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 O(√1-p/k₂) on Stress and O((1-p)√k₂*ρ(A*)) 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.