arXiv subjects

Cun-Hui Zhang

Publications and source records attributed to Cun-Hui Zhang.

At least 55 records · Page 3Linked to original sources

arXiv2014

Compressed Sensing with Very Sparse Gaussian Random Projections

We study the use of very sparse random projections for compressed sensing (sparse signal recovery) when the signal entries can be either positive or negative. In our setting, the entries of a Gaussian design matrix are randomly sparsified so that only a very small fraction of the entries are nonzero. Our proposed decoding algorithm is simple and efficient in that the major cost is one linear scan of the coordinates. We have developed two estimators: (i) the {\em tie estimator}, and (ii) the {\em absolute minimum estimator}. Using only the tie estimator, we are able to recover a $K$-sparse signal of length $N$ using $1.551 eK \log K/δ$ measurements (where $δ\leq 0.05$ is the confidence). Using only the absolute minimum estimator, we can detect the support of the signal using $eK\log N/δ$ measurements. For a particular coordinate, the absolute minimum estimator requires fewer measurements (i.e., with a constant $e$ instead of $1.551e$). Thus, the two estimators can be combined to form an even more practical decoding framework. Prior studies have shown that existing one-scan (or roughly one-scan) recovery algorithms using sparse matrices would require substantially more (e.g., one order of magnitude) measurements than L1 decoding by linear programming, when the nonzero entries of signals can be either negative or positive. In this paper, following a known experimental setup, we show that, at the same number of measurements, the recovery accuracies of our proposed method are (at least) similar to the standard L1 decoding.

stat.ME↗

arXiv2014

Optimality of Graphlet Screening in High Dimensional Variable Selection

Consider a linear regression model where the design matrix X has n rows and p columns. We assume (a) p is much large than n, (b) the coefficient vector beta is sparse in the sense that only a small fraction of its coordinates is nonzero, and (c) the Gram matrix G = X'X is sparse in the sense that each row has relatively few large coordinates (diagonals of G are normalized to 1). The sparsity in G naturally induces the sparsity of the so-called graph of strong dependence (GOSD). We find an interesting interplay between the signal sparsity and the graph sparsity, which ensures that in a broad context, the set of true signals decompose into many different small-size components of GOSD, where different components are disconnected. We propose Graphlet Screening (GS) as a new approach to variable selection, which is a two-stage Screen and Clean method. The key methodological innovation of GS is to use GOSD to guide both the screening and cleaning. Compared to m-variate brute-forth screening that has a computational cost of p^m, the GS only has a computational cost of p (up to some multi-log(p) factors) in screening. We measure the performance of any variable selection procedure by the minimax Hamming distance. We show that in a very broad class of situations, GS achieves the optimal rate of convergence in terms of the Hamming distance. Somewhat surprisingly, the well-known procedures subset selection and the lasso are rate non-optimal, even in very simple settings and even when their tuning parameters are ideally set.

math.ST↗

arXiv2014

On Tensor Completion via Nuclear Norm Minimization

Many problems can be formulated as recovering a low-rank tensor. Although an increasingly common task, tensor recovery remains a challenging problem because of the delicacy associated with the decomposition of higher order tensors. To overcome these difficulties, existing approaches often proceed by unfolding tensors into matrices and then apply techniques for matrix completion. We show here that such matricization fails to exploit the tensor structure and may lead to suboptimal procedure. More specifically, we investigate a convex optimization approach to tensor completion by directly minimizing a tensor nuclear norm and prove that this leads to an improved sample size requirement. To establish our results, we develop a series of algebraic and probabilistic techniques such as characterization of subdifferetial for tensor nuclear norm and concentration inequalities for tensor martingales, which may be of independent interests and could be useful in other tensor related problems.

stat.ML↗

arXiv2014

Multivariate Analysis of Nonparametric Estimates of Large Correlation Matrices

We study concentration in spectral norm of nonparametric estimates of correlation matrices. We work within the confine of a Gaussian copula model. Two nonparametric estimators of the correlation matrix, the sine transformations of the Kendall's tau and Spearman's rho correlation coefficient, are studied. Expected spectrum error bound is obtained for both the estimators. A general large deviation bound for the maximum spectral error of a collection of submatrices of a given dimension is also established. These results prove that when both the number of variables and sample size are large, the spectral error of the nonparametric estimators is of no greater order than that of the latent sample covariance matrix, at least when compared with some of the sharpest known error bounds for the later. As an application, we establish the minimax optimal convergence rate in the estimation of high-dimensional bandable correlation matrices via tapering off of these nonparametric estimators. An optimal convergence rate for sparse principal component analysis is also established as another example of possible applications of the main results.

