LSH on the Hypercube Revisited

TL;DR

This paper revisits the application of locality sensitive hashing (LSH) for nearest-neighbor search in high-dimensional binary hypercube spaces, providing a clear presentation inspired by recent work, without introducing new methods.

Contribution

The paper offers a concise overview of LSH techniques specifically for the hypercube setting, emphasizing clarity over novel contributions.

Findings

Clarifies LSH scheme for hypercube points

Highlights the simplicity of the approach

Inspired by recent related work

Abstract

LSH (locality sensitive hashing) had emerged as a powerful technique in nearest-neighbor search in high dimensions [IM98, HIM12]. Given a point set $P$ in a metric space, and given parameters $r$ and $\varepsilon > 0$, the task is to preprocess the point set, such that given a query point $q$, one can quickly decide if $q$ is in distance at most $\leq r$ or $\geq (1+\varepsilon)r$ from the point set $P$. Once such a near-neighbor data-structure is available, one can reduce the general nearest-neighbor search to logarithmic number of queries in such structures [IM98, Har01, HIM12]. In this note, we revisit the most basic settings, where $P$ is a set of points in the binary hypercube $\{0,1\}^d$, under the $L_1$/Hamming metric, and present a short description of the LSH scheme in this case. We emphasize that there is no new contribution in this note, except (maybe) the presentation itself, which is inspired by the authors recent work [HM17].

Equations

$$\beta =\frac{1}{1+\epsilon }\in (0,1),\phi =1-\exp \left(-\frac{1}{r}\right)\approx \frac{1}{r},T=\beta \ln n,\text{and}L=O(n^{\beta }\log n).$$

$$(1-\phi {)}^{\Delta }\le (1-\phi {)}^{(1+\epsilon )rT}={\left(e^{-1/r}\right)}^{(1+\epsilon )rT}=\exp \left(-(1+\epsilon )\beta \ln n\right)=\frac{1}{n},$$

$$\displaystyle\mathop{\mathbf{E}}\left[{\bigl{.}\left|{\ell_{i}}\right|}\right]=\sum_{{\mathsf{x}}\in{P}_{\geq}}{\mathbf{Pr}}_{\bigl{.}{{M}}_{i}}\left[{{{M}}_{i}{\mathsf{x}}={{M}}_{i}{\mathsf{q}}}\right]\leq\left|{{P}_{\geq}}\right|\frac{1}{n}\leq 1.$$

Full text

In this note, we revisit the most basic settings, where $P$ is a set of points in the binary hypercube $\left\{{0,1}\right\}^{d}$, under the $L_{1}$/Hamming metric, and present a short description of the LSH scheme in this case. We emphasize that there is no new contribution in this note, except (maybe) the presentation itself, which is inspired by the authors recent work [HM17].

1 Locality sensitive hashing revisited

1.1 Preliminaries

Definition 1.1.

Consider a sequence ${{{m}}}$ of $k$, not necessarily distinct, integers i_{1},i_{2},\ldots,i_{k}\in\left\llbracket d\right\rrbracket, where \left\llbracket d\right\rrbracket=\left\{{1,\ldots,d}\right\}. For a point ${\mathsf{x}}=({{x}}_{1},\ldots,{{x}}_{d})\in{\mathbb{R}}^{d}$, its projection by ${{{m}}}$, denoted by ${{{m}}}{\mathsf{x}}$ is the point $\left({{{x}}_{i_{1}},\ldots,{{x}}_{i_{k}}}\right)\in{\mathbb{R}}^{k}$. Similarly, the projection of a point set ${P}\subseteq{\mathbb{R}}^{d}$ by ${{{m}}}$ is the point set ${{{m}}}{P}=\left\{{{{m}}}{\mathsf{x}}\;\middle|\;{\mathsf{x}}\in{P}\right\}$.

Given two sequences ${{{m}}}=i_{1},\ldots,i_{k}$ and ${{{u}}}=j_{1},\ldots,j_{k^{\prime}}$, let ${{{m}}}|{{{u}}}$ denote the concatenated sequence ${{{m}}}|{{{u}}}=i_{1},\ldots,i_{k},j_{1},\ldots,j_{k^{\prime}}.$ Given a probability ${\varphi}$, a natural way to create such a projection, is to include the $i$th coordinate, for $i=1,\ldots,d$, with probability ${\varphi}$. Let ${\mathcal{D}}_{{\varphi}}$ denote the distribution of such sequences of indices.

