@article{BachocBlanchardNeuvial2018, author = {Bachoc, Francois and Blanchard, Gilles and Neuvial, Pierre}, title = {On the post selection inference constant under restricted isometry properties}, series = {Electronic journal of statistics}, volume = {12}, journal = {Electronic journal of statistics}, number = {2}, publisher = {Institute of Mathematical Statistics}, address = {Cleveland}, issn = {1935-7524}, doi = {10.1214/18-EJS1490}, pages = {3736 -- 3757}, year = {2018}, abstract = {Uniformly valid confidence intervals post model selection in regression can be constructed based on Post-Selection Inference (PoSI) constants. PoSI constants are minimal for orthogonal design matrices, and can be upper bounded in function of the sparsity of the set of models under consideration, for generic design matrices. In order to improve on these generic sparse upper bounds, we consider design matrices satisfying a Restricted Isometry Property (RIP) condition. We provide a new upper bound on the PoSI constant in this setting. This upper bound is an explicit function of the RIP constant of the design matrix, thereby giving an interpolation between the orthogonal setting and the generic sparse setting. We show that this upper bound is asymptotically optimal in many settings by constructing a matching lower bound.}, language = {en} } @article{BeinruckerDoganBlanchard2016, author = {Beinrucker, Andre and Dogan, Urun and Blanchard, Gilles}, title = {Extensions of stability selection using subsamples of observations and covariates}, series = {Statistics and Computing}, volume = {26}, journal = {Statistics and Computing}, publisher = {Springer}, address = {Dordrecht}, issn = {0960-3174}, doi = {10.1007/s11222-015-9589-y}, pages = {1059 -- 1077}, year = {2016}, abstract = {We introduce extensions of stability selection, a method to stabilise variable selection methods introduced by Meinshausen and Buhlmann (J R Stat Soc 72:417-473, 2010). We propose to apply a base selection method repeatedly to random subsamples of observations and subsets of covariates under scrutiny, and to select covariates based on their selection frequency. We analyse the effects and benefits of these extensions. Our analysis generalizes the theoretical results of Meinshausen and Buhlmann (J R Stat Soc 72:417-473, 2010) from the case of half-samples to subsamples of arbitrary size. We study, in a theoretical manner, the effect of taking random covariate subsets using a simplified score model. Finally we validate these extensions on numerical experiments on both synthetic and real datasets, and compare the obtained results in detail to the original stability selection method.}, language = {en} } @book{Blanchard2011, author = {Blanchard, Gilles}, title = {Komplexit{\"a}tsanalyse in Statistik und Lerntheorie : Antrittsvorlesung 2011-05-04}, publisher = {Univ.-Bibl.}, address = {Potsdam}, year = {2011}, abstract = {Gilles Blanchards Vortrag gew{\"a}hrt Einblicke in seine Arbeiten zur Entwicklung und Analyse statistischer Eigenschaften von Lernalgorithmen. In vielen modernen Anwendungen, beispielsweise bei der Schrifterkennung oder dem Spam- Filtering, kann ein Computerprogramm auf der Basis vorgegebener Beispiele automatisch lernen, relevante Vorhersagen f{\"u}r weitere F{\"a}lle zu treffen. Mit der mathematischen Analyse der Eigenschaften solcher Methoden besch{\"a}ftigt sich die Lerntheorie, die mit der Statistik eng zusammenh{\"a}ngt. Dabei spielt der Begriff der Komplexit{\"a}t der erlernten Vorhersageregel eine wichtige Rolle. Ist die Regel zu einfach, wird sie wichtige Einzelheiten ignorieren. Ist sie zu komplex, wird sie die vorgegebenen Beispiele "auswendig" lernen und keine Verallgemeinerungskraft haben. Blanchard wird erl{\"a}utern, wie Mathematische Werkzeuge dabei helfen, den richtigen Kompromiss zwischen diesen beiden Extremen zu finden.