Papers › DiffRed: Dimensionality Reduction guided by stable rank
DiffRed: Dimensionality Reduction guided by stable rank
Prarabdh Shukla, Gagan Raj Gupta, Kunal Dutta
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.
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