Radial Isotropic Position via an Implicit Newton's Method

2025-04-08

AI Summary

This paper addresses radial isotropization of high-dimensional datasets: computing an ε-approximate Forster transform—a reversible linear map (R) such that the second-moment matrix of the normalized points ({R a_i / |R a_i|_2}) lies within ([exp(-varepsilon) I, exp(varepsilon) I]). To overcome the (O(n^3 + n^2 d^{omega-1})) time bottleneck of classical cutting-plane methods, we propose the first algorithm combining implicit-constraint Newton’s method with graph-Laplacian sparsification, built upon Barthe’s potential function. Our approach leverages fast matrix–vector multiplication and smoothness analysis to ensure numerical stability. Under a polynomially bounded aspect-ratio assumption, the algorithm outputs an ε-approximate solution with high probability in (Oig(nd^{omega-1} (n/varepsilon)^{o(1)}ig)) time—nearly matching the theoretical lower bound on objective evaluations—and achieves a substantial improvement in runtime efficiency.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Placing a dataset $A = {mathbf{a}_i}_{i in [n]} subset mathbb{R}^d$ in radial isotropic position, i.e., finding an invertible $mathbf{R} in mathbb{R}^{d imes d}$ such that the unit vectors ${(mathbf{R} mathbf{a}_i) |mathbf{R} mathbf{a}_i|_2^{-1}}_{i in [n]}$ are in isotropic position, is a powerful tool with applications in functional analysis, communication complexity, coding theory, and the design of learning algorithms. When the transformed dataset has a second moment matrix within a $exp(pm epsilon)$ factor of a multiple of $mathbf{I}_d$, we call $mathbf{R}$ an $epsilon$-approximate Forster transform. We give a faster algorithm for computing approximate Forster transforms, based on optimizing an objective defined by Barthe [Barthe98]. When the transform has a polynomially-bounded aspect ratio, our algorithm uses $O(nd^{omega - 1}(frac n epsilon)^{o(1)})$ time to output an $epsilon$-approximate Forster transform with high probability, when one exists. This is almost the natural limit of this approach, as even evaluating Barthe's objective takes $O(nd^{omega - 1})$ time. Previously, the state-of-the-art runtime in this regime was based on cutting-plane methods, and scaled at least as $approx n^3 + n^2 d^{omega - 1}$. We also provide explicit estimates on the aspect ratio in the smoothed analysis setting, and show that our algorithm similarly improves upon those in the literature. To obtain our results, we develop a subroutine of potential broader interest: a reduction from almost-linear time sparsification of graph Laplacians to the ability to support almost-linear time matrix-vector products. We combine this tool with new stability bounds on Barthe's objective to implicitly implement a box-constrained Newton's method [CMTV17, ALOW17].

Problem

• Computing approximate Forster transforms efficiently
• Optimizing Barthe's objective for radial isotropic positioning
• Improving runtime for dataset transformation algorithms

Innovation

• Implements implicit Newton's method
• Optimizes Barthe's objective function
• Reduces graph Laplacian sparsification time