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The Power of Recursive Embeddings for $\ell_p$ Metrics

Robert Krauthgamer, Nir Petruschka, Shay Sapir

TL;DR

The paper addresses embedding and algorithmic tasks in $\ell_p$ spaces with $p>2$ by introducing a recursive, reductions-based framework that leverages intermediate spaces via the Mazur map to outperform direct $\ell_p$ to $\ell_2$ embeddings. Through double recursion, the authors achieve state-of-the-art Lipschitz decomposition for finite subsets of $\ell_p$ and $\ell_\infty^d$, efficient approximate nearest neighbor search in $\ell_p$, and improved finite-$\ell_p$ embedding bounds into $\ell_2$ for the regime $3<p<3\sqrt{e}$, highlighting the power of problem-specific, non-oblivious embeddings. Key contributions include (i) a double-recursive Lipschitz decomposition with bounds $\beta^*_n(\ell_p)=O(p^4\sqrt{\log n})$ and $\beta^*(\ell_p^d)=O((\min\{p, \log d\})^4 \sqrt{d})$, (ii) a recursive ANN framework achieving $c=O(p^{2.387})$-approx with poly$(d\log n)$ query time and $n^{O(\log p)}$ space, (iii) an improved $c_2^n(\ell_p)$ bound for $3<p<3\sqrt{e}$, and (iv) connections to Lipschitz extension and spanner construction. This recursive, reduction-based approach opens avenues for applying similar techniques to other metric spaces and practical tasks such as NNS, offering deeper theoretical insight and potential practical impact on high-dimensional metric embedding problems.

Abstract

Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in $\ell_p$ spaces, $p>2$, yielding state-of-the-art results for Lipschitz decomposition, for Nearest Neighbor Search, and for embedding into $\ell_2$. In a nutshell, our method composes metric embeddings by viewing them as reductions between problems, and thereby obtains a new reduction that is substantially more effective than the known reduction that employs a single embedding. We in fact apply this method recursively, oftentimes using double recursion, which further amplifies the gap from a single embedding.

The Power of Recursive Embeddings for $\ell_p$ Metrics

TL;DR

The paper addresses embedding and algorithmic tasks in spaces with by introducing a recursive, reductions-based framework that leverages intermediate spaces via the Mazur map to outperform direct to embeddings. Through double recursion, the authors achieve state-of-the-art Lipschitz decomposition for finite subsets of and , efficient approximate nearest neighbor search in , and improved finite- embedding bounds into for the regime , highlighting the power of problem-specific, non-oblivious embeddings. Key contributions include (i) a double-recursive Lipschitz decomposition with bounds and , (ii) a recursive ANN framework achieving -approx with poly query time and space, (iii) an improved bound for , and (iv) connections to Lipschitz extension and spanner construction. This recursive, reduction-based approach opens avenues for applying similar techniques to other metric spaces and practical tasks such as NNS, offering deeper theoretical insight and potential practical impact on high-dimensional metric embedding problems.

Abstract

Metric embedding is a powerful tool used extensively in mathematics and computer science. We devise a new method of using metric embeddings recursively, which turns out to be particularly effective in spaces, , yielding state-of-the-art results for Lipschitz decomposition, for Nearest Neighbor Search, and for embedding into . In a nutshell, our method composes metric embeddings by viewing them as reductions between problems, and thereby obtains a new reduction that is substantially more effective than the known reduction that employs a single embedding. We in fact apply this method recursively, oftentimes using double recursion, which further amplifies the gap from a single embedding.

Paper Structure

This paper contains 9 sections, 10 theorems, 23 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Lipschitz Decomposition
  3. Nearest Neighbor Search
  4. Low-Distortion Embeddings
  5. Preliminaries
  6. Lipschitz Decomposition of $\ell_p$ Metrics
  7. Nearest Neighbor Search
  8. Embedding Finite $\ell_p$ Metrics into $\ell_2$
  9. Future Directions

Key Result

Theorem 1.2

Let $p \geq 2$ and $d \geq 1$. Then for every $n$-point metric $\mathcal{C} \subset \ell_{p}^{d}$ and $\Delta>0$, there exists an $(O(p^4\sqrt{\min\{\log {n}, d\}}), \Delta)$-Lipschitz decomposition.

Figures (2)

  • Figure 1: The distortion of embedding from $\ell_p$, $p>3$ into $\ell_2$ shown by depicting the exponent of $\log n$ in NR25 (blue) compared with our bound in \ref{['thm:improved-c_2-distortion']} (red).
  • Figure 2: An illustration of \ref{['claim:ANN_Mazur_Map']}. For the purpose of this illustration, the $\ell_p$ and $\ell_t$ balls are depicted using a Euclidean circle, and $x$ is assumed to lie at the origin of $\ell_p$. Given a query point $q$, an approximated solution $x$ is found in $\ell_p$ using $A_{base}$. The Mazur map $M^x$ is then applied, after which a solution $M^x(z)$ is found in $\ell_t$ using $A_x$. Finally, the inverse map is applied to obtain an improved solution $z$ in $\ell_p$.

Theorems & Definitions (31)

  • Definition 1.1: Lipschitz decomposition Bartal96
  • Theorem 1.2
  • Corollary 1.3
  • proof
  • Corollary 1.4
  • proof
  • Remark 1.5
  • Remark 1.6
  • Definition 1.7: Approximate Near Neighbor
  • Theorem 1.8
  • ...and 21 more