Report

NICTA Machine Learning Research Group Seminar APPROXIMATE NEAREST NEIGHBOR QUERIES WITH A TINY INDEX 6/26/14 Wei Wang, University of New South Wales 1 Outline 2 Overview of Our Research SRS: c-Approximate Nearest Neighbor with a tiny index [PVLDB 2015] Conclusions NICTA Machine Learning Research Group Seminar 6/26/14 Research Projects 3 Similarity Query Processing Keyword search on (Semi-) Structured Data Graph Succinct Data Structures NICTA Machine Learning Research Group Seminar 6/26/14 NN and c-ANN Queries 4 c-ANN = x x D = {a, b, x} aka. (1+ε)-ANN NN = a Definitions A set of points, D = ∪i=1n {Oi}, in d-dimensional Euclidean space (d is large, e.g., hundreds) Given a query point q, find the closest point, O*, in D Relaxed version: Return a c-ANN point: i.e., its distance to q is at most c*Dist(O*, q) May return a c-ANN point with at least constant probability NICTA Machine Learning Research Group Seminar 6/26/14 Applications and Challenges 5 Applications Feature vectors: Data Mining, Multimedia DB Fundamental geometric problem: “post-office problem” Quantization in coding/compression … Challenges Curse of Dimensionality / Concentration of Measure Hard Large to find algorithms sub-linear in n and polynomial in d data size: 1KB for a single point with 256 dims NICTA Machine Learning Research Group Seminar 6/26/14 Existing Solutions 6 NN: Linear scan is (practically) the best approach using linear space & time O(d5log(n)) query time, O(n2d+δ) space O(dn1-ε(d)) query time, O(dn) space Linear scan: O(dn/B) I/Os, O(dn) space (1+ε)-ANN LSH is the best approach using sub-quadratic space O(log(n) + 1/ε(d-1)/2) query time, O(n*log(1/ε)) space Probabilistic test remove exponential dependency on d Fast JLT: O(d*log(d) + ε-3log2(n)) query time, O(nmax(2, ε^-2)) space LSH-based: Õ(dnρ+o(1)) query time, Õ(n1+ρ+o(1) + nd) space ρ= 1/(1+ε) + oc(1) NICTA Machine Learning Research Group Seminar 6/26/14 Approximate NN for Multimedia Retrieval 7 Cover-tree Spill-tree Reduce to NN search with Hamming distance Dimensionality reduction (e.g., PCA) Quantization-based approaches (e.g., CK-Means) NICTA Machine Learning Research Group Seminar 6/26/14 LSH is the best approach using sub-quadratic space Locality Sensitive Hashing (LSH) 8 Equality search Index: store o into bucket h(o) Query: retrieve every o in the bucket h(q), verify if o = q LSH ∀h∈LSH-family, Pr[ h(q) = h(o) ] ∝ 1/Dist(q, o) h :: Rd Z technically, dependent on r “Near-by” points (blue) have more chance of colliding with q than “faraway” points (red) NICTA Machine Learning Research Group Seminar 6/26/14 LSH: Indexing & Query Processing 9 Index For a fixed r sig(o) = ⟨h1(o), h2(o), …, hk(o)⟩ store o into bucket sig(o) Reduce Query Cost Iteratively increase r Query Const Succ. Prob. Search with a fixed r Retrieve and “verify” points in the bucket sig(q) Repeat this L times (boosting) Galloping search to find the first Incurs additional cost + good r 2 only c quality guarantee NICTA Machine Learning Research Group Seminar 6/26/14 Locality Sensitive Hashing (LSH) 10 Standard LSH c2-ANN binary search on R(ci, ci+1)-NN problems LSH on external memory O((dn/B)0.5) query, O((dn/B)1.5) space LSB-forest [SIGMOD’09, TODS’10]: different reduction from c2-ANN to a R(ci, ci+1)-NN problem O(n*log(n)/B) query, A C2LSH [SIGMOD’12]: O(n*log(n)/B) space Do not use composite hash keys Perform fine-granular counting number of collisions in m LSH projections SRS O(n/B) query, (Ours) O(n/B) space NICTA Machine Learning Research Group Seminar 6/26/14 Weakness of