TY - JOUR A1 - Blanchard, Gilles A1 - Flaska, Marek A1 - Handy, Gregory A1 - Pozzi, Sara A1 - Scott, Clayton T1 - Classification with asymmetric label noise: Consistency and maximal denoising JF - Electronic journal of statistics N2 - 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. KW - Classification KW - label noise KW - mixture proportion estimation KW - surrogate loss KW - consistency Y1 - 2016 U6 - https://doi.org/10.1214/16-EJS1193 SN - 1935-7524 VL - 10 SP - 2780 EP - 2824 PB - Institute of Mathematical Statistics CY - Cleveland ER - TY - JOUR A1 - Mieth, Bettina A1 - Kloft, Marius A1 - Rodriguez, Juan Antonio A1 - Sonnenburg, Soren A1 - Vobruba, Robin A1 - Morcillo-Suarez, Carlos A1 - Farre, Xavier A1 - Marigorta, Urko M. A1 - Fehr, Ernst A1 - Dickhaus, Thorsten A1 - Blanchard, Gilles A1 - Schunk, Daniel A1 - Navarro, Arcadi A1 - Müller, Klaus-Robert T1 - Combining Multiple Hypothesis Testing with Machine Learning Increases the Statistical Power of Genome-wide Association Studies JF - Scientific reports N2 - The standard approach to the analysis of genome-wide association studies (GWAS) is based on testing each position in the genome individually for statistical significance of its association with the phenotype under investigation. To improve the analysis of GWAS, we propose a combination of machine learning and statistical testing that takes correlation structures within the set of SNPs under investigation in a mathematically well-controlled manner into account. The novel two-step algorithm, COMBI, first trains a support vector machine to determine a subset of candidate SNPs and then performs hypothesis tests for these SNPs together with an adequate threshold correction. Applying COMBI to data from a WTCCC study (2007) and measuring performance as replication by independent GWAS published within the 2008-2015 period, we show that our method outperforms ordinary raw p-value thresholding as well as other state-of-the-art methods. COMBI presents higher power and precision than the examined alternatives while yielding fewer false (i.e. non-replicated) and more true (i.e. replicated) discoveries when its results are validated on later GWAS studies. More than 80% of the discoveries made by COMBI upon WTCCC data have been validated by independent studies. Implementations of the COMBI method are available as a part of the GWASpi toolbox 2.0. Y1 - 2016 U6 - https://doi.org/10.1038/srep36671 SN - 2045-2322 VL - 6 PB - Nature Publ. Group CY - London ER - TY - INPR A1 - Blanchard, Gilles A1 - Krämer, Nicole T1 - Convergence rates of kernel conjugate gradient for random design regression N2 - We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient algorithm, where regularization against overfitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L^2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 8 KW - nonparametric regression KW - reproducing kernel Hilbert space KW - conjugate gradient KW - partial least squares KW - minimax convergence rates Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-94195 SN - 2193-6943 VL - 5 IS - 8 PB - Universitätsverlag Potsdam CY - Potsdam ER - TY - JOUR A1 - Blanchard, Gilles A1 - Kraemer, Nicole T1 - Convergence rates of Kernel Conjugate Gradient for random design regression JF - Analysis and applications N2 - We prove statistical rates of convergence for kernel-based least squares regression from i.i.d. data using a conjugate gradient (CG) algorithm, where regularization against over-fitting is obtained by early stopping. This method is related to Kernel Partial Least Squares, a regression method that combines supervised dimensionality reduction with least squares projection. Following the setting introduced in earlier related literature, we study so-called "fast convergence rates" depending on the regularity of the target regression function (measured by a source condition in terms of the kernel integral operator) and on the effective dimensionality of the data mapped into the kernel space. We obtain upper bounds, essentially matching known minimax lower bounds, for the L-2 (prediction) norm as well as for the stronger Hilbert norm, if the true regression function belongs to the reproducing kernel Hilbert space. If the latter assumption is not fulfilled, we obtain similar convergence rates for appropriate norms, provided additional unlabeled data are available. KW - Nonparametric regression KW - reproducing kernel Hilbert space KW - conjugate gradient KW - partial least squares KW - minimax convergence rates Y1 - 2016 U6 - https://doi.org/10.1142/S0219530516400017 SN - 0219-5305 SN - 1793-6861 VL - 14 SP - 763 EP - 794 PB - World Scientific CY - Singapore ER - TY - JOUR A1 - Beinrucker, Andre A1 - Dogan, Urun A1 - Blanchard, Gilles T1 - Extensions of stability selection using subsamples of observations and covariates JF - Statistics and Computing N2 - 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. KW - Variable selection KW - Stability selection KW - Subsampling Y1 - 2016 U6 - https://doi.org/10.1007/s11222-015-9589-y SN - 0960-3174 SN - 1573-1375 VL - 26 SP - 1059 EP - 1077 PB - Springer CY - Dordrecht ER - TY - INPR A1 - Blanchard, Gilles A1 - Mücke, Nicole T1 - Optimal rates for regularization of statistical inverse learning problems N2 - We consider a statistical inverse learning problem, where we observe the image of a function f through a linear operator A at i.i.d. random design points X_i, superposed with an additional noise. The distribution of the design points is unknown and can be very general. We analyze simultaneously the direct (estimation of Af) and the inverse (estimation of f) learning problems. In this general framework, we obtain strong and weak minimax optimal rates of convergence (as the number of observations n grows large) for a large class of spectral regularization methods over regularity classes defined through appropriate source conditions. This improves on or completes previous results obtained in related settings. The optimality of the obtained rates is shown not only in the exponent in n but also in the explicit dependence of the constant factor in the variance of the noise and the radius of the source condition set. T3 - Preprints des Instituts für Mathematik der Universität Potsdam - 5 (2016) 5 KW - statistical inverse problem KW - minimax rate KW - kernel method Y1 - 2016 U6 - http://nbn-resolving.de/urn/resolver.pl?urn:nbn:de:kobv:517-opus4-89782 SN - 2193-6943 VL - 5 IS - 5 PB - Universitätsverlag Potsdam CY - Potsdam ER -