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Register a new userThe LASSO risk for gaussian matrices
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Added by
Andrea Montanari
Publication date
2010
fields
Informatics Engineering
and research's language is
English
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We consider the problem of learning a coefficient vector x_0in R^N from noisy linear observation y=Ax_0+w in R^n. In many contexts (ranging from model selection to image processing) it is desirable to construct a sparse estimator x. In this case, a popular approach consists in solving an L1-penalized least squares problem known as the LASSO or Basis Pursuit DeNoising (BPDN). For sequences of matrices A of increasing dimensions, with independent gaussian entries, we prove that the normalized risk of the LASSO converges to a limit, and we obtain an explicit expression for this limit. Our result is the first rigorous derivation of an explicit formula for the asymptotic mean square error of the LASSO for random instances. The proof technique is based on the analysis of AMP, a recently developed efficient algorithm, that is inspired from graphical models ideas. Simulations on real data matrices suggest that our results can be relevant in a broad array of practical applications.
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The Lasso is a method for high-dimensional regression, which is now commonly used when the number of covariates $p$ is of the same order or larger than the number of observations $n$. Classical asymptotic normality theory is not applicable for this model due to two fundamental reasons: $(1)$ The regularized risk is non-smooth; $(2)$ The distance between the estimator $bf widehat{theta}$ and the true parameters vector $bf theta^star$ cannot be neglected. As a consequence, standard perturbative arguments that are the traditional basis for asymptotic normality fail. On the other hand, the Lasso estimator can be precisely characterized in the regime in which both $n$ and $p$ are large, while $n/p$ is of order one. This characterization was first obtained in the case of standard Gaussian designs, and subsequently generalized to other high-dimensional estimation procedures. Here we extend the same characterization to Gaussian correlated designs with non-singular covariance structure. This characterization is expressed in terms of a simpler ``fixed design model. We establish non-asymptotic bounds on the distance between distributions of various quantities in the two models, which hold uniformly over signals $bf theta^star$ in a suitable sparsity class, and values of the regularization parameter. As applications, we study the distribution of the debiased Lasso, and show that a degrees-of-freedom correction is necessary for computing valid confidence intervals.
This paper studies the problem of accurately recovering a sparse vector $beta^{star}$ from highly corrupted linear measurements $y = X beta^{star} + e^{star} + w$ where $e^{star}$ is a sparse error vector whose nonzero entries may be unbounded and $w$ is a bounded noise. We propose a so-called extended Lasso optimization which takes into consideration sparse prior information of both $beta^{star}$ and $e^{star}$. Our first result shows that the extended Lasso can faithfully recover both the regression as well as the corruption vector. Our analysis relies on the notion of extended restricted eigenvalue for the design matrix $X$. Our second set of results applies to a general class of Gaussian design matrix $X$ with i.i.d rows $oper N(0, Sigma)$, for which we can establish a surprising result: the extended Lasso can recover exact signed supports of both $beta^{star}$ and $e^{star}$ from only $Omega(k log p log n)$ observations, even when the fraction of corruption is arbitrarily close to one. Our analysis also shows that this amount of observations required to achieve exact signed support is indeed optimal.
We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.
We provide a unifying view of statistical information measures, multi-way Bayesian hypothesis testing, loss functions for multi-class classification problems, and multi-distribution $f$-divergences, elaborating equivalence results between all of these objects, and extending existing results for binary outcome spaces to more general ones. We consider a generalization of $f$-divergences to multiple distributions, and we provide a constructive equivalence between divergences, statistical information (in the sense of DeGroot), and losses for multiclass classification. A major application of our results is in multi-class classification problems in which we must both infer a discriminant function $gamma$---for making predictions on a label $Y$ from datum $X$---and a data representation (or, in the setting of a hypothesis testing problem, an experimental design), represented as a quantizer $mathsf{q}$ from a family of possible quantizers $mathsf{Q}$. In this setting, we characterize the equivalence between loss functions, meaning that optimizing either of two losses yields an optimal discriminant and quantizer $mathsf{q}$, complementing and extending earlier results of Nguyen et. al. to the multiclass case. Our results provide a more substantial basis than standard classification calibration results for comparing different losses: we describe the convex losses that are consistent for jointly choosing a data representation and minimizing the (weighted) probability of error in multiclass classification problems.
The lasso procedure is ubiquitous in the statistical and signal processing literature, and as such, is the target of substantial theoretical and applied research. While much of this research focuses on the desirable properties that lasso possesses---predictive risk consistency, sign consistency, correct model selection---all of it has assumes that the tuning parameter is chosen in an oracle fashion. Yet, this is impossible in practice. Instead, data analysts must use the data twice, once to choose the tuning parameter and again to estimate the model. But only heuristics have ever justified such a procedure. To this end, we give the first definitive answer about the risk consistency of lasso when the smoothing parameter is chosen via cross-validation. We show that under some restrictions on the design matrix, the lasso estimator is still risk consistent with an empirically chosen tuning parameter.
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