Definition 1.2.

Let ${\mathcal{D}}_{{\varphi}}^{T}$ denote the distribution resulting from concatenating $t$ independent sequences sampled from ${\mathcal{D}}_{{\varphi}}$. The length of a sampled sequence is $d{T}$.

Observe that for a point ${\mathsf{x}}\in\left\{{0,1}\right\}^{d}$, and ${{M}}\in{\mathcal{D}}_{{\varphi}}^{T}$, the projection ${{M}}{\mathsf{x}}$ might be higher dimensional than the original point ${\mathsf{x}}$, as it might contain repeated coordinates of the original point.

1.2 Algorithm

Input.

The input is a set ${P}$ of $n$ points in the hypercube $\left\{{0,1}\right\}^{d}$, and parameters ${r}$ and ${\varepsilon}$.

Preprocessing.

We set parameters as follows:

[TABLE]

We randomly and independently pick ${L}$ sequences ${{M}}_{1},\ldots,{{M}}_{L}\in{\mathcal{D}}_{{\varphi}}^{T}.$ Next, the algorithm computes the point sets ${Q}_{i}={{M}}_{i}{P}_{i}$, for $i=1,\ldots,{L}$, and stores them each in a hash table, denoted by ${D}_{i}$, for $i=1,\ldots,{L}$.

Answering a query.

Given a query point ${\mathsf{q}}\in\left\{{0,1}\right\}^{d}$, the algorithm computes ${\mathsf{q}}_{i}={{M}}_{i}{\mathsf{q}}$, for $i=1,\ldots,{L}$. From each ${D}_{i}$, the algorithm retrieves a list $\ell_{i}$ of all the points that collide with ${\mathsf{q}}_{i}$. The algorithm scans the points in the lists $\ell_{1},\ldots,\ell_{L}$. If any of these points is in Hamming distance smaller than $(1+{\varepsilon}){r}$, the algorithm returns it as the desired near-neighbor (and stops). Otherwise, the algorithm returns that all the points in ${P}$ are in distance at least ${r}$ from ${\mathsf{q}}$.

1.3 Analysis

1.3.1 Correctness

Lemma 1.3**.**

*Let $K$ be a set of ${r}$ marked/forbidden coordinates. The probability that a sequence ${{M}}=({{{m}}}_{1},\ldots,{{{m}}}_{T})$ sampled from ${\mathcal{D}}_{{\varphi}}^{T}$ does not sample any of the coordinates of $K$ is $1/n^{{\beta}}.$ This probability increases if $K$ contains fewer coordinates. *

Proof:

*For any $i$, the probability that ${{{m}}}_{i}$ does not contain any of these coordinates is $(1-{\varphi})^{r}=\left({e^{-1/{r}}}\right)^{{r}}=1/e.$ Since this experiment is repeated ${T}$ times, the probability is $e^{-{T}}=e^{-{\beta}\ln n}=n^{-{\beta}}.$    *

Lemma 1.4**.**

We have the following:

1. (A)

Let ${\mathsf{x}}$ be the nearest-neighbor to ${\mathsf{q}}$ in ${P}$. If $\left\|{{\mathsf{q}}-{\mathsf{x}}}\right\|_{1}\leq{r}$ then, with high probability, the data-structure returns a point that is in distance $\leq(1+{\varepsilon}){r}$ from ${\mathsf{q}}$. 2. (B)

In expectation, the total number of points in $\ell_{1},\ldots,\ell_{L}$ that are in distance $\geq(1+{\varepsilon}){r}$ from ${\mathsf{q}}$ is $\leq{L}$.

Proof:

(A) The good event here is that ${\mathsf{x}}$ and ${\mathsf{q}}$ collide under one of the sequences of ${{M}}_{1},\ldots,{{M}}_{L}$. However, the probability that ${{M}}_{i}{\mathsf{x}}={{M}}_{i}{\mathsf{q}}$ is at least $1/n^{{\beta}}$, by Lemma 1.3, as this is the probability that ${{M}}_{i}$ does not sample any of the (at most ${r}$) coordinates where ${\mathsf{x}}$ and ${\mathsf{q}}$ disagree. As such, the probability that all ${L}$ data-structures fail (i.e., none of the lists $\ell_{1},\ldots,\ell_{L}$ contains ${\mathsf{x}}$), is at most $(1-1/n^{{\beta}})^{{L}}<1/n^{O(1)}$, as ${L}=O\left({n^{{\beta}}\log n}\right)$.

