Nearest Neighbors for Matrix Estimation Interpreted as Blind Regression for Latent Variable Model
Abstract
We consider the setup of nonparametric {\em blind regression} for estimating the entries of a large $m \times n$ matrix, when provided with a small, random fraction of noisy measurements. We assume that all rows $u \in [m]$ and columns $i \in [n]$ of the matrix are associated to latent features $x_{\text{row}}(u)$ and $x_{\text{col}}(i)$ respectively, and the $(u,i)$th entry of the matrix, $A(u, i)$ is equal to $f(x_{\text{row}}(u), x_{\text{col}}(i))$ for a latent function $f$. Given noisy observations of a small, random subset of the matrix entries, our goal is to estimate the unobserved entries of the matrix as well as to "denoise" the observed entries. As the main result of this work, we introduce a nearestneighborbased estimation algorithm, and establish its consistency when the underlying latent function $f$ is Lipschitz, the underlying latent space is a bounded diameter Polish space, and the random fraction of observed entries in the matrix is at least $\max \left( m^{1 + \delta}, n^{1/2 + \delta} \right)$, for any $\delta > 0$. As an important byproduct, our analysis sheds light into the performance of the classical collaborative filtering algorithm for matrix completion, which has been widely utilized in practice. Experiments with the MovieLens and Netflix datasets suggest that our algorithm provides a principled improvement over basic collaborative filtering and is competitive with matrix factorization methods. Our algorithm has a natural extension to the setting of tensor completion via flattening the tensor to matrix. When applied to the setting of image inpainting, which is a $3$order tensor, we find that our approach is competitive with respect to stateofart tensor completion algorithms across benchmark images.
 Publication:

arXiv eprints
 Pub Date:
 May 2017
 arXiv:
 arXiv:1705.04867
 Bibcode:
 2017arXiv170504867L
 Keywords:

 Mathematics  Statistics Theory;
 62G08;
 62G99
 EPrint:
 27 pages, 3 figures. To appear in IEEE Transactions on Information Theory