We consider a sequence space model of statistical linear inverse problems where we need to estimate a function f from indirect noisy observations. Let a finite set of linear estimators be given. Our aim is to mimic the estimator in that has the smallest risk on the true f . Under general conditions, we show that this can be achieved by simple minimization of unbiased risk estimator, provided the singular values of the operator of the inverse problem decrease as a power law. The main result is a nonasymptotic oracle inequality that is shown to be asymptotically exact. This inequality can be also used to obtain sharp minimax adaptive results. In particular, we apply it to show that minimax adaptation on ellipsoids in multivariate anisotropic case is realized by minimization of unbiased risk estimator without any loss of efficiency with respect to optimal non-adaptive procedures. Mathematics Subject Classifications: 62G05, 62G20 Key Words: Statistical inverse problems, Oracle inequaliti...

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Principal component analysis (PCA) is a classical method for dimensionality reduction based on extracting the dominant eigenvectors of the sample covariance matrix. However, PCA is well known to behave poorly in the “large p, small n ” setting, in which the problem dimension p is comparable to or larger than the sample size n. This paper studies PCA in this high-dimensional regime, but under the additional assumption that the maximal eigenvector is sparse, say with at most k non-zero components. We analyze two computationally tractable methods for recovering the support of this maximal eigenvector: (a) a simple diagonal cut-off method, which transitions from success to failure as a function of the order parameter θdia(n, p, k) = n/[k 2 log(p − k)]; and (b) a more sophisticated semidefinite programming (SDP) relaxation, which succeeds once the order parameter θsdp(n, p, k) = n/[k log(p − k)] is larger than a critical threshold. Our results thus highlight an interesting trade-off between computational and statistical efficiency in high-dimensional inference.

We study the information-theoretic limits of exactly recovering the support of a sparse signal using noisy projections defined by various classes of measurement matrices. Our analysis is high-dimensional in nature, in which the number of observations n, the ambient signal dimension p, and the signal sparsity k are all allowed to tend to infinity in a general manner. This paper makes two novel contributions. First, we provide sharper necessary conditions for exact support recovery using general (non-Gaussian) dense measurement matrices. Combined with previously known sufficient conditions, this result yields a sharp characterization of when the optimal decoder can recover a signal with linear sparsity (k = Θ(p)) using a linear scaling of observations (n = Θ(p)) in the presence of noise. Our second contribution is to prove necessary conditions on the number of observations n required for asymptotically reliable recovery using a class of γ-sparsified measurement matrices, where the measurement sparsity γ(n, p, k) G (0, 1] corresponds to the fraction of non-zero entries per row. Our analysis allows general scaling of the quadruplet (n, p, k, γ), and reveals three different regimes, corresponding to whether measurement sparsity has no effect, a minor effect, or a dramatic effect on the information-theoretic limits of the subset recovery problem.

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Starting from the observation of an R n-Gaussian vector of mean f and covariance matrix σ 2 In (In is the identity matrix), we propose a method for building a Euclidean confidence ball around f, with prescribed probability of coverage. For each n, we describe its nonasymptotic property and show its optimality with respect to some criteria. 1. Introduction. In the present paper

Information-theoretic limits on sparsity recovery in the high-dimensional and noisy setting

In this paper we propose a general methodology, based on multiple testing, for testing that the mean of a Gaussian vector in R n belongs to a convex set. We show that the test achieves its nominal level, and characterize a class of vectors over which the tests achieve a prescribed power. In the functional regression model this general methodology is applied to test some qualitative hypotheses on the regression function. For example, we test that the regression function is positive, increasing, convex, or more generally, satisfies a differential inequality. Uniform separation rates over classes of smooth functions are established and a comparison with other results in the literature is provided. A simulation study evaluates some of the procedures for testing monotonicity. 1. Introduction. 1.1. The statistical framework. We consider the following regression model: Yi = F(xi) + σεi,