math.ST↗

arXiv2013

Sparse Recovery with Very Sparse Compressed Counting

Compressed sensing (sparse signal recovery) often encounters nonnegative data (e.g., images). Recently we developed the methodology of using (dense) Compressed Counting for recovering nonnegative K-sparse signals. In this paper, we adopt very sparse Compressed Counting for nonnegative signal recovery. Our design matrix is sampled from a maximally-skewed p-stable distribution (0 0, it suffices to use M= K/(1-exp(-gK) log N measurements, so that all coordinates can be recovered in one scan of the coordinates. If g = 1 (i.e., dense design), then M = K log N. If g= 1/K or 2/K (i.e., very sparse design), then M = 1.58K log N or M = 1.16K log N. This means the design matrix can be indeed very sparse at only a minor inflation of the sample complexity. Interestingly, as p->1, the required number of measurements is essentially M = 2.7K log N, provided g= 1/K. It turns out that this result is a general worst-case bound.

stat.ME↗

arXiv2013

Adaptive Threshold Estimation by FDR

This paper addresses the following simple question about sparsity. For the estimation of an $n$-dimensional mean vector $\boldsymbolθ$ in the Gaussian sequence model, is it possible to find an adaptive optimal threshold estimator in a full range of sparsity levels where nonadaptive optimality can be achieved by threshold estimators? We provide an explicit affirmative answer as follows. Under the squared loss, adaptive minimaxity in strong and weak $\ell_p$ balls with $0\le p<2$ is achieved by a class of smooth threshold estimators with the threshold level of the Benjamini-Hochberg FDR rule or its a certain approximation, provided that the minimax risk is between $n^{-δ_n}$ and $δ_n n$ for some $δ_n\to 0$. For $p=0$, this means adaptive minimaxity in $\ell_0$ balls when $1\le \|\boldsymbolθ\|_0\ll n$. The class of smooth threshold estimators includes the soft and firm threshold estimators but not the hard threshold estimator. The adaptive minimaxity in such a wide range is a delicate problem since the same is not true for the FDR hard threshold estimator at certain threshold and nominal FDR levels. The above adaptive minimaxity of the FDR smooth-threshold estimator is established by proving a stronger notion of adaptive ratio optimality for the soft threshold estimator in the sense that the risk for the FDR threshold level is uniformly within an infinitesimal fraction of the risk for the optimal threshold level for each unknown vector, when the minimum risk of nonadaptive soft threshold estimator is between $n^{-δ_n}$ and $δ_n n$. It is an interesting consequence of this adaptive ratio optimality that the FDR smooth-threshold estimator outperforms the sample mean in the common mean model $θ_i=μ$ when $|μ|<n^{-1/2}$.

math.ST↗

arXiv2013

Semi-Penalized Inference with Direct False Discovery Rate Control in High-Dimensions

We propose a new method, semi-penalized inference with direct false discovery rate control (SPIDR), for variable selection and confidence interval construction in high-dimensional linear regression. SPIDR first uses a semi-penalized approach to constructing estimators of the regression coefficients. We show that the SPIDR estimator is ideal in the sense that it equals an ideal least squares estimator with high probability under a sparsity and other suitable conditions. Consequently, the SPIDR estimator is asymptotically normal. Based on this distributional result, SPIDR determines the selection rule by directly controlling false discovery rate. This provides an explicit assessment of the selection error. This also naturally leads to confidence intervals for the selected coefficients with a proper confidence statement. We conduct simulation studies to evaluate its finite sample performance and demonstrate its application on a breast cancer gene expression data set. Our simulation studies and data example suggest that SPIDR is a useful method for high-dimensional statistical inference in practice.

stat.ME↗

arXiv2013

Sparse Matrix Inversion with Scaled Lasso

We propose a new method of learning a sparse nonnegative-definite target matrix. Our primary example of the target matrix is the inverse of a population covariance or correlation matrix. The algorithm first estimates each column of the target matrix by the scaled Lasso and then adjusts the matrix estimator to be symmetric. The penalty level of the scaled Lasso for each column is completely determined by data via convex minimization, without using cross-validation. We prove that this scaled Lasso method guarantees the fastest proven rate of convergence in the spectrum norm under conditions of weaker form than those in the existing analyses of other $\ell_1$ regularized algorithms, and has faster guaranteed rate of convergence when the ratio of the $\ell_1$ and spectrum norms of the target inverse matrix diverges to infinity. A simulation study demonstrates the computational feasibility and superb performance of the proposed method. Our analysis also provides new performance bounds for the Lasso and scaled Lasso to guarantee higher concentration of the error at a smaller threshold level than previous analyses, and to allow the use of the union bound in column-by-column applications of the scaled Lasso without an adjustment of the penalty level. In addition, the least squares estimation after the scaled Lasso selection is considered and proven to guarantee performance bounds similar to that of the scaled Lasso.