}, language = {de} } @article{BlanchardCarpentierGutzeit2018, author = {Blanchard, Gilles and Carpentier, Alexandra and Gutzeit, Maurilio}, title = {Minimax Euclidean separation rates for testing convex hypotheses in R-d}, series = {Electronic journal of statistics}, volume = {12}, journal = {Electronic journal of statistics}, number = {2}, publisher = {Institute of Mathematical Statistics}, address = {Cleveland}, issn = {1935-7524}, doi = {10.1214/18-EJS1472}, pages = {3713 -- 3735}, year = {2018}, abstract = {We consider composite-composite testing problems for the expectation in the Gaussian sequence model where the null hypothesis corresponds to a closed convex subset C of R-d. We adopt a minimax point of view and our primary objective is to describe the smallest Euclidean distance between the null and alternative hypotheses such that there is a test with small total error probability. In particular, we focus on the dependence of this distance on the dimension d and variance 1/n giving rise to the minimax separation rate. In this paper we discuss lower and upper bounds on this rate for different smooth and non-smooth choices for C.}, language = {en} } @article{BlanchardDelattreRoquain2014, author = {Blanchard, Gilles and Delattre, Sylvain and Roquain, Etienne}, title = {Testing over a continuum of null hypotheses with False Discovery Rate control}, series = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, volume = {20}, journal = {Bernoulli : official journal of the Bernoulli Society for Mathematical Statistics and Probability}, number = {1}, publisher = {International Statistical Institute}, address = {Voorburg}, issn = {1350-7265}, doi = {10.3150/12-BEJ488}, pages = {304 -- 333}, year = {2014}, abstract = {We consider statistical hypothesis testing simultaneously over a fairly general, possibly uncountably infinite, set of null hypotheses, under the assumption that a suitable single test (and corresponding p-value) is known for each individual hypothesis. We extend to this setting the notion of false discovery rate (FDR) as a measure of type I error. Our main result studies specific procedures based on the observation of the p-value process. Control of the FDR at a nominal level is ensured either under arbitrary dependence of p-values, or under the assumption that the finite dimensional distributions of the p-value process have positive correlations of a specific type (weak PRDS). Both cases generalize existing results established in the finite setting. Its interest is demonstrated in several non-parametric examples: testing the mean/signal in a Gaussian white noise model, testing the intensity of a Poisson process and testing the c.d.f. of i.i.d. random variables.}, language = {en} } @article{BlanchardDickhausRoquainetal.2014, author = {Blanchard, Gilles and Dickhaus, Thorsten and Roquain, Etienne and Villers, Fanny}, title = {On least favorable configurations for step-up-down tests}, series = {Statistica Sinica}, volume = {24}, journal = {Statistica Sinica}, number = {1}, publisher = {Statistica Sinica, Institute of Statistical Science, Academia Sinica}, address = {Taipei}, issn = {1017-0405}, doi = {10.5705/ss.2011.205}, pages = {1 -- U31}, year = {2014}, language = {en} } @article{BlanchardFlaskaHandyetal.2016, author = {Blanchard, Gilles and Flaska, Marek and Handy, Gregory and Pozzi, Sara and Scott, Clayton}, title = {Classification with asymmetric label noise: Consistency and maximal denoising}, series = {Electronic journal of statistics}, volume = {10}, journal = {Electronic journal of statistics}, publisher = {Institute of Mathematical Statistics}, address = {Cleveland}, issn = {1935-7524}, doi = {10.1214/16-EJS1193}, pages = {2780 -- 2824}, year = {2016}, abstract = {In many real-world classification problems, the labels of training examples are randomly corrupted. Most previous theoretical work on classification with label noise assumes that the two classes are separable, that the label noise is independent of the true class label, or that the noise proportions for each class are known. In this work, we give conditions that are necessary and sufficient for the true class-conditional distributions to be identifiable. These conditions are weaker than those analyzed previously, and allow for the classes to be nonseparable and the noise levels to be asymmetric and unknown. The conditions essentially state that a majority of the observed labels are correct and that the true class-conditional distributions are "mutually irreducible," a concept we introduce that limits the similarity of the two distributions. For any label noise problem, there is a unique pair of true class-conditional distributions satisfying the proposed conditions, and we argue that this pair corresponds in a certain sense to maximal denoising of the observed distributions. Our results are facilitated by a connection to "mixture proportion estimation," which is the problem of estimating the maximal proportion of one distribution that is present in another. We establish a novel rate of convergence result for mixture proportion estimation, and apply this to obtain consistency of a discrimination rule based on surrogate loss minimization. Experimental results on benchmark data and a nuclear particle classification problem demonstrate the efficacy of our approach.