solution given the super-resolved auto-correlation function, iii) we recover the amplitudes of the atoms given their locations and the measured auto-correlation function. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. Along with the algorithm, we derive its performance bound with a theoretical analysis and propose a set of enhancements to improve its computational complexity and noise resilience. Finally, we demonstrate the benefits of the proposed method via a comparison against Charge Flipping, a notable algorithm in crystallography.

cs.IT↗

arXiv2018

Streaming PCA and Subspace Tracking: The Missing Data Case

For many modern applications in science and engineering, data are collected in a streaming fashion carrying time-varying information, and practitioners need to process them with a limited amount of memory and computational resources in a timely manner for decision making. This often is coupled with the missing data problem, such that only a small fraction of data attributes are observed. These complications impose significant, and unconventional, constraints on the problem of streaming Principal Component Analysis (PCA) and subspace tracking, which is an essential building block for many inference tasks in signal processing and machine learning. This survey article reviews a variety of classical and recent algorithms for solving this problem with low computational and memory complexities, particularly those applicable in the big data regime with missing data. We illustrate that streaming PCA and subspace tracking algorithms can be understood through algebraic and geometric perspectives, and they need to be adjusted carefully to handle missing data. Both asymptotic and non-asymptotic convergence guarantees are reviewed. Finally, we benchmark the performance of several competitive algorithms in the presence of missing data for both well-conditioned and ill-conditioned systems.

stat.ML↗

arXiv2018

Phase Retrieval via Polytope Optimization: Geometry, Phase Transitions, and New Algorithms

We study algorithms for solving quadratic systems of equations based on optimization methods over polytopes. Our work is inspired by a recently proposed convex formulation of the phase retrieval problem, which estimates the unknown signal by solving a simple linear program over a polytope constructed from the measurements. We present a sharp characterization of the high-dimensional geometry of the aforementioned polytope under Gaussian measurements. This characterization allows us to derive asymptotically exact performance guarantees for PhaseMax, which also reveal a phase transition phenomenon with respect to its sample complexity. Moreover, the geometric insights gained from our analysis lead to a new nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm, which we call PhaseLamp. We show that this new algorithm has superior recovery performance over the original PhaseMax method. Finally, as yet another variation on the theme of performing phase retrieval via polytope optimization, we propose a weighted version of PhaseLamp and demonstrate, through numerical simulations, that it outperforms several state-of-the-art algorithms under both generic Gaussian measurements as well as more realistic Fourier-type measurements that arise in phase retrieval applications.

cs.IT↗

arXiv2017

Scaling Limit: Exact and Tractable Analysis of Online Learning Algorithms with Applications to Regularized Regression and PCA

We present a framework for analyzing the exact dynamics of a class of online learning algorithms in the high-dimensional scaling limit. Our results are applied to two concrete examples: online regularized linear regression and principal component analysis. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measures of the target feature vector and its estimates provided by the algorithms will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE can be efficiently obtained. These solutions lead to precise predictions of the performance of the algorithms, as many practical performance metrics are linear functionals of the joint empirical measures. In addition to characterizing the dynamic performance of online learning algorithms, our asymptotic analysis also provides useful insights. In particular, in the high-dimensional limit, and due to exchangeability, the original coupled dynamics associated with the algorithms will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight for nonconvex optimization problems may prove an interesting line of future research.

cs.LG↗

arXiv2017

The Scaling Limit of High-Dimensional Online Independent Component Analysis

We analyze the dynamics of an online algorithm for independent component analysis in the high-dimensional scaling limit. As the ambient dimension tends to infinity, and with proper time scaling, we show that the time-varying joint empirical measure of the target feature vector and the estimates provided by the algorithm will converge weakly to a deterministic measured-valued process that can be characterized as the unique solution of a nonlinear PDE. Numerical solutions of this PDE, which involves two spatial variables and one time variable, can be efficiently obtained. These solutions provide detailed information about the performance of the ICA algorithm, as many practical performance metrics are functionals of the joint empirical measures. Numerical simulations show that our asymptotic analysis is accurate even for moderate dimensions. In addition to providing a tool for understanding the performance of the algorithm, our PDE analysis also provides useful insight. In particular, in the high-dimensional limit, the original coupled dynamics associated with the algorithm will be asymptotically "decoupled", with each coordinate independently solving a 1-D effective minimization problem via stochastic gradient descent. Exploiting this insight to design new algorithms for achieving optimal trade-offs between computational and statistical efficiency may prove an interesting line of future research.

cs.LG↗

arXiv2017

Phase Retrieval via Linear Programming: Fundamental Limits and Algorithmic Improvements

