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PAC-Bayesian estimation of low-rank matrices

Abstract : The first two parts of the thesis study pseudo-Bayesian estimation for the problem of matrix completion and quantum tomography. A novel low-rank inducing prior distribution is proposed for each problem. The statistical performance is examined: in each case we provide the rate of convergence of the pseudo-Bayesian estimator. Our analysis relies on PAC-Bayesian oracle inequalities. We also propose an MCMC algorithm to compute our estimator. The numerical behavior is tested on simulated and real data sets. The last part of the thesis studies the lifelong learning problem, a scenario of transfer learning, where information is transferred from one learning task to another. We propose an online formalization of the lifelong learning problem. Then, a meta-algorithm is proposed for lifelong learning. It relies on the idea of exponentially weighted aggregation. We provide a regret bound on this strategy. One of the nice points of our analysis is that it makes no assumption on the learning algorithm used within each task. Some applications are studied in details: finite subset of relevant predictors, single index model, dictionary learning.
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Submitted on : Friday, June 30, 2017 - 7:16:11 PM
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  • HAL Id : tel-01552128, version 1


The Tien Mai. PAC-Bayesian estimation of low-rank matrices. Statistics [math.ST]. Université Paris Saclay (COmUE), 2017. English. ⟨NNT : 2017SACLG001⟩. ⟨tel-01552128⟩



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