math.ST↗

arXiv2013

Compressed Counting Meets Compressed Sensing

Compressed sensing (sparse signal recovery) has been a popular and important research topic in recent years. By observing that natural signals are often nonnegative, we propose a new framework for nonnegative signal recovery using Compressed Counting (CC). CC is a technique built on maximally-skewed p-stable random projections originally developed for data stream computations. Our recovery procedure is computationally very efficient in that it requires only one linear scan of the coordinates. Our analysis demonstrates that, when 0 0 and C=pi/2 when p=0.5. In particular, when p->0 the required number of measurements is essentially M=K\log N, where K is the number of nonzero coordinates of the signal.

stat.ME↗

arXiv2013

Oracle inequalities for the lasso in the Cox model

We study the absolute penalized maximum partial likelihood estimator in sparse, high-dimensional Cox proportional hazards regression models where the number of time-dependent covariates can be larger than the sample size. We establish oracle inequalities based on natural extensions of the compatibility and cone invertibility factors of the Hessian matrix at the true regression coefficients. Similar results based on an extension of the restricted eigenvalue can be also proved by our method. However, the presented oracle inequalities are sharper since the compatibility and cone invertibility factors are always greater than the corresponding restricted eigenvalue. In the Cox regression model, the Hessian matrix is based on time-dependent covariates in censored risk sets, so that the compatibility and cone invertibility factors, and the restricted eigenvalue as well, are random variables even when they are evaluated for the Hessian at the true regression coefficients. Under mild conditions, we prove that these quantities are bounded from below by positive constants for time-dependent covariates, including cases where the number of covariates is of greater order than the sample size. Consequently, the compatibility and cone invertibility factors can be treated as positive constants in our oracle inequalities.

math.ST↗

arXiv2013

Exact Sparse Recovery with L0 Projections

Many applications concern sparse signals, for example, detecting anomalies from the differences between consecutive images taken by surveillance cameras. This paper focuses on the problem of recovering a K-sparse signal x in N dimensions. In the mainstream framework of compressed sensing (CS), the vector x is recovered from M non-adaptive linear measurements y = xS, where S (of size N x M) is typically a Gaussian (or Gaussian-like) design matrix, through some optimization procedure such as linear programming (LP). In our proposed method, the design matrix S is generated from an $α$-stable distribution with $α\approx 0$. Our decoding algorithm mainly requires one linear scan of the coordinates, followed by a few iterations on a small number of coordinates which are "undetermined" in the previous iteration. Comparisons with two strong baselines, linear programming (LP) and orthogonal matching pursuit (OMP), demonstrate that our algorithm can be significantly faster in decoding speed and more accurate in recovery quality, for the task of exact spare recovery. Our procedure is robust against measurement noise. Even when there are no sufficient measurements, our algorithm can still reliably recover a significant portion of the nonzero coordinates. To provide the intuition for understanding our method, we also analyze the procedure by assuming an idealistic setting. Interestingly, when K=2, the "idealized" algorithm achieves exact recovery with merely 3 measurements, regardless of N. For general K, the required sample size of the "idealized" algorithm is about 5K.

stat.ML↗

arXiv2012

Calibrated Elastic Regularization in Matrix Completion

This paper concerns the problem of matrix completion, which is to estimate a matrix from observations in a small subset of indices. We propose a calibrated spectrum elastic net method with a sum of the nuclear and Frobenius penalties and develop an iterative algorithm to solve the convex minimization problem. The iterative algorithm alternates between imputing the missing entries in the incomplete matrix by the current guess and estimating the matrix by a scaled soft-thresholding singular value decomposition of the imputed matrix until the resulting matrix converges. A calibration step follows to correct the bias caused by the Frobenius penalty. Under proper coherence conditions and for suitable penalties levels, we prove that the proposed estimator achieves an error bound of nearly optimal order and in proportion to the noise level. This provides a unified analysis of the noisy and noiseless matrix completion problems. Simulation results are presented to compare our proposal with previous ones.

math.ST↗

arXiv2012

Confidence Intervals for Low-Dimensional Parameters in High-Dimensional Linear Models

