DiffRed: Dimensionality reduction guided by stable rank

Prarabdh Shukla, Gagan Raj Gupta, Kunal Dutta
Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR 238:3430-3438, 2024.

Abstract

In this work, we propose a novel dimensionality reduction technique, \textit{DiffRed}, which first projects the data matrix, A, along first $k_1$ principal components and the residual matrix $A^{*}$ (left after subtracting its $k_1$-rank approximation) along $k_2$ Gaussian random vectors. We evaluate \emph{M1}, the distortion of mean-squared pair-wise distance, and \emph{Stress}, the normalized value of RMS of distortion of the pairwise distances. We rigorously prove that \textit{DiffRed} achieves a general upper bound of $O\left(\sqrt{\frac{1-p}{k_2}}\right)$ on \emph{Stress} and $O\left(\frac{1-p}{\sqrt{k_2*\rho(A^{*})}}\right)$ on \emph{M1} where $p$ is the fraction of variance explained by the first $k_1$ principal components and $\rho(A^{*})$ is the \textit{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 \textit{DiffRed} achieves near zero \emph{M1} and much lower values of \emph{Stress} as compared to the well-known dimensionality reduction techniques. In particular, \textit{DiffRed} can map a 6 million dimensional dataset to 10 dimensions with 54% lower \emph{Stress} than PCA.

Cite this Paper

BibTeX
@InProceedings{pmlr-v238-shukla24a, title = { {D}iff{R}ed: Dimensionality reduction guided by stable rank }, author = {Shukla, Prarabdh and Raj Gupta, Gagan and Dutta, Kunal}, booktitle = {Proceedings of The 27th International Conference on Artificial Intelligence and Statistics}, pages = {3430--3438}, year = {2024}, editor = {Dasgupta, Sanjoy and Mandt, Stephan and Li, Yingzhen}, volume = {238}, series = {Proceedings of Machine Learning Research}, month = {02--04 May}, publisher = {PMLR}, pdf = {https://proceedings.mlr.press/v238/shukla24a/shukla24a.pdf}, url = {https://proceedings.mlr.press/v238/shukla24a.html}, abstract = { In this work, we propose a novel dimensionality reduction technique, \textit{DiffRed}, which first projects the data matrix, A, along first $k_1$ principal components and the residual matrix $A^{*}$ (left after subtracting its $k_1$-rank approximation) along $k_2$ Gaussian random vectors. We evaluate \emph{M1}, the distortion of mean-squared pair-wise distance, and \emph{Stress}, the normalized value of RMS of distortion of the pairwise distances. We rigorously prove that \textit{DiffRed} achieves a general upper bound of $O\left(\sqrt{\frac{1-p}{k_2}}\right)$ on \emph{Stress} and $O\left(\frac{1-p}{\sqrt{k_2*\rho(A^{*})}}\right)$ on \emph{M1} where $p$ is the fraction of variance explained by the first $k_1$ principal components and $\rho(A^{*})$ is the \textit{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 \textit{DiffRed} achieves near zero \emph{M1} and much lower values of \emph{Stress} as compared to the well-known dimensionality reduction techniques. In particular, \textit{DiffRed} can map a 6 million dimensional dataset to 10 dimensions with 54% lower \emph{Stress} than PCA. } }
Endnote
%0 Conference Paper %T DiffRed: Dimensionality reduction guided by stable rank %A Prarabdh Shukla %A Gagan Raj Gupta %A Kunal Dutta %B Proceedings of The 27th International Conference on Artificial Intelligence and Statistics %C Proceedings of Machine Learning Research %D 2024 %E Sanjoy Dasgupta %E Stephan Mandt %E Yingzhen Li %F pmlr-v238-shukla24a %I PMLR %P 3430--3438 %U https://proceedings.mlr.press/v238/shukla24a.html %V 238 %X In this work, we propose a novel dimensionality reduction technique, \textit{DiffRed}, which first projects the data matrix, A, along first $k_1$ principal components and the residual matrix $A^{*}$ (left after subtracting its $k_1$-rank approximation) along $k_2$ Gaussian random vectors. We evaluate \emph{M1}, the distortion of mean-squared pair-wise distance, and \emph{Stress}, the normalized value of RMS of distortion of the pairwise distances. We rigorously prove that \textit{DiffRed} achieves a general upper bound of $O\left(\sqrt{\frac{1-p}{k_2}}\right)$ on \emph{Stress} and $O\left(\frac{1-p}{\sqrt{k_2*\rho(A^{*})}}\right)$ on \emph{M1} where $p$ is the fraction of variance explained by the first $k_1$ principal components and $\rho(A^{*})$ is the \textit{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 \textit{DiffRed} achieves near zero \emph{M1} and much lower values of \emph{Stress} as compared to the well-known dimensionality reduction techniques. In particular, \textit{DiffRed} can map a 6 million dimensional dataset to 10 dimensions with 54% lower \emph{Stress} than PCA.
APA
Shukla, P., Raj Gupta, G. & Dutta, K.. (2024). DiffRed: Dimensionality reduction guided by stable rank . Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, in Proceedings of Machine Learning Research 238:3430-3438 Available from https://proceedings.mlr.press/v238/shukla24a.html.

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