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Suppose we are given a large matrix $A=(a_{i,j})$ that cannot be stored inmemory but is in a disk or is presented in a data stream. However, we need tocompute a matrix decomposition of the entry-wisely transformed matrix,$f(A):=(f(a_{i,j}))$ for some function $f$. Is it possible to do it in a spaceefficient way? Many machine learning applications indeed need to deal with suchlarge transformed matrices, for example word embedding method in NLP needs towork with the pointwise mutual information (PMI) matrix, while the entrywisetransformation makes it difficult to apply known linear algebraic tools.Existing approaches for this problem either need to store the whole matrix andperform the entry-wise transformation afterwards, which is space consuming orinfeasible, or need to redesign the learning method, which is applicationspecific and requires substantial remodeling.  In this paper, we first propose a space-efficient sketching algorithm forcomputing the product of a given small matrix with the transformed matrix. Itworks for a general family of transformations with provable small error boundsand thus can be used as a primitive in downstream learning tasks. We then applythis primitive to a concrete application: low-rank approximation. We show thatour approach obtains small error and is efficient in both space and time. Wecomplement our theoretical results with experiments on synthetic and real data.

Sketching Transformed Matrices with Applications to Natural Language Processing