The purpose of this paper is to propose methodologies for statistical inference of low-dimensional parameters with high-dimensional data. We focus on constructing confidence intervals for individual coefficients and linear combinations of several of them in a linear regression model, although our ideas are applicable in a much broad context. The theoretical results presented here provide sufficient conditions for the asymptotic normality of the proposed estimators along with a consistent estimator for their finite-dimensional covariance matrices. These sufficient conditions allow the number of variables to far exceed the sample size. The simulation results presented here demonstrate the accuracy of the coverage probability of the proposed confidence intervals, strongly supporting the theoretical results.

stat.ME↗

arXiv2012

Pick Up Sticks

We apply a classic result in Theoretical Computer Science to solve a sticky problem, and give a neat and slick quick proof of half of it

math.CO↗

arXiv2012

Correlated variables in regression: clustering and sparse estimation

We consider estimation in a high-dimensional linear model with strongly correlated variables. We propose to cluster the variables first and do subsequent sparse estimation such as the Lasso for cluster-representatives or the group Lasso based on the structure from the clusters. Regarding the first step, we present a novel and bottom-up agglomerative clustering algorithm based on canonical correlations, and we show that it finds an optimal solution and is statistically consistent. We also present some theoretical arguments that canonical correlation based clustering leads to a better-posed compatibility constant for the design matrix which ensures identifiability and an oracle inequality for the group Lasso. Furthermore, we discuss circumstances where cluster-representatives and using the Lasso as subsequent estimator leads to improved results for prediction and detection of variables. We complement the theoretical analysis with various empirical results.

stat.ME↗

arXiv2012

One Permutation Hashing for Efficient Search and Learning

Recently, the method of b-bit minwise hashing has been applied to large-scale linear learning and sublinear time near-neighbor search. The major drawback of minwise hashing is the expensive preprocessing cost, as the method requires applying (e.g.,) k=200 to 500 permutations on the data. The testing time can also be expensive if a new data point (e.g., a new document or image) has not been processed, which might be a significant issue in user-facing applications. We develop a very simple solution based on one permutation hashing. Conceptually, given a massive binary data matrix, we permute the columns only once and divide the permuted columns evenly into k bins; and we simply store, for each data vector, the smallest nonzero location in each bin. The interesting probability analysis (which is validated by experiments) reveals that our one permutation scheme should perform very similarly to the original (k-permutation) minwise hashing. In fact, the one permutation scheme can be even slightly more accurate, due to the "sample-without-replacement" effect. Our experiments with training linear SVM and logistic regression on the webspam dataset demonstrate that this one permutation hashing scheme can achieve the same (or even slightly better) accuracies compared to the original k-permutation scheme. To test the robustness of our method, we also experiment with the small news20 dataset which is very sparse and has merely on average 500 nonzeros in each data vector. Interestingly, our one permutation scheme noticeably outperforms the k-permutation scheme when k is not too small on the news20 dataset. In summary, our method can achieve at least the same accuracy as the original k-permutation scheme, at merely 1/k of the original preprocessing cost.

cs.LG↗

arXiv2012

Scaled Sparse Linear Regression

Scaled sparse linear regression jointly estimates the regression coefficients and noise level in a linear model. It chooses an equilibrium with a sparse regression method by iteratively estimating the noise level via the mean residual square and scaling the penalty in proportion to the estimated noise level. The iterative algorithm costs little beyond the computation of a path or grid of the sparse regression estimator for penalty levels above a proper threshold. For the scaled lasso, the algorithm is a gradient descent in a convex minimization of a penalized joint loss function for the regression coefficients and noise level. Under mild regularity conditions, we prove that the scaled lasso simultaneously yields an estimator for the noise level and an estimated coefficient vector satisfying certain oracle inequalities for prediction, the estimation of the noise level and the regression coefficients. These inequalities provide sufficient conditions for the consistency and asymptotic normality of the noise level estimator, including certain cases where the number of variables is of greater order than the sample size. Parallel results are provided for the least squares estimation after model selection by the scaled lasso. Numerical results demonstrate the superior performance of the proposed methods over an earlier proposal of joint convex minimization.

stat.ML↗

arXiv2012

A General Framework of Dual Certificate Analysis for Structured Sparse Recovery Problems

This paper develops a general theoretical framework to analyze structured sparse recovery problems using the notation of dual certificate. Although certain aspects of the dual certificate idea have already been used in some previous work, due to the lack of a general and coherent theory, the analysis has so far only been carried out in limited scopes for specific problems. In this context the current paper makes two contributions. First, we introduce a general definition of dual certificate, which we then use to develop a unified theory of sparse recovery analysis for convex programming. Second, we present a class of structured sparsity regularization called structured Lasso for which calculations can be readily performed under our theoretical framework. This new theory includes many seemingly loosely related previous work as special cases; it also implies new results that improve existing ones even for standard formulations such as L1 regularization.

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