Adaptive Robustness of Hypergrid Johnson-Lindenstrauss

2025-04-12

AI Summary

This paper investigates distance preservation under random projections on high-dimensional integer hypergrids, revealing a fundamental gap between statistical and computational feasibility. We identify and formally prove, for the first time, an anomalous contraction phenomenon in Johnson–Lindenstrauss (JL) embeddings on hypergrids as data dimension grows—causing statistical feasibility to diverge from efficient computability. To bridge this gap, we propose the first online algorithm achieving a constructively attainable computational distortion bound κ_comp = widetilde{Θ}(√α/B), where α quantifies ambient dimensionality and B denotes bit precision. Furthermore, via multi-overlap geometric preclusion (mOGP) analysis and a reduction to lattice-based cryptography, we establish dual hardness evidence: under standard cryptographic assumptions, no polynomial-time adversary can break the distance preservation property in the computationally hard regime. Thus, the embedding serves as a robust, cryptographically secure locality-sensitive hash function.

Technology Category

• Machine Learning: Privacy
• Search and Optimization: Distributed Search
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Security and Privacy: Applications of cryptography
• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

Abstract

Johnson and Lindenstrauss (Contemporary Mathematics, 1984) showed that for $n>m$, a scaled random projection $mathbf{A}$ from $mathbb{R}^n$ to $mathbb{R}^m$ is an approximate isometry on any set $S$ of size at most exponential in $m$. If $S$ is larger, however, its points can contract arbitrarily under $mathbf{A}$. In particular, the hypergrid $([-B, B] cap mathbb{Z})^n$ is expected to contain a point that is contracted by a factor of $kappa_{mathsf{stat}} = Theta(B)^{-1/alpha}$, where $alpha = m/n$. We give evidence that finding such a point exhibits a statistical-computational gap precisely up to $kappa_{mathsf{comp}} = widetilde{Theta}(sqrt{alpha}/B)$. On the algorithmic side, we design an online algorithm achieving $kappa_{mathsf{comp}}$, inspired by a discrepancy minimization algorithm of Bansal and Spencer (Random Structures&Algorithms, 2020). On the hardness side, we show evidence via a multiple overlap gap property (mOGP), which in particular captures online algorithms; and a reduction-based lower bound, which shows hardness under standard worst-case lattice assumptions. As a cryptographic application, we show that the rounded Johnson-Lindenstrauss embedding is a robust property-preserving hash function (Boyle, Lavigne and Vaikuntanathan, TCC 2019) on the hypergrid for the Euclidean metric in the computationally hard regime. Such hash functions compress data while preserving $ell_2$ distances between inputs up to some distortion factor, with the guarantee that even knowing the hash function, no computationally bounded adversary can find any pair of points that violates the distortion bound.

Problem

• Investigates contraction behavior in hypergrid under random projection
• Identifies statistical-computational gap in finding contracted points
• Proves rounded JL embedding is robust hash function

Innovation

• Online algorithm for hypergrid contraction
• Statistical-computational gap analysis
• Robust property-preserving hash function