(B) Let ${P}_{\geq}$ be the set of points in ${P}$ that are in distance $\geq(1+{\varepsilon}){r}$ from ${\mathsf{q}}$. For a point ${\mathsf{v}}\in{P}_{\geq}$, with $\Delta=\left\|{{\mathsf{v}}-{\mathsf{q}}}\right\|_{1}$, we have that the probability for ${{M}}\in{\mathcal{D}}_{{\varphi}}^{T}$ misses all the $\Delta$ coordinates, where ${\mathsf{v}}$ and ${\mathsf{q}}$ differ, is

[TABLE]

as ${\varphi}=1-e^{-1/{r}}$, ${T}={\beta}\ln n$, and ${\beta}=1/(1+{\varepsilon})$. But then, for any $i$, we have

[TABLE]

*As such, the total number of far points in the lists is at most ${L}\cdot 1={L}$, implying the claim.  *

1.3.2 Running time

For each $i$, the query computes ${{M}}_{i}{\mathsf{q}}$ and that takes $O(d{T})=O(d\log n)$ time. Repeated ${L}$ times, this takes $O({L}d\log n)$ time overall. Let $X$ be the random variable that is the number of points in the extracted lists that are in distance $\geq(1+{\varepsilon}){r}$ from the query point. The time to scan the lists is O\left({\bigl{.}d(X+1)}\right), since the algorithm stops as soon as it finds a near point. As such, by Lemma 1.4 (B), the expected query time is $O({L}d\log n+{L}d)=O\left({dn^{1/(1+{\varepsilon})}\log^{2}n}\right).$

1.3.3 Improving the performance (a bit)

Observe that for ${{M}}\in{\mathcal{D}}_{{\varphi}}^{T}$, and any two points ${\mathsf{x}},{\mathsf{v}}\in\left\{{0,1}\right\}^{d}$, all the algorithm cares about is whether ${{M}}{\mathsf{x}}={{M}}{\mathsf{v}}$. As such, if a coordinate is probed many times by ${{M}}$, we might as well probe this coordinate only once. In particular, for a sequence ${{M}}\in{\mathcal{D}}_{{\varphi}}^{T}$, let ${{M}}^{\prime}={\mathrm{uniq}}\left({{{M}}}\right)$ be the projection sequence resulting from removing replications in ${{M}}$. Significantly, ${{M}}^{\prime}$ is only of length $\leq d$, and as such, computing ${{M}}^{\prime}{\mathsf{x}}$, for a point ${\mathsf{x}}$, takes only $O(d)$ time. It is not hard to verify that one can also sample directly ${\mathrm{uniq}}\left({{{M}}}\right)$, for ${{M}}\in{\mathcal{D}}_{{\varphi}}^{T}$, in $O(d)$ time. This improves the query and processing by a logarithmic factor.

1.3.4 Result

Theorem 1.5**.**

Given a set $P$ of $n$ points in $\left\{{0,1}\right\}^{d}$, and parameters ${r},{\varepsilon}$, one can preprocess $P$ in $O(dn^{1+1/(1+{\varepsilon})}\log n)$ time and space, such that given a query point ${\mathsf{q}}$, the algorithm returns, in expected $O(dn^{1/(1+{\varepsilon})}\log n)$ time, one of the following:

1. (A)

a point ${\mathsf{x}}\in P$ such that $\left\|{{\mathsf{q}}-{\mathsf{x}}}\right\|_{1}\leq(1+{\varepsilon}){r}$, or 2. (B)

the distance of ${\mathsf{q}}$ from $P$ is larger than ${r}$.

*The algorithm may return either result if the distance of ${\mathsf{q}}$ from $P$ is in the range $[r,(1+{\varepsilon})r]$. The algorithm succeeds with high probability (per query). *

One can also get a high-probability guarantee on the query time. For a parameter $\delta>0$, create $O(\log\delta^{-1})$ LSH data-structures as above. Perform the query as above, except that when the query time exceeds (say) twice the expected time, move on to redo the query in the next LSH data-structure. The probability that the query had failed on one of these LSH data-structures is $\leq 1/2$, by Markov’s inequality. As such, overall, the query time becomes $O(dn^{1/(1+{\varepsilon})}\log n\log\delta^{-1})$, with probability $\geq 1-\delta$.