[Paper Review] Randomized Dimension Reduction on Massive Data

Stoyan Georgiev, Sayan Mukherjee|arXiv (Cornell University)|Nov 7, 2012

Sparse and Compressive Sensing Techniques44 references3 citations

TL;DR

This paper proposes a randomized dimension reduction framework that leverages adaptive randomized SVD to efficiently solve generalized eigendecomposition problems in PCA, Sliced Inverse Regression (SIR), and Locality Preserving Projections (LPP), achieving both computational speedups and improved statistical accuracy through implicit regularization, validated on real and simulated data.

ABSTRACT

Scalability of statistical estimators is of increasing importance in modern applications and dimension reduction is often used to extract relevant information from data. A variety of popular dimension reduction approaches can be framed as symmetric generalized eigendecomposition problems. In this paper we outline how taking into account the low rank structure assumption implicit in these dimension reduction approaches provides both computational and statistical advantages. We adapt recent randomized low-rank approximation algorithms to provide efficient solutions to three dimension reduction methods: Principal Component Analysis (PCA), Sliced Inverse Regression (SIR), and Localized Sliced Inverse Regression (LSIR). A key observation in this paper is that randomization serves a dual role, improving both computational and statistical performance. This point is highlighted in our experiments on real and simulated data.

Motivation & Objective

• Address the scalability challenge of dimension reduction in massive, high-dimensional datasets by integrating statistical and numerical efficiency.
• Overcome computational bottlenecks in traditional generalized eigendecomposition methods used in supervised and unsupervised dimension reduction.
• Develop a unified framework that leverages low-rank structure and randomization to improve both runtime and statistical performance.
• Demonstrate that randomization provides dual benefits—computational acceleration and implicit regularization enhancing out-of-sample prediction accuracy.

Proposed method

• Adapt adaptive randomized SVD to infer the number of singular vectors and iterations based on statistical criteria, reducing computational cost while preserving accuracy.
• Apply the randomized SVD as a core engine to approximate the generalized eigendecomposition in PCA, SIR, and LSIR, enabling scalable computation on large datasets.
• Construct low-rank approximations of the data and weight matrices (e.g., $ ilde{X}^T ilde{W} ilde{X}$) without explicitly forming dense matrices, using sparse nearest-neighbor structures.
• Use power iterations and random projections to efficiently compute low-rank approximations of large, structured matrices arising in dimension reduction.
• Integrate the randomized SVD with full SVD and back-transformation steps to recover the final projection basis in a numerically stable way.
• Extend the framework to manifold learning via Locality Preserving Projections (LPP), where the graph Laplacian is approximated using randomized SVD on sparse neighborhood matrices.

Experimental results

Research questions

• RQ1 Can randomized low-rank approximation improve both computational efficiency and statistical accuracy in dimension reduction for massive datasets?
• RQ2 To what extent does randomization implicitly regularize estimators, reducing out-of-sample prediction error in supervised dimension reduction?
• RQ3 How can adaptive selection of rank and iteration count in randomized SVD be guided by statistical criteria rather than fixed parameters?
• RQ4 Can the proposed framework be generalized to unsupervised and semi-supervised dimension reduction methods like LPP and Laplacian Eigenmaps?
• RQ5 Does the randomized approach maintain or improve the theoretical convergence properties of classical eigendecomposition-based methods?

Key findings

• The proposed randomized SVD algorithm achieves significant computational speedups while maintaining high accuracy on both simulated and real-world datasets.
• Randomized estimators for SIR and LSIR show improved out-of-sample prediction accuracy compared to classical methods, attributed to implicit regularization from the approximation process.
• The adaptive SVD component automatically determines the optimal number of singular vectors and iterations based on data-driven statistical criteria, reducing manual tuning.
• For LPP, the method efficiently approximates the graph Laplacian without explicitly constructing the full weight matrix, enabling scalable manifold learning on large datasets.
• Empirical results demonstrate that the randomized approach reduces runtime by orders of magnitude while preserving or enhancing predictive performance in dimension reduction tasks.
• The framework successfully generalizes to multiple dimension reduction methods, including PCA, SIR, LSIR, and LPP, through a unified computational pipeline based on randomized SVD.