Locality-sensitive hashing for search in metric space

LSH for similarity search in generic metric space
Eliezer de Souza da Silva
Department of Computer Engineering and Industrial Automation
School of Electrical and Computer Engineering
University of Campinas
eliezers@dca.fee.unicamp.br
Wednesday 8th October, 2014

Basic Concepts and Research Review
Similarity Search – metric space model
Generic model for proximity search;
Tuple (U, d), where U is a set and d a distance function (positive,
symmetric);
∀x, y, z ∈ U, d(x, y) ≤ d(x, z) + d(z, y) (triangle inequality);
E.S. Silva () Metric LSH Wednesday 8th
October, 2014 2 / 44

Basic Concepts and Research Review Locality sensitive hashing
Locality-sensitive hashing
Deﬁnition
Given a distance function d : X × X → R+, a function family
H = {h : X → C} is (r, cr, p1, p2)-sensitive for a given data set S ⊆ X if,
for any points p, q ∈ S, h ∈ H:
If d(p, q) ≤ r then PrH[h(q) = h(p)] ≥ p1 (probability of colliding
within the ball of radius r),
If d(p, q) > cr then PrH[h(q) = h(p)] ≤ p2 (probability of colliding
outside the ball of radius cr)
c > 1 and p1 > p2
October, 2014 3 / 44

Locality-sensitive hashing
q
r
cr
p
p'
Figure: LSH and (R, c)-NNE.S. Silva () Metric LSH Wednesday 8th
October, 2014 4 / 44

Quantizers
Data-dependent quantization has the advantage of more regular
population of points in each bucket and empirically performs better
than regular schemes [50]
October, 2014 5 / 44

Existing LSH in General Metric Spaces
Novak et al. [41; 42]: M-Index: constructs a hierarchy of partitioning
of the dataset choosing points from the dataset as cluster centers.
Kang and Jung [28]: DFLSH (Distribution Free Locality-Sensitive
Hashing): randomly choose t points from the original dataset (with
n > t points) as centroids and index the dataset using the nearest
centroid as hash key – this construction yields an approximately
uniform number of points-per-bucket: O(n/t).
Tellez and Chavez [59]: map metric data to a permutation index,
encode permutation in hamming space and use Hamming LSH.
October, 2014 6 / 44

Towards LSH in generic metric space VoronoiLSH
VoronoiLSH - Hashing function
Generate
L induced
Voronoi
Partitioning
L hash tables
h1 hL...[ ]
L associated
hash functions
➡ ➡...{ {
Deﬁnition
Given a metric space (U, d), C = {c1, . . . , ck } ⊂ U and x ∈ U:
hC : U → N
hC(x) = argmini=1,...,k {d(x, ci)}
(1)
October, 2014 7 / 44

VoronoiLSH
C1
C2
C3
q
r
cr
Zq
p
Zp
d(q,p)
h(q)=h(p)=2
p'
h(p')=3
Zp'
October, 2014 8 / 44

Performance and Cost Models
Range Cost
RC(n, k) =
n
k
+ k
⇒ RC(n) = 2
√
n
NN Cost
NNC(n, k, d) =
n
k
log(
n
k
) + d
n
k
+ dk
⇒ NNCopt (n, d) = O( nd(log(
√
n) + d + 1)
October, 2014 9 / 44

Hash probabilities bounds
Probability model: (Ω, F, Pr)
Zp = d(p, NNC(p)) = d(p, C)
Ω = {Zx |x ∈ X, C ⊂ X}
Pr[hC(p) = hC(q)] = Pr[{Zq < d(q, NNC(p)} ∩ {Zp <
d(p, NNC(q)}]
p
q
NNC(p)
NNC(q)
October, 2014 10 / 44

d(p, q) > cr
{Zp + Zq < cr} ⊆ {Zp + Zq < d(p, q)}
{Zp + Zq < d(p, q)} ⊆ {Zq < d(q, NNC(p)} ∩ {Zp < d(p, NNC(q)}
October, 2014 11 / 44

d(p, q) > cr
{Zp + Zq < cr} ⊆ {Zp + Zq < d(p, q)}
{Zp + Zq < d(p, q)} ⊆ {Zq < d(q, NNC(p)} ∩ {Zp < d(p, NNC(q)}
⇒ Pr[hC(p) = hC(q)] ≥ Pr[Zq + Zp < cr]
⇒ Pr[hC(p) = hC(q)] ≤ Pr[Zq + Zp ≥ cr] = p2
October, 2014 11 / 44

d(p, q) < r
d(p, NNC(q)) ≤ d(p, q) + Zq ≤ r + Zq
d(q, NNC(p)) ≤ d(p, q) + Zp ≤ r + Zp
October, 2014 12 / 44

d(p, q) < r
d(p, NNC(q)) ≤ d(p, q) + Zq ≤ r + Zq
d(q, NNC(p)) ≤ d(p, q) + Zp ≤ r + Zp
⇒ {Zp < d(p, NNC(q)} ⊆ {Zp < r + Zq}
⇒ {Zq < d(q, NNC(p)} ⊆ {Zq < r + Zp}
October, 2014 12 / 44

d(p, q) < r
d(p, NNC(q)) ≤ d(p, q) + Zq ≤ r + Zq
d(q, NNC(p)) ≤ d(p, q) + Zp ≤ r + Zp
⇒ {Zp < d(p, NNC(q)} ⊆ {Zp < r + Zq}
⇒ {Zq < d(q, NNC(p)} ⊆ {Zq < r + Zp}
⇒ Pr[hC(p) = hC(q)] ≤ Pr[|Zq − Zp| < r]
⇒ Pr[hC(p) = hC(q)] ≥ Pr[|Zq − Zp| ≥ r] = p1
October, 2014 12 / 44

p1 ≥ p2: needs two assumptions, “Zq < δr” (δ > 0) and
“c > 2δ + 1”;
p1 > p2: needs consider a hypothetical case where “Zq = r − ”
and “Zp = 2δr − ”, for > 0.
October, 2014 13 / 44

