Robust Low-Rank Matrix Estimation

• DOI: 10.48550/arxiv.1603.09071
• Publication Date: 2016-01-01
• Authors: Andreas Elsener; Sara van de Geer
• Keywords: Rank (graph theory); Matrix norm; Matrix (chemical analysis); Matrix Completion; Low-rank approximation

Many results have been proved for various nuclear norm penalized estimators of the uniform sampling matrix completion problem. However, most these are not robust: in cases quadratic loss function and its modifications used. We consider robust using two well-known functions: absolute value Huber loss. Under several conditions on sparsity problem (i.e. rank parameter matrix) regularity risk sharp non-sharp oracle inequalities shown to hold with high probability. As a consequence, asymptotic behavior is derived. Similar error bounds obtained under assumption weak sparsity, i.e. case where assumed be only approximately low-rank. In all our we high-dimensional setting. this case, means that assume $n\leq pq$. Finally, simulations confirm theoretical results.