DiffRed: Dimensionality Reduction guided by stable rank

In this work, we propose a novel dimensionality reduction technique, DiffRed, which first projects the data matrix, A, along first k1 principal components and the residual matrix A* (left after subtracting its k1-rank approximation) along k2 Gaussian random vectors. We evaluate M1, the distortion of mean-squared pair-wise distance, and Stress, the normalized value of RMS of distortion of the pairwise distances. We rigorously prove that DiffRed achieves a general upper bound of Oleft(√frac1-pk2right) on Stress and Oleft(frac(1-p)√k2*ρ(A*)right) on M1 where p is the fraction of variance explained by the first k1 principal components and ρ(A*) is the stable rank of A*. These bounds are tighter than the currently known results for Random maps. Our extensive experiments on a variety of real-world datasets demonstrate that DiffRed achieves near zero M1 and much lower values of Stress as compared to the well-known dimensionality reduction techniques. In particular, DiffRed can map a 6 million dimensional dataset to 10 dimensions with 54% lower Stress than PCA.