Sparse Matrix Factorization

We investigate the problem of factorizing a matrix into several sparse matrices and propose an algorithm for this under randomness and sparsity assumptions. This problem can be viewed as a simplification of the deep learning problem where finding a factorization corresponds to finding edges in different layers and values of hidden units. We prove that under certain assumptions for a sparse linear deep network with $n$ nodes in each layer, our algorithm is able to recover the structure of the network and values of top layer hidden units for depths up to $\tilde O(n^{1/6})$. We further discuss the relation among sparse matrix factorization, deep learning, sparse recovery and dictionary learning.Comment: 20 page

Expectile Matrix Factorization for Skewed Data Analysis

Matrix factorization is a popular approach to solving matrix estimation problems based on partial observations. Existing matrix factorization is based on least squares and aims to yield a low-rank matrix to interpret the conditional sample means given the observations. However, in many real applications with skewed and extreme data, least squares cannot explain their central tendency or tail distributions, yielding undesired estimates. In this paper, we propose \emph{expectile matrix factorization} by introducing asymmetric least squares, a key concept in expectile regression analysis, into the matrix factorization framework. We propose an efficient algorithm to solve the new problem based on alternating minimization and quadratic programming. We prove that our algorithm converges to a global optimum and exactly recovers the true underlying low-rank matrices when noise is zero. For synthetic data with skewed noise and a real-world dataset containing web service response times, the proposed scheme achieves lower recovery errors than the existing matrix factorization method based on least squares in a wide range of settings.Comment: 8 page main text with 5 page supplementary documents, published in AAAI 201

Is Simple Better? Revisiting Non-linear Matrix Factorization for Learning Incomplete Ratings

Matrix factorization techniques have been widely used as a method for collaborative filtering for recommender systems. In recent times, different variants of deep learning algorithms have been explored in this setting to improve the task of making a personalized recommendation with user-item interaction data. The idea that the mapping between the latent user or item factors and the original features is highly nonlinear suggest that classical matrix factorization techniques are no longer sufficient. In this paper, we propose a multilayer nonlinear semi-nonnegative matrix factorization method, with the motivation that user-item interactions can be modeled more accurately using a linear combination of non-linear item features. Firstly, we learn latent factors for representations of users and items from the designed multilayer nonlinear Semi-NMF approach using explicit ratings. Secondly, the architecture built is compared with deep-learning algorithms like Restricted Boltzmann Machine and state-of-the-art Deep Matrix factorization techniques. By using both supervised rate prediction task and unsupervised clustering in latent item space, we demonstrate that our proposed approach achieves better generalization ability in prediction as well as comparable representation ability as deep matrix factorization in the clustering task.Comment: version