Approximate nearest neighbor search for $ell_p$-spaces ($2 < p < infty$) via embeddings

While the problem of approximate nearest neighbor search has been well-studied for Euclidean space and $\ell_1$, few non-trivial algorithms are known for $\ell_p$ when ($2 < p < \infty$). In this paper, we revisit this fundamental problem and present approximate nearest-neighbor search algorithms which give the first non-trivial approximation factor guarantees in this setting.

💡 Research Summary

The paper addresses the long‑standing gap in algorithms for approximate nearest‑neighbor (ANN) search in high‑dimensional ℓₚ spaces when p > 2. While Euclidean (ℓ₂) and ℓ₁ spaces have been extensively studied, the regime p > 2 has resisted non‑trivial approximation guarantees. The authors propose two complementary embedding‑based approaches that enable ANN structures originally designed for ℓ∞ or ℓ₂ to be applied to ℓₚ with provable distortion bounds and query‑time guarantees.

1. ℓₚ → ℓ∞ embedding via Fréchet max‑stable variables
The first technique draws d independent Fréchet random variables Z₁,…,Z_d (with tail Pr