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- 1. 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
- 2. 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
- 3. 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 3 / 44
- 4. Basic Concepts and Research Review Locality sensitive hashing Locality-sensitive hashing q r cr p p' Figure: LSH and (R, c)-NNE.S. Silva () Metric LSH Wednesday 8th October, 2014 4 / 44
- 5. Basic Concepts and Research Review Locality sensitive hashing Quantizers Data-dependent quantization has the advantage of more regular population of points in each bucket and empirically performs better than regular schemes [50] E.S. Silva () Metric LSH Wednesday 8th October, 2014 5 / 44
- 6. Basic Concepts and Research Review Locality sensitive hashing 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. E.S. Silva () Metric LSH Wednesday 8th October, 2014 6 / 44
- 7. 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) E.S. Silva () Metric LSH Wednesday 8th October, 2014 7 / 44
- 8. Towards LSH in generic metric space VoronoiLSH VoronoiLSH C1 C2 C3 q r cr Zq p Zp d(q,p) h(q)=h(p)=2 p' h(p')=3 Zp' E.S. Silva () Metric LSH Wednesday 8th October, 2014 8 / 44
- 9. Towards LSH in generic metric space VoronoiLSH 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) E.S. Silva () Metric LSH Wednesday 8th October, 2014 9 / 44
- 10. Towards LSH in generic metric space VoronoiLSH 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) E.S. Silva () Metric LSH Wednesday 8th October, 2014 10 / 44
- 11. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds 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)} E.S. Silva () Metric LSH Wednesday 8th October, 2014 11 / 44
- 12. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 11 / 44
- 13. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds d(p, q) < r d(p, NNC(q)) ≤ d(p, q) + Zq ≤ r + Zq d(q, NNC(p)) ≤ d(p, q) + Zp ≤ r + Zp E.S. Silva () Metric LSH Wednesday 8th October, 2014 12 / 44
- 14. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds 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} E.S. Silva () Metric LSH Wednesday 8th October, 2014 12 / 44
- 15. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 12 / 44
- 16. Towards LSH in generic metric space VoronoiLSH Hash probabilities bounds 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. E.S. Silva () Metric LSH Wednesday 8th October, 2014 13 / 44
- 17. 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 14 / 44
- 18. Towards LSH in generic metric space VoronoiPlexLSH 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
- 19. Towards LSH in generic metric space VoronoiPlexLSH 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 16 / 44
- 20. Towards LSH in generic metric space Parallel VoronoiLSH Parallel VoronoiLSH Dataﬂow programming distributed computation; Computing stages distributed in processors and nodes; Message-passing interface. E.S. Silva () Metric LSH Wednesday 8th October, 2014 17 / 44
- 21. 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). E.S. Silva () Metric LSH Wednesday 8th October, 2014 18 / 44
- 22. 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) E.S. Silva () Metric LSH Wednesday 8th October, 2014 19 / 44
- 23. Results Experimental results 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 E.S. Silva () Metric LSH Wednesday 8th October, 2014 20 / 44
- 24. Results Experimental results 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) E.S. Silva () Metric LSH Wednesday 8th October, 2014 21 / 44
- 25. Results Experimental results Large scale experiment (c) Query time / Recall (d) Parallel efﬁciency E.S. Silva () Metric LSH Wednesday 8th October, 2014 22 / 44
- 26. 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; E.S. Silva () Metric LSH Wednesday 8th October, 2014 23 / 44
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