Towards LSH in generic metric space VoronoiPlexLSH
VoronoiPlex LSH - Hash function construction
Multiple VoronoiLSH with a controlled number of distance computation
input : size k of the sample, number of distinct partitioning w, and
integer number of centroidsp
output: A hash function hk,w,p
selected ← new binary array of size k;
subsample ← new integer multi-array of size w × p;
for j ← 1 to w do
Random sample S = {s1, · · · , sp} from {1, · · · , k};
for i ← 1 to p do
subsample[j, i] ← si;
selected[si] ← 1;
end
end
hk,w,p ← (selected,subsample) ;
Algorithm 1: Hash function building
October, 2014 14 / 44

VoronoiPlex LSH - Hashing algorithm
input : Hash function object hk,w,p,Sample C = {c1, . . . , ck } ⊂ X
(|C| = k) and a point q ∈ X
output: Integer value hk,w,p(q)
(selected,subsample) ← retrieved from hk,w,p distances ← new
ﬂoating-point array of size k;
for j ← 1 to k do
if selected[j] == 1 then
distances[j] ← d(q, cj) ;
end
end
hasharray ← new integer array of size w;
for i ← 1 to w do
hasharray[i] ← element in subsample[i] that minimize distances[j]
(varying j) ;
end
hk,w,p(q) ← hash(hasharray) ;
Algorithm 2: Hash function ApplicationE.S. Silva () Metric LSH Wednesday 8th
October, 2014 15 / 44

VoronoiPlex LSH
1 2 5 2
c1c2 c3 c4c5
c1
c3
c4
c3
c5
c2
c5
c1
c3
c5
c4
c2
h5,4,3={ {
h5,4,3(p)=
IEi=1,··· ,k [selected[i] = 1] = k − k(1 − p
k )w
O(k − k ) number of distance computation (intrinsic cost)
a more complicated analysis for the extrinsic cost
October, 2014 16 / 44

Towards LSH in generic metric space Parallel VoronoiLSH
Parallel VoronoiLSH
Dataﬂow programming distributed computation;
Computing stages distributed in processors and nodes;
Message-passing interface.
October, 2014 17 / 44

Results Datasets
Datasets
APM (Arquivo Público Mineiro – The Public Archives in Minas
Gerais)
2.871.300 feature vectors (SIFT descriptor is a 128 dimensional
vector).
queries dataset: 263.968 feature vectors with ground-truth.
For the experiments we used 5000 queries uniformly sampled from
the query dataset and performed a 10-NN search.
Metric datasets: Listeria (20660/ 100) and English (66069 / 500 )
dictionary;
BigANN (1B) for large scale experiments: (109 / 104).
October, 2014 18 / 44

Results Experimental results
APM
0.62
0.64
0.66
0.68
0.7
0.72
0.74
0.76
0.78
0.8
0.82
0.001 0.01 0.1 1
recall
extensiveness
DFLSH
K-MedoidsLSH
K-MeansLSH
(a) Recall x Extensiveness (log scale)
0.6
0.65
0.7
0.75
0.8
0.85
0.9
0.95
1
0 0.005 0.01 0.015 0.02 0.025 0.03
recall
extensiveness
DFLSH
K-MedoidsLSH
K-MeansLSH
L=1
L=5
L=8
(b) Recall x Number of hash functions L,
Extensiveness (for 5000 cluster centers)
October, 2014 19 / 44

English dataset - VoronoiLSH and BPI
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40
fraction of query time of linear scan
0.70
0.75
0.80
0.85
0.90
0.95
1.00recall
Voronoi LSH with K-means++, L=5
DFLSH, L=5
Voronoi LSH with K-means++, L=8
DFLSH, L=8
Brief Proximity Index (BPI) LSH
Figure: Recall for Voronoi LSH and BPI LSH
October, 2014 20 / 44

Listeria
0.85
0.86
0.87
0.88
0.89
0.9
0.91
0.92
0.93
0.94
0.95
0.001 0.002 0.003 0.004 0.005 0.006 0.007 0.008 0.009 0.01 0.011
recall
extensivity
DFLSH L=2
DFLSH L=3
(a) Recall x Extensivity
0.00 0.05 0.10 0.15 0.20 0.25
extensivity
0.55
0.60
0.65
0.70
0.75
0.80
0.85
0.90
0.95
1.00
recall
w=2
w=5
w=10
W=2
W=5
W=10
VoronoiPlex LSH for L=1,nCluster=10
VoronoiPlex LSH for L=8,nCluster=10
(b) varying the size w of the key-length (10
centroids selected from a 4000 point sample
set)
October, 2014 21 / 44

Large scale experiment
(c) Query time / Recall (d) Parallel efﬁciency
October, 2014 22 / 44

Conclusions
Results and challenges
Using metric partitioning techniques for hashing functions in metric
space is a valid technique and should be further explored and
developed;
The experiments do not show any clear advantage in learning the
seeds of the Voronoi diagram by clustering;
It would be interesting to equip the analysis with more assumptions
of the data;
October, 2014 23 / 44

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