[Paper Review] Simple average-case lower bounds for approximate near-neighbor from isoperimetric inequalities

Yitong Yin|arXiv (Cornell University)|Feb 17, 2016

Face and Expression Recognition17 references3 citations

TL;DR

This paper establishes the first average-case cell-probe lower bounds for deterministic or Las Vegas randomized algorithms solving the approximate near-neighbor (ANN) problem in high-dimensional Hamming space. By linking ANN complexity to isoperimetric inequalities in metric spaces and using a strengthened richness lemma via cell-sampling, it proves an Ω(d / log(swγ²/(nd))) lower bound on expected cell-probe complexity, matching the strongest known worst-case bounds.

ABSTRACT

We prove an $Ω(d/\log \frac{sw}{nd})$ lower bound for the average-case cell-probe complexity of deterministic or Las Vegas randomized algorithms solving approximate near-neighbor (ANN) problem in $d$-dimensional Hamming space in the cell-probe model with $w$-bit cells, using a table of size $s$. This lower bound matches the highest known worst-case cell-probe lower bounds for any static data structure problems. This average-case cell-probe lower bound is proved in a general framework which relates the cell-probe complexity of ANN to isoperimetric inequalities in the underlying metric space. A tighter connection between ANN lower bounds and isoperimetric inequalities is established by a stronger richness lemma proved by cell-sampling techniques.

Motivation & Objective

• To close the gap in understanding average-case complexity for deterministic or Las Vegas randomized algorithms in approximate near-neighbor (ANN) search.
• To establish tight lower bounds for ANN in high-dimensional Hamming space under uniform and independent input distributions.
• To develop a general framework connecting cell-probe complexity of ANN to isoperimetric properties of the underlying metric space.
• To extend the richness lemma technique to average-case settings using cell-sampling, enabling stronger lower bounds.
• To match the highest known worst-case cell-probe lower bounds in the average-case regime for ANN.

Proposed method

• Formalizes a general framework linking ANN cell-probe complexity to isoperimetric inequalities in metric spaces.
• Introduces a strengthened richness lemma via cell-sampling techniques to handle variable query time in Las Vegas and deterministic algorithms.
• Applies the framework to d-dimensional Hamming space {0,1}^d with uniform input distribution over databases and queries.
• Uses Chernoff bounds and Harper’s theorem to establish expansion properties of λ-neighborhoods under uniform measure.
• Derives lower bounds by combining expansion properties with the generalized richness lemma, yielding Ω(d / log(swγ²/(nd))) for expected cell probes.
• Extends the approach to ℓ∞-norm spaces using a distribution μ with heavy-tailed coordinates, proving sw = n^{Ω(ρ/t)} lower bound under specific parameter regimes.

Experimental results

Research questions

• RQ1 Can tight average-case cell-probe lower bounds be established for deterministic or Las Vegas randomized algorithms in approximate near-neighbor search?
• RQ2 How does the cell-probe complexity of ANN relate to isoperimetric properties of the underlying metric space?
• RQ3 Can the richness lemma technique be adapted to average-case complexity where query time varies per input?
• RQ4 What is the optimal trade-off between space, word size, and query time for ANN in high-dimensional Hamming space under uniform input distributions?
• RQ5 Can the framework be extended to other norms, such as ℓ∞, with non-uniform input distributions?

Key findings

• An Ω(d / log(swγ²/(nd))) lower bound is established for the expected cell-probe complexity of deterministic or Las Vegas randomized algorithms solving (γ,λ)-ANN in d-dimensional Hamming space.
• The lower bound matches the highest known worst-case cell-probe lower bounds for static data structure problems, closing a long-standing gap in average-case complexity.
• The framework proves that λ-neighborhoods in Hamming space are (Φ,Ψ)-expanding under uniform measure with Φ=Ψ=2^{Ω(d/γ²)}, enabling strong lower bounds via isoperimetry.
• For ℓ∞-norm spaces with a specific heavy-tailed distribution μ, the lower bound sw = n^{Ω(ρ/t)} is derived under conditions where t ≤ ρ and w ≤ n^{1−2δ}.
• The result shows that even with randomization and approximation, ANN remains hard in high dimensions unless space grows super-polynomially in n for fixed t.
• The proof technique via cell-sampling and strengthened richness lemma enables the first average-case lower bounds for algorithms with variable query time, extending prior work limited to fixed worst-case complexity.