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Author: Blanchard, Gilles × Version: publishedVersion ×

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Now showing 1 - 3 of 3
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    Convergence analysis of Tikhonov regularization for non-linear statistical inverse problems
    (Ithaca, NY : Cornell University Library, 2020) Rastogi, Abhishake; Blanchard, Gilles; Mathé, Peter
    We study a non-linear statistical inverse problem, where we observe the noisy image of a quantity through a non-linear operator at some random design points. We consider the widely used Tikhonov regularization (or method of regularization) approach to estimate the quantity for the non-linear ill-posed inverse problem. The estimator is defined as the minimizer of a Tikhonov functional, which is the sum of a data misfit term and a quadratic penalty term. We develop a theoretical analysis for the minimizer of the Tikhonov regularization scheme using the concept of reproducing kernel Hilbert spaces. We discuss optimal rates of convergence for the proposed scheme, uniformly over classes of admissible solutions, defined through appropriate source conditions.
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    In search on non-Gaussian components of a high-dimensional distribution
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2006) Blanchard, Gilles; Kawanabe, Motoaki; Sugiyama, Masashi; Spokoiny, Vladimir; Müller, Klaus-Robert
    Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new em linear method to identify the "non-Gaussian subspace'' within a very general semi-parametric framework. Our proposed method, called NGCA (Non-Gaussian Component Analysis), is essentially based on the fact that we can construct a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector which belongs to the low dimensional non-Gaussian target subspace up to an estimation error. By applying this operator to a family of different nonlinear functions, one obtains a family of different vectors lying in a vicinity of the target space. As a final step, the target space itself is estimated by applying PCA to this family of vectors. We show that this procedure is consistent in the sense that the estimaton error tends to zero at a parametric rate, uniformly over the family, Numerical examples demonstrate the usefulness of our method
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    Semi-supervised novelty detection
    (Berlin : Weierstraß-Institut für Angewandte Analysis und Stochastik, 2009) Blanchard, Gilles; Lee, Gyemin; Scott, Clayton
    A common setting for novelty detection assumes that labeled examples from the nominal class are available, but that labeled examples of novelties are unavailable. The standard (inductive) approach is to declare novelties where the nominal density is low, which reduces the problem to density level set estimation. In this paper, we consider the setting where an unlabeled and possibly contaminated sample is also available at learning time. We argue that novelty detection in this semi-supervised setting is naturally solved by a general reduction to a binary classification problem. In particular, a detector with a desired false positive rate can be achieved through a reduction to Neyman-Pearson classification. Unlike the inductive approach, semi-supervised novelty detection (SSND) yields detectors that are optimal (e.g., statistically consistent) regardless of the distribution on novelties. Therefore, in novelty detection, unlabeled data have a substantial impact on the theoretical properties of the decision rule. We validate the practical utility of SSND with an extensive experimental study. We also show that SSND provides distribution-free, learning-theoretic solutions to two well known problems in hypothesis testing. First, our results provide a general solution to the general two-sample problem, that is, the problem of determining whether two random samples arise from the same distribution. Second, a specialization of SSND coincides with the standard $p$-value approach to multiple testing under the so-called random effects model. Unlike standard rejection regions based on thresholded $p$-values, the general SSND framework allows for adaptation to arbitrary alternative distributions.