}, language = {en} } @article{BlanchardHoffmannReiss2018, author = {Blanchard, Gilles and Hoffmann, Marc and Reiss, Markus}, title = {Optimal adaptation for early stopping in statistical inverse problems}, series = {SIAM/ASA Journal on Uncertainty Quantification}, volume = {6}, journal = {SIAM/ASA Journal on Uncertainty Quantification}, number = {3}, publisher = {Society for Industrial and Applied Mathematics}, address = {Philadelphia}, issn = {2166-2525}, doi = {10.1137/17M1154096}, pages = {1043 -- 1075}, year = {2018}, abstract = {For linear inverse problems Y = A mu + zeta, it is classical to recover the unknown signal mu by iterative regularization methods ((mu) over cap,(m) = 0,1, . . .) and halt at a data-dependent iteration tau using some stopping rule, typically based on a discrepancy principle, so that the weak (or prediction) squared-error parallel to A((mu) over cap (()(tau)) - mu)parallel to(2) is controlled. In the context of statistical estimation with stochastic noise zeta, we study oracle adaptation (that is, compared to the best possible stopping iteration) in strong squared- error E[parallel to((mu) over cap (()(tau)) - mu)parallel to(2)]. For a residual-based stopping rule oracle adaptation bounds are established for general spectral regularization methods. The proofs use bias and variance transfer techniques from weak prediction error to strong L-2-error, as well as convexity arguments and concentration bounds for the stochastic part. Adaptive early stopping for the Landweber method is studied in further detail and illustrated numerically.}, language = {en} } @article{BlanchardHoffmannReiss2018, author = {Blanchard, Gilles and Hoffmann, Marc and Reiss, Markus}, title = {Early stopping for statistical inverse problems via truncated SVD estimation}, series = {Electronic journal of statistics}, volume = {12}, journal = {Electronic journal of statistics}, number = {2}, publisher = {Institute of Mathematical Statistics}, address = {Cleveland}, issn = {1935-7524}, doi = {10.1214/18-EJS1482}, pages = {3204 -- 3231}, year = {2018}, abstract = {We consider truncated SVD (or spectral cut-off, projection) estimators for a prototypical statistical inverse problem in dimension D. Since calculating the singular value decomposition (SVD) only for the largest singular values is much less costly than the full SVD, our aim is to select a data-driven truncation level (m) over cap is an element of {1, . . . , D} only based on the knowledge of the first (m) over cap singular values and vectors. We analyse in detail whether sequential early stopping rules of this type can preserve statistical optimality. Information-constrained lower bounds and matching upper bounds for a residual based stopping rule are provided, which give a clear picture in which situation optimal sequential adaptation is feasible. Finally, a hybrid two-step approach is proposed which allows for classical oracle inequalities while considerably reducing numerical complexity.}, language = {en} } @article{BlanchardKawanabeSugiyamaetal.2006, author = {Blanchard, Gilles and Kawanabe, Motoaki and Sugiyama, Masashi and Spokoiny, Vladimir G. and M{\"u}ller, Klaus-Robert}, title = {In search of non-Gaussian components of a high-dimensional distribution}, issn = {1532-4435}, year = {2006}, abstract = {Finding non-Gaussian components of high-dimensional data is an important preprocessing step for efficient information processing. This article proposes a new 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 based on a linear operator which, to any arbitrary nonlinear (smooth) function, associates a vector belonging 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}, language = {en} }