A recently proposed convex formulation of the phase retrieval problem estimates the unknown signal by solving a simple linear program. This new scheme, known as PhaseMax, is computationally efficient compared to standard convex relaxation methods based on lifting techniques. In this paper, we present an exact performance analysis of PhaseMax under Gaussian measurements in the large system limit. In contrast to previously known performance bounds in the literature, our results are asymptotically exact and they also reveal a sharp phase transition phenomenon. Furthermore, the geometrical insights gained from our analysis led us to a novel nonconvex formulation of the phase retrieval problem and an accompanying iterative algorithm based on successive linearization and maximization over a polytope. This new algorithm, which we call PhaseLamp, has provably superior recovery performance over the original PhaseMax method.

cs.IT↗

arXiv2017

Fundamental Limits of PhaseMax for Phase Retrieval: A Replica Analysis

We consider a recently proposed convex formulation, known as the PhaseMax method, for solving the phase retrieval problem. Using the replica method from statistical mechanics, we analyze the performance of PhaseMax in the high-dimensional limit. Our analysis predicts the \emph{exact} asymptotic performance of PhaseMax. In particular, we show that a sharp phase transition phenomenon takes place, with a simple analytical formula characterizing the phase transition boundary. This result shows that the oversampling ratio required by existing performance bounds in the literature can be significantly reduced. Numerical results confirm the validity of our replica analysis, showing that the theoretical predictions are in excellent agreement with the actual performance of the algorithm, even for moderate signal dimensions.

cs.IT↗

arXiv2017

Sparse Representation in Fourier and Local Bases Using ProSparse: A Probabilistic Analysis

Finding the sparse representation of a signal in an overcomplete dictionary has attracted a lot of attention over the past years. This paper studies ProSparse, a new polynomial complexity algorithm that solves the sparse representation problem when the underlying dictionary is the union of a Vandermonde matrix and a banded matrix. Unlike our previous work which establishes deterministic (worst-case) sparsity bounds for ProSparse to succeed, this paper presents a probabilistic average-case analysis of the algorithm. Based on a generating-function approach, closed-form expressions for the exact success probabilities of ProSparse are given. The success probabilities are also analyzed in the high-dimensional regime. This asymptotic analysis characterizes a sharp phase transition phenomenon regarding the performance of the algorithm.

cs.IT↗

arXiv2017

Multiprocessor Approximate Message Passing with Column-Wise Partitioning

Solving a large-scale regularized linear inverse problem using multiple processors is important in various real-world applications due to the limitations of individual processors and constraints on data sharing policies. This paper focuses on the setting where the matrix is partitioned column-wise. We extend the algorithmic framework and the theoretical analysis of approximate message passing (AMP), an iterative algorithm for solving linear inverse problems, whose asymptotic dynamics are characterized by state evolution (SE). In particular, we show that column-wise multiprocessor AMP (C-MP-AMP) obeys an SE under the same assumptions when the SE for AMP holds. The SE results imply that (i) the SE of C-MP-AMP converges to a state that is no worse than that of AMP and (ii) the asymptotic dynamics of C-MP-AMP and AMP can be identical. Moreover, for a setting that is not covered by SE, numerical results show that damping can improve the convergence performance of C-MP-AMP.

cs.IT↗

arXiv2016

Understanding Symmetric Smoothing Filters: A Gaussian Mixture Model Perspective

Many patch-based image denoising algorithms can be formulated as applying a smoothing filter to the noisy image. Expressed as matrices, the smoothing filters must be row normalized so that each row sums to unity. Surprisingly, if we apply a column normalization before the row normalization, the performance of the smoothing filter can often be significantly improved. Prior works showed that such performance gain is related to the Sinkhorn-Knopp balancing algorithm, an iterative procedure that symmetrizes a row-stochastic matrix to a doubly-stochastic matrix. However, a complete understanding of the performance gain phenomenon is still lacking. In this paper, we study the performance gain phenomenon from a statistical learning perspective. We show that Sinkhorn-Knopp is equivalent to an Expectation-Maximization (EM) algorithm of learning a Gaussian mixture model of the image patches. By establishing the correspondence between the steps of Sinkhorn-Knopp and the EM algorithm, we provide a geometrical interpretation of the symmetrization process. This observation allows us to develop a new denoising algorithm called Gaussian mixture model symmetric smoothing filter (GSF). GSF is an extension of the Sinkhorn-Knopp and is a generalization of the original smoothing filters. Despite its simple formulation, GSF outperforms many existing smoothing filters and has a similar performance compared to several state-of-the-art denoising algorithms.

cs.CV↗