Average-Distortion Sketching

TL;DR

This work introduces average-distortion sketching, a distribution-aware relaxation of worst-case metric sketching, to compress metric-space points while preserving average pairwise distances with respect to a fixed distribution μ. For ℓ_p spaces with p>2, the authors construct average-distortion sketches that achieve a constant distortion using 2^{O(p/c)}·log^2(dΔ) bits, outperforming worst-case sketch bounds and enabling improved approximate nearest neighbor structures. They establish an asymmetric sketch variant to reduce space on one input, and provide a lower-bound argument in a certificate model, suggesting exponential dependence on p/c may be necessary. By connecting average-distortion sketches to NN data structures and data-dependent hashing ideas, the paper demonstrates practical improvements for large-p regimes and lays out open problems regarding broader metric classes and tighter bounds. Overall, the results indicate average-distortion sketching can surpass traditional worst-case barriers and offer scalable NN solutions tailored to data distributions.

Abstract

We introduce average-distortion sketching for metric spaces. As in (worst-case) sketching, these algorithms compress points in a metric space while approximately recovering pairwise distances. The novelty is studying average-distortion: for any fixed (yet, arbitrary) distribution $μ$ over the metric, the sketch should not over-estimate distances, and it should (approximately) preserve the average distance with respect to draws from $μ$. The notion generalizes average-distortion embeddings into $\ell_1$ [Rabinovich '03, Kush-Nikolov-Tang '21] as well as data-dependent locality-sensitive hashing [Andoni-Razenshteyn '15, Andoni-Naor-Nikolov-et-al. '18], which have been recently studied in the context of nearest neighbor search. $\bullet$ For all $p \in (2, \infty)$ and any $c$ larger than a fixed constant, we give an average-distortion sketch for $([Δ]^d, \ell_p)$ with approximation $c$ and bit-complexity $\text{poly}(2^{p/c} \cdot \log(dΔ))$, which is provably impossible in (worst-case) sketching. $\bullet$ As an application, we improve on the approximation of sublinear-time data structures for nearest neighbor search over $\ell_p$ (for large $p > 2$). The prior best approximation was $O(p)$ [Andoni-Naor-Nikolov-et-al. '18, Kush-Nikolov-Tang '21], and we show it can be any $c$ larger than a fixed constant (irrespective of $p$) by using $n^{O(p/c)}$ space. We give some evidence that $2^{Ω(p/c)}$ space may be necessary by giving a lower bound on average-distortion sketches which produce a certain probabilistic certificate of farness (which our sketches crucially rely on).

Figures (4)

• Figure 1: Single-Scale Sketch for $([\Delta]^d, \ell_{p})$
• Figure 2: Accompanying diagram for the proof of Lemma \ref{['lem:bad-events']}. The events are labeled as nodes, leading to, either the cases considered in Lemma \ref{['lem:bad-events']}, or two nodes labeled $\mathsf{FAR}$ where the sketch will output $\mathsf{FAR}$. Edges are labelled "T" or "F", corresponding to whether the events in nodes hold (in the case "T") or do not hold (in the case "F").
• Figure 3: Core-Preprocess Subroutine.
• Figure 4: Core-Query Subroutine.

Theorems & Definitions (53)

• Definition 1.1: Average-Distortion Sketches
• Theorem 1: Average-Distortion Sketching for $\ell_p$
• Theorem 2: Approximate Nearest Neighbor in $\ell_p$
• Lemma 2.1
• Remark 2.2
• Lemma 2.3
• Lemma 2.4
• proof
• proof : Proof of \ref{['lem:close']}
• Lemma 2.5