arXiv subjects

Edgar Dobriban

Publications and source records attributed to Edgar Dobriban.

95 records · Page 6Linked to original sources

arXiv2015

Regularity Properties for Sparse Regression

Statistical and machine learning theory has developed several conditions ensuring that popular estimators such as the Lasso or the Dantzig selector perform well in high-dimensional sparse regression, including the restricted eigenvalue, compatibility, and $\ell_q$ sensitivity properties. However, some of the central aspects of these conditions are not well understood. For instance, it is unknown if these conditions can be checked efficiently on any given data set. This is problematic, because they are at the core of the theory of sparse regression. Here we provide a rigorous proof that these conditions are NP-hard to check. This shows that the conditions are computationally infeasible to verify, and raises some questions about their practical applications. However, by taking an average-case perspective instead of the worst-case view of NP-hardness, we show that a particular condition, $\ell_q$ sensitivity, has certain desirable properties. This condition is weaker and more general than the others. We show that it holds with high probability in models where the parent population is well behaved, and that it is robust to certain data processing steps. These results are desirable, as they provide guidance about when the condition, and more generally the theory of sparse regression, may be relevant in the analysis of high-dimensional correlated observational data.

math.ST↗

arXiv2015

High-Dimensional Asymptotics of Prediction: Ridge Regression and Classification

We provide a unified analysis of the predictive risk of ridge regression and regularized discriminant analysis in a dense random effects model. We work in a high-dimensional asymptotic regime where $p, n \to \infty$ and $p/n \to γ\in (0, \, \infty)$, and allow for arbitrary covariance among the features. For both methods, we provide an explicit and efficiently computable expression for the limiting predictive risk, which depends only on the spectrum of the feature-covariance matrix, the signal strength, and the aspect ratio $γ$. Especially in the case of regularized discriminant analysis, we find that predictive accuracy has a nuanced dependence on the eigenvalue distribution of the covariance matrix, suggesting that analyses based on the operator norm of the covariance matrix may not be sharp. Our results also uncover several qualitative insights about both methods: for example, with ridge regression, there is an exact inverse relation between the limiting predictive risk and the limiting estimation risk given a fixed signal strength. Our analysis builds on recent advances in random matrix theory.

math.ST↗

arXiv2015

Efficient Computation of Limit Spectra of Sample Covariance Matrices

Consider an $n \times p$ data matrix $X$ whose rows are independently sampled from a population with covariance $Σ$. When $n,p$ are both large, the eigenvalues of the sample covariance matrix are substantially different from those of the true covariance. Asymptotically, as $n,p \to \infty$ with $p/n \to γ$, there is a deterministic mapping from the population spectral distribution (PSD) to the empirical spectral distribution (ESD) of the eigenvalues. The mapping is characterized by a fixed-point equation for the Stieltjes transform. We propose a new method to compute numerically the output ESD from an arbitrary input PSD. Our method, called Spectrode, finds the support and the density of the ESD to high precision; we prove this for finite discrete distributions. In computational experiments it outperforms existing methods by several orders of magnitude in speed and accuracy. We apply Spectrode to compute expectations and contour integrals of the ESD. These quantities are often central in applications of random matrix theory (RMT). We illustrate that Spectrode is directly useful in statistical problems, such as estimation and hypothesis testing for covariance matrices. Our proposal may make it more convenient to use asymptotic RMT in aspects of high-dimensional data analysis.

math.NA↗

arXiv2015

Optimal Multiple Testing Under a Gaussian Prior on the Effect Sizes

We develop a new method for frequentist multiple testing with Bayesian prior information. Our procedure finds a new set of optimal p-value weights called the Bayes weights. Prior information is relevant to many multiple testing problems. Existing methods assume fixed, known effect sizes available from previous studies. However, the case of uncertain information is usually the norm. For a Gaussian prior on effect sizes, we show that finding the optimal weights is a non-convex problem. Despite the non-convexity, we give an efficient algorithm that solves this problem nearly exactly. We show that our method can discover new loci in genome-wide association studies. On several data sets it compares favorably to other methods. Open source code is available.

stat.ME↗

arXiv2012

Certifying the restricted isometry property is hard

This paper is concerned with an important matrix condition in compressed sensing known as the restricted isometry property (RIP). We demonstrate that testing whether a matrix satisfies RIP is NP-hard. As a consequence of our result, it is impossible to efficiently test for RIP provided P \neq NP.

math.FA↗