Finding the sparsest solution to underdetermined systems of linear equations y = Φx is NP-hard in general. We show here that for systems with ‘typical’/‘random ’ Φ, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra. Our proposal, Stagewise Orthogonal Matching Pursuit (StOMP), successively transforms the signal into a negligible residual. Starting with initial residual r0 = y, at the s-th stage it forms the ‘matched filter ’ Φ T rs−1, identifies all coordinates with amplitudes exceeding a specially-chosen threshold, solves a least-squares problem using the selected coordinates, and subtracts the leastsquares fit, producing a new residual. After a fixed number of stages (e.g. 10), it stops. In contrast to Orthogonal Matching Pursuit (OMP), many coefficients can enter the model at each stage in StOMP while only one enters per stage in OMP; and StOMP takes a fixed number of stages (e.g. 10), while OMP can take many (e.g. n). StOMP runs much faster than competing proposals for sparse solutions, such as ℓ1 minimization and OMP, and so is attractive for solving large-scale problems. We use phase diagrams to compare algorithm performance. The problem of recovering a k-sparse vector x0 from (y, Φ) where Φ is random n × N and y = Φx0 is represented by a point (n/N, k/n)

This paper extends the theory of false discovery rates (FDR) pioneered by Benjamini and Hochberg (1995). We develop a framework in which the False Discovery Proportion (FDP) – the number of false rejections divided by the number of rejections – is treated as a stochastic process. After obtaining the limiting distribution of the process, we demonstrate the validitiy of a class of procedures for controlling the False Discovery Rate (the expected FDP). We construct a confidence envelope for the whole FDP process. From these envelopes we derive confidence thresholds, for controlling the quantiles of the distribution of the FDP as well as controlling the number of false discoveries. We also

This paper explores a class of empirical Bayes methods for level-dependent threshold selection in wavelet shrinkage. The prior considered for each wavelet coefficient is a mixture of an atom of probability at zero and a heavy-tailed density. The mixing weight, or sparsity parameter, for each level of the transform is chosen by marginal maximum likelihood. If estimation

Abstract: This paper studies oracle properties of ℓ1-penalized least squares in nonparametric regression setting with random design. We show that the penalized least squares estimator satisfies sparsity oracle inequalities, i.e., bounds in terms of the number of non-zero components of the oracle vector. The results are valid even when the dimension of the model is (much) larger than the sample size and the regression matrix is not positive definite. They can be applied to high-dimensional linear regression, to nonparametric adaptive regression estimation and to the problem of aggregation of arbitrary estimators.

An empirical Bayes approach to the estimation of possibly sparse sequences observed in Gaussian white noise is set out and investigated. The prior considered is a mixture of an atom of probability at zero and a heavy-tailed density, with the mixing weight chosen by marginal maximum likelihood, in the hope of adapting between sparse and dense sequences. If estimation is then carried out using the posterior median, this is a random thresholding procedure. Other thresholding rules using the same threshold can also be used. Probability bounds on the threshold chosen by the marginal maximum likelihood approach lead to overall bounds on the risk of the method over the class of signal sequences of length n with normalized ` p norm bounded by , for > 0 and 0 < p 2: Estimation error is measured by mean q loss, for 0 < q 2: For all p and q in (0; 2], the method achieves the optimal estimation rate as n ! 1 and ! 0 at various rates, and in this sense adapts automatically to the sparseness or otherwise of the underlying signal. In addition the risk is uniformly bounded over all signals. If the posterior mean is used as the estimator, the results still hold for q > 1: Simulations show excellent performance. Computationally, the method is tractable and essentially of O(n) complexity, and software is available. The extension to a modi ed thresholding method relevant to the wavelet estimation of derivatives of functions is also considered.

This paper considers regularizing a covariance matrix of p variables estimated from n observations, by hard thresholding. We show that the thresholded estimate is consistent in the operator norm as long as the true covariance matrix is sparse in a suitable sense, the variables are Gaussian or sub-Gaussian, and (log p)/n → 0, and obtain explicit rates. The results are uniform over families of covariance matrices which satisfy a fairly natural notion of sparsity. We discuss an intuitive resampling scheme for threshold selection and prove a general cross-validation result that justifies this approach. We also compare thresholding to other covariance estimators in simulations and on an example from climate data. 1. Introduction. Estimation

We predict the onset of personal bankruptcy using least squares regression. Although well publicized, only 2,244 bankruptcies occur in our data set of 2.9 million months of credit-card activity. We use stepwise selection to find predictors from a mix of payment history, debt load, demographics, and their interactions. This combination of rare responses and over 67,000 possible predictors leads to a challenging modeling question: How does one separate coincidental from useful predictors? We show that three modifications turn stepwise regression into an effective methodology for predicting bankruptcy. Our version of stepwise regression (1) organizes calculations to accommodate interactions, (2) exploits modern decision theoretic criteria to choose predictors, and (3) conservatively estimates p-values to handle sparse data and a binary response. Omitting any one of these leads to poor performance. A final step in our procedure calibrates regression predictions. With these modifications, stepwise regression predicts bankruptcy as well, if not better, than recently developed data-mining tools. When sorted, the largest 14,000 resulting predictions hold 1000 of the 1800 bankruptcies hidden in a validation sample of 2.3 million observations. If the cost of missing a bankruptcy is 200 times that of a false positive, our predictions incur less than 2/3 of the costs of classification errors produced by the tree-based classifier C4.5. Key Phrases: AIC, Cp, Bonferroni, calibration, hard thresholding, risk inflation criterion (RIC),

We consider the problem of classifying among a set of M hypotheses via distributed noisy sensors. The sensors can collaborate over a communication network and the task is to arrive at a consensus about the event after exchanging messages. We apply a variant of belief propagation as a strategy for collaboration to arrive at a solution to the distributed classification problem. We show that the message evolution can be re-formulated as the evolution of a linear dynamical system, which is primarily characterized by network connectivity. We show that a consensus to the centralized MAP estimate can almost always reached by the sensors for any arbitrary network. We then extend these results in several directions. First, we demonstrate that these results continue to hold with quantization of the messages, which is appealing from the point of view of finite bit rates supportable between links. We then demonstrate robustness against packet losses, which implies that optimal decisions can be achieved with asynchronous transmissions as well. Next, we present an account of energy requirements for distributed detection and demonstrate significant improvement over conventional decentralized detection. Finally, extensions to distributed estimation are described. 1

Recent methods for estimating sparse undirected graphs for real-valued data in high dimensional problems rely heavily on the assumption of normality. We show how to use a semiparametric Gaussian copula—or “nonparanormal”—for high dimensional inference. Just as additive models extend linear models by replacing linear functions with a set of one-dimensional smooth functions, the nonparanormal extends the normal by transforming the variables by smooth functions. We derive a method for estimating the nonparanormal, study the method’s theoretical properties, and show that it works well in many examples.

An important issue raised by Efron [7] in the context of large-scale multiple comparisons is that in many applications the usual assumption that the null distribution is known is incorrect, and seemingly negligible differences in the null may result in large differences in subsequent studies. This suggests that a careful study of estimation of the null is indispensable. In this paper, we consider the problem of estimating a null normal distri-bution, and a closely related problem, estimation of the proportion of non-null effects. We develop an approach based on the empirical characteristic function and Fourier analysis. The estimators are shown to be uniformly consistent over a wide class of parameters. Numerical performance of the estimators is investigated using both simulated and real data. In particular, we apply our