Beyond Worst-Case Dimensionality Reduction for Sparse Vectors

2025-02-27

AI Summary

Traditional worst-case analysis for dimensionality reduction of sparse vectors is overly conservative. Method: We propose two new paradigms—average-case and nonnegative sparse settings—and integrate linear/nonlinear embeddings, ℓₚ-distance preservation, compressed sensing, and probabilistic lower bound construction inspired by the birthday paradox. Contributions/Results: We establish the first average-case lower bound for s-sparse vectors in s² dimensions; prove that nonnegativity enables information-theoretically optimal ℓ∞-preserving dimensionality reduction with O(s log |X|) dimensions; and demonstrate that nonlinear embeddings synergizing with nonnegativity yield exponential improvements. For nonnegative s-sparse data, our framework achieves (1±ε)-ℓₚ distance preservation in O(s log(|X|s)/ε²) dimensions and ℓ∞-preserving reduction in O(s log |X|) dimensions—strictly improving upon the generic sparse vector lower bound of O(s log d).

Technology Category

• Machine Learning: Dimensionality Reduction/Feature Selection
• Search and Optimization: Non-convex Optimization
• Data Mining & Knowledge Management: Data Compression

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Security and Privacy: Large-scale security measurements
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems

Abstract

We study beyond worst-case dimensionality reduction for $s$-sparse vectors. Our work is divided into two parts, each focusing on a different facet of beyond worst-case analysis: We first consider average-case guarantees. A folklore upper bound based on the birthday-paradox states: For any collection $X$ of $s$-sparse vectors in $mathbb{R}^d$, there exists a linear map to $mathbb{R}^{O(s^2)}$ which emph{exactly} preserves the norm of $99%$ of the vectors in $X$ in any $ell_p$ norm (as opposed to the usual setting where guarantees hold for all vectors). We give lower bounds showing that this is indeed optimal in many settings: any oblivious linear map satisfying similar average-case guarantees must map to $Omega(s^2)$ dimensions. The same lower bound also holds for a wide class of smooth maps, including `encoder-decoder schemes', where we compare the norm of the original vector to that of a smooth function of the embedding. These lower bounds reveal a separation result, as an upper bound of $O(s log(d))$ is possible if we instead use arbitrary (possibly non-smooth) functions, e.g., via compressed sensing algorithms. Given these lower bounds, we specialize to sparse emph{non-negative} vectors. For a dataset $X$ of non-negative $s$-sparse vectors and any $p ge 1$, we can non-linearly embed $X$ to $O(slog(|X|s)/epsilon^2)$ dimensions while preserving all pairwise distances in $ell_p$ norm up to $1pm epsilon$, with no dependence on $p$. Surprisingly, the non-negativity assumption enables much smaller embeddings than arbitrary sparse vectors, where the best known bounds suffer exponential dependence. Our map also guarantees emph{exact} dimensionality reduction for $ell_{infty}$ by embedding into $O(slog |X|)$ dimensions, which is tight. We show that both the non-linearity of $f$ and the non-negativity of $X$ are necessary, and provide downstream algorithmic improvements.

Problem

• Beyond worst-case dimensionality reduction for sparse vectors
• Optimal average-case guarantees for sparse vector embeddings
• Non-linear dimensionality reduction for non-negative sparse vectors

Innovation

• Average-case dimensionality reduction
• Non-linear embedding for sparse vectors
• Exact dimensionality reduction for non-negative vectors