LSH for similarity search in generic metric space

LSH for similarity search in generic metric space

Eliezer de Souza da Silva

Department of Computer Engineering and Industrial AutomationSchool of Electrical and Computer Engineering

University of [email protected]

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 familyH = 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 collidingwithin the ball of radius r),If d(p,q) > cr then PrH [h(q) = h(p)] ≤ p2 (probability of collidingoutside the ball of radius cr)c > 1 and p1 > p2



Locality-sensitive hashing

qr

crp

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 regularpopulation of points in each bucket and empirically performs betterthan regular schemes [50]



Existing LSH in General Metric Spaces

Novak et al. [41; 42]: M-Index : constructs a hierarchy of partitioningof the dataset choosing points from the dataset as cluster centers.Kang and Jung [28]: DFLSH (Distribution Free Locality-SensitiveHashing): randomly choose t points from the original dataset (withn > t points) as centroids and index the dataset using the nearestcentroid as hash key – this construction yields an approximatelyuniform 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.


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 → NhC(x) = argmini=1,...,kd(x , ci)

(1)



VoronoiLSH

C1

C2

C3

qr

cr

Zqp

Zp

d(q,p)

h(q)=h(p)=2

p'

h(p')=3

Zp'



Performance and Cost Models

Range CostRC(n, k) =

nk

+ k

⇒ RC(n) = 2√

n

NN CostNNC(n, k ,d) =

nk

log(nk

) + dnk

+ dk

⇒ NNCopt (n,d) = O(

√nd(log(

√n) + d + 1)



Hash probabilities bounds

Probability model: (Ω,F ,Pr)

Zp = d(p,NNC(p)) = d(p,C)

Ω = Zx |x ∈ X ,C ⊂ XPr [hC(p) 6= hC(q)] = Pr [Zq < d(q,NNC(p) ∩ Zp <d(p,NNC(q)]

pq

NNC(p)

NNC(q)




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) 6= hC(q)] ≥ Pr [Zq + Zp < cr ]

⇒ Pr [hC(p) = hC(q)] ≤ Pr [Zq + Zp ≥ cr ] = p2




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) 6= hC(q)] ≤ Pr [|Zq − Zp| < r ]

⇒ Pr [hC(p) = hC(q)] ≥ Pr [|Zq − Zp| ≥ r ] = p1




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


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 , andinteger number of centroidsp

output: A hash function hk ,w ,pselected← 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;

endendhk ,w ,p ← (selected,subsample) ;

Algorithm 1: Hash function building



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← newfloating-point array of size k ;for j ← 1 to k do

if selected[j] == 1 thendistances[j]← d(q, cj) ;

endendhasharray← new integer array of size w ;for i ← 1 to w do

hasharray[i]← element in subsample[i] that minimize distances[j](varying j) ;

endhk ,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

c1c3c4

c3c5c2

c5c1c3

c5c4c2

h5,4,3= h5,4,3(p)=

IEi=1,··· ,k [selected[i] = 1] = k − k(1− pk )w

O(k − kε) number of distance computation (intrinsic cost)a more complicated analysis for the extrinsic cost


Towards LSH in generic metric space Parallel VoronoiLSH

Parallel VoronoiLSH

Dataflow programming distributed computation;Computing stages distributed in processors and nodes;Message-passing interface.


Results Datasets

Datasets

APM (Arquivo Público Mineiro – The Public Archives in MinasGerais)

2.871.300 feature vectors (SIFT descriptor is a 128 dimensionalvector).queries dataset: 263.968 feature vectors with ground-truth.For the experiments we used 5000 queries uniformly sampled fromthe 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).


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

DFLSHK-MedoidsLSHK-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 0.03recall

extensiveness

DFLSHK-MedoidsLSHK-MeansLSH

L=1

L=5

L=8

(b) Recall x Number of hash functions L,Extensiveness (for 5000 cluster centers)



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

1.00re

call

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



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

reca

ll

extensivity

DFLSH L=2DFLSH 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

reca

ll

w=2

w=5

w=10

W=2W=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 (10centroids selected from a 4000 point sampleset)



Large scale experiment

(c) Query time / Recall (d) Parallel efficiency


Conclusions

Results and challenges

Using metric partitioning techniques for hashing functions in metricspace is a valid technique and should be further explored anddeveloped;The experiments do not show any clear advantage in learning theseeds of the Voronoi diagram by clustering;It would be interesting to equip the analysis with more assumptionsof the data;


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LSH for similarity search in generic metric space

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