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.
Complete list of metadatas

Cited literature [161 references]  Display  Hide  Download

https://pastel.archives-ouvertes.fr/tel-01552128
Contributor : Abes Star <>
Submitted on : Friday, June 30, 2017 - 7:16:11 PM
Last modification on : Wednesday, April 24, 2019 - 3:05:20 AM
Long-term archiving on : Monday, January 22, 2018 - 11:07:27 PM

File

70723_MAI_2017_archivage.pdf
Version validated by the jury (STAR)

Identifiers

  • HAL Id : tel-01552128, version 1

Citation

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

Share

Metrics

Record views

1594

Files downloads

218