Journal: Mathematical Statistics and Learning

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Publisher

European Mathematical Society

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ISSN

2520-2316
2520-2324

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2021 - 20222
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Publications 1 - 2 of 2
  • Tensor denoising with trend filtering
    Item type: Journal Article
    Ortelli, Francesco; van de Geer, Sara (2022)
    Mathematical Statistics and Learning
    We extend the notion of trend filtering to tensors by considering the kth-order Vitali variation – a discretized version of the integral of the absolute value of the kth-order total derivative. We prove adaptive ℓ0-rates and not-so-slow ℓ1-rates for tensor denoising with trend filtering. For k={1,2,3,4} we prove that the d-dimensional margin of a d-dimensional tensor can be estimated at the ℓ0-rate n−1, up to logarithmic terms, if the underlying tensor is a product of (k−1)th-order polynomials on a constant number of hyperrectangles. For general k we prove the ℓ1-rate of estimation n−H(d)+2k−12H(d)+2k−1, up to logarithmic terms, where H(d) is the dth harmonic number. Thanks to an ANOVA-type of decomposition we can apply these results to the lower dimensional margins of the tensor to prove bounds for denoising the whole tensor. Our tools are interpolating tensors to bound the effective sparsity for ℓ0-rates, mesh grids for ℓ1-rates and, in the background, the projection arguments by Dalalyan, Hebiri, and Lederer (2017).
  • Distribution-free robust linear regression
    Item type: Journal Article
    Mourtada, Jaouad; Vaškevičius, Tomas; Zhivotovskiy, Nikita (2021)
    Mathematical Statistics and Learning
    We study random design linear regression with no assumptions on the distribution of the covariates and with a heavy-tailed response variable. In this distribution-free regression setting, we show that boundedness of the conditional second moment of the response given the covariates is a necessary and sufficient condition for achieving non-trivial guarantees. As a starting point, we prove an optimal version of the classical in-expectation bound for the truncated least squares estimator due to Györfi, Kohler, Krzyzak, and Walk. ˙ However, we show that this procedure fails with constant probability for some distributions despite its optimal in-expectation performance. Then, combining the ideas of truncated least squares, median-ofmeans procedures, and aggregation theory, we construct a non-linear estimator achieving excess risk of order d/n with the optimal sub-exponential tail. While existing approaches to linear regression for heavy-tailed distributions focus on proper estimators that return linear functions, we highlight that the improperness of our procedure is necessary for attaining non-trivial guarantees in the distribution-free setting.
Publications 1 - 2 of 2