Ext-Memory LSH Methods 11 Existing methods uses super-linear space Thousands (or more) of hash tables needed if rigorous People resorts to hashing into binary code (and using Hamming distance) for multimedia retrieval Can only handle c, where c = x2, for integer x ≥ 2 To enable reusing the hash table (merging buckets) Valuable information lost (due to quantization) Update? (changes to n, and c) Dataset Size Audio, 40MB LSB-forest 1500 MB C2LSH+ 127 MB NICTA Machine Learning Research Group Seminar SRS (Ours) 2 MB 6/26/14 SRS: Our Proposed Method 12 Solving c-ANN queries with O(n) query time and O(n) space with constant probability Constants hidden in O() is very small Early-termination condition is provably effective Advantages: Small index Rich-functionality Simple Central idea: c-ANN query in d dims kNN query in m-dims with filtering Model the distribution of m “stable random projections” NICTA Machine Learning Research Group Seminar 6/26/14 2-stable Random Projection 13 Let D be the 2-stable random projection = standard Normal distribution N(0, 1) For two i.i.d. random variables A ~ D, B ~ D, then x*A + y*B ~ (x2+y2)1/2 * D Illustration V r1 V r 2 Seminar NICTA Machine Learning Research Group 6/26/14 Dist(O) and ProjDist(O) and Their Relationship Proj(O) 14 O O Proj(Q) r1 Q Q O in d dims m 2-stable random projections r2 z1≔⟨V, r1⟩ ~ N(0, ǁvǁ) z2≔⟨V, r2⟩ ~ N(0, ǁvǁ) z12+z22 ~ ǁvǁ2 *χ2m Dist(O) (z1, … zm) in m dims ProjDist(O) i.e., scaled Chi-squared distribution of m degrees of freedom Ψm(x): cdf of the standardχ2m distribution NICTA Machine Learning Research Group Seminar 6/26/14 LSH-like Property 15 Intuitive idea: If Dist(O1) ≪ Dist(O2) then ProjDist(O1) < ProjDist(O2) with high probability But the inverse is NOT true NN object in the projected space is most likely not the NN object in the original space with few projections, as Many far-away objects projected before the NN/cNN objects But we can bound the expected number of such cases! (say T) Solution Perform incremental k-NN search on the projected space till accessing T objects + Early termination test NICTA Machine Learning Research Group Seminar 6/26/14 Indexing 16 Finding the minimum m Input Output n, c, T ≔ max # of points to access by the algorithm m : # of 2-stable random projections T’ ≤ T: a better bound on T m = O(n/T). We use T = O(n), so m = O(1) to achieve linear space index Generate m 2-stable random projections n projected points in a m-dimensional space Index these projections using any index that supports incremental kNN search, e.g., R-tree Space cost: O(m * n) = O(n) NICTA Machine Learning Research Group Seminar 6/26/14 SRS-αβ(T, c, pτ) 17 Early-termination test: c 2 * ProjDist2 (ok ) m p 2 Dist (omin ) Compute proj(Q) Do incremental kNN search from proj(Q) // stopping condition α for k = 1 to T Compute Dist(Ok) Maintain Omin = argmin1≤i≤k Dist(Oi) If early-termination test (c, pτ) = TRUE // stopping condition β BREAK Return Omin c = 4, d = 256, m = 6, T = 0.00242n, B = 1024, pτ=0.18 Index = 0.0059n, Query = 0.0084n, succ prob = 0.13 Main Theorem: NICTA Machine Research p Group Seminar SRS-αβ returns a c-NN point withLearning probability with 6/26/14 O(n) I/O cost τ-f(m,c) Variations of SRS-αβ(T, c, pτ) 18 Compute proj(Q) Do incremental kNN search from proj(Q) for k = 1 to T // stopping condition α Compute Dist(Ok) Maintain Omin = argmin1≤i≤k Dist(Oi) If early-termination test (c, pτ) = TRUE // stopping condition β 1. 2. 