Machine Learning at the Flatiron Institute Seminar: Lawrence Saul

Title: A geometrical connection between sparse and low-rank matrices and its uses for machine learning

Abstract: Many problems in high dimensional data analysis can be formulated as a search for structure in large matrices. One important type of structure is sparsity; for example, when a matrix is sparse, with a large number of zero elements, it can be stored in a highly compressed format. Another type of structure is linear dependence; when a matrix is low-rank, it can be expressed as the product of two smaller matrices. It is well known that neither one of these structures implies the other. But can one find more subtle connections by looking beyond the canonical decompositions of linear algebra?

In this talk, I will consider when a sparse nonnegative matrix can be recovered from a real-valued matrix of significantly lower rank. Of particular interest is the setting where the positive elements of the sparse matrix encode the similarities of nearby points on a low-dimensional manifold. The recovery can then be posed as a problem in manifold learning—namely, how to learn a similarity-preserving mapping of high-dimensional inputs into a lower-dimensional space. I will describe an algorithm for this problem based on a generalized low-rank decomposition of sparse matrices. This decomposition has the interesting property that it can be encoded by a neural network with one layer of rectified linear units; since the algorithm discovers this encoding, it can also be viewed as a layerwise primitive for deep learning. Finally, I will apply the algorithm to data sets where vector magnitudes and small cosine distances have interpretable meanings (e.g., the brightness of an image, the similarity to other words). On these data sets, the algorithm is able to discover much lower dimensional representations that preserve these meanings.