3. BREAK Return Omin SRS-α SRS-β SRS-αβ(T, c’, pτ) Better quality; query cost is O(T) Best quality; query cost bounded by O(n); handles c = 1 Better quality; query cost bounded by O(T) All with success probability least p6/26/14 NICTA Machine Learning Research GroupatSeminar τ Other Results 19 Can be easily extended to support top-k c-ANN queries (k > 1) No previous known guarantee on the correctness of returned results We guarantee the correctness with probability at least pτ, if SRS-αβ stops due to early-termination condition ≈100% in practice (97% in theory) NICTA Machine Learning Research Group Seminar 6/26/14 Analysis 20 NICTA Machine Learning Research Group Seminar 6/26/14 Stopping Condition α 21 “near” point: the NN point its distance ≕ r “far” points: points whose distance > c * r Then for any κ > 0 and any o: Pr[ProjDist(o)≤κ*r | o is a near point] ≥ ψm(κ2) Pr[ProjDist(o)≤κ*r | o is a far point] ≤ ψm(κ2/c2) Both because ProjDist2(o)/Dist2(o) ~ χ2m Pr[the NN point projected beforeκ*r] ≥ ψm(κ2) Pr[# of bad points projected before κ*r < T] > (1 - ψm(κ2/c2)) * (n/T) Choose κ such that P1 + P2 – 1 > 0 Feasible due to good concentration bound for χ2m NICTA Machine Learning Research Group Seminar 6/26/14 P1 P2 Choosing κ 22 κ*r NICTA Machine Learning Research Group Seminar Let c = 4 Mode = m – 2 Blue: 4 Red: 4*(c2) = 64 6/26/14 ProjDist(OT): Case I Consider cases where both conditions hold (re. near and far points) P1 + P2 – 1 probability 23 ProjDist(o) in m-dims ProjDist(OT) NICTA Machine Learning Research Group Seminar Omin = the NN point 6/26/14 ProjDist(OT): Case II Consider cases where both conditions hold (re. near and far points) P1 + P2 – 1 probability 24 ProjDist(o) in m-dims ProjDist(OT) Omin = a cNN point NICTA Machine Learning Research Group Seminar 6/26/14 Early-termination Condition (β) 25 Omit the proof here Also relies on the fact that the squared sum of m projected distances follows a scaled χ2m distribution Key to Handle the case where c = 1 Returns the NN point with guaranteed probability Impossible to handle by LSH-based methods Guarantees the correctness of top-k cANN points returned when stopped by this condition No such guarantee by any previous method NICTA Machine Learning Research Group Seminar 6/26/14 Experiment Setup 26 Algorithms LSB-forest [SIGMOD’09, TODS’10] C2LSH [SIGMOD’12] SRS-* [VLDB’15] Data Measures Index size, query cost, result quality, success probability NICTA Machine Learning Research Group Seminar 6/26/14 Datasets 27 5.6PB 369GB 16GB NICTA Machine Learning Research Group Seminar 6/26/14 Tiny Image Dataset (8M pts, 384 dims) 28 Fastest: SRS-αβ, Slowest: C2LSH Quality the other way around SRS-αhas comparable quality with C2LSH yet has much lower cost. SRS-* dominates LSBforest NICTA Machine Learning Research Group Seminar 6/26/14 Approximate Nearest Neighbor 29 Empirically better than the theoretic guarantee With 15% I/Os of linear scan, returns NN with probability 71% With 62% I/Os of linear scan, returns NN with probability 99.7% NICTA Machine Learning Research Group Seminar 6/26/14 Large Dataset (0.45 Billion) 30 NICTA Machine Learning Research Group Seminar 6/26/14 Summary 31 c-ANN queries in arbitrarily high dim space kNN query in low dim space Our index size is approximately d/m of the size of the data file Opens up a new direction in c-ANN queries in highdimensional space Find efficient solution to kNN problem in 6-10 dimensional space NICTA Machine Learning Research Group Seminar 6/26/14 Q&A 32 Similarity Query Processing Project Homepage: http://www.cse.unsw.edu.au/~weiw/project/simjoin.html NICTA Machine Learning Research Group Seminar 6/26/14