cobinatorial algorithms for nearest neighbors, near-duplicates and small world design - yury...
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![Page 1: Cobinatorial Algorithms for Nearest Neighbors, Near-Duplicates and Small World Design - Yury Lifshits - SODA 2009](https://reader034.vdocuments.net/reader034/viewer/2022052521/53ff97c88d7f724c088b46a7/html5/thumbnails/1.jpg)
Combinatorial Algorithmsfor Nearest Neighbors, Near-Duplicates
and Small-World Design
Yury LifshitsYahoo! Research
Shengyu ZhangThe Chinese University of Hong Kong
SODA 2009
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 1 / 29
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Similarity Search: an ExampleInput: Set of objects
Task: Preprocess it
Query: New object
Task: Find the most
similar one in the dataset
Most similar
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 2 / 29
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Similarity Search: an ExampleInput: Set of objects
Task: Preprocess it
Query: New object
Task: Find the most
similar one in the dataset
Most similar
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 2 / 29
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Similarity Search: an ExampleInput: Set of objects
Task: Preprocess it
Query: New object
Task: Find the most
similar one in the dataset
Most similar
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 2 / 29
![Page 5: Cobinatorial Algorithms for Nearest Neighbors, Near-Duplicates and Small World Design - Yury Lifshits - SODA 2009](https://reader034.vdocuments.net/reader034/viewer/2022052521/53ff97c88d7f724c088b46a7/html5/thumbnails/5.jpg)
Similarity Search
Search space: object domain U, similarityfunction σ
Input: database S = {p1, . . . ,pn} ⊆ UQuery: q ∈ UTask: find argmax σ(pi,q)
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Nearest Neighbors in TheorySphere Rectangle Tree Orchard’s Algorithm k-d-B tree
Geometric near-neighbor access tree Excludedmiddle vantage point forest mvp-tree Fixed-height
fixed-queries tree AESA Vantage-pointtree LAESA R∗-tree Burkhard-Keller tree BBD tree
Navigating Nets Voronoi tree Balanced aspect ratio
tree Metric tree vps-tree M-treeLocality-Sensitive Hashing SS-tree
R-tree Spatial approximation treeMulti-vantage point tree Bisector tree mb-tree Cover
tree Hybrid tree Generalized hyperplane tree Slim tree
Spill Tree Fixed queries tree X-tree k-d tree Balltree
Quadtree Octree Post-office treeYury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 4 / 29
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Revision: Basic Assumptions
In theory:Triangle inequalityDoubling dimension is o(logn)
Typical web dataset has separation effect
For almost all i, j : 1/2 ≤ d(pi,pj) ≤ 1
Classic methods fail:Branch and bound algorithms visit every objectDoubling dimension is at least logn/2
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 5 / 29
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Revision: Basic Assumptions
In theory:Triangle inequalityDoubling dimension is o(logn)
Typical web dataset has separation effect
For almost all i, j : 1/2 ≤ d(pi,pj) ≤ 1
Classic methods fail:Branch and bound algorithms visit every objectDoubling dimension is at least logn/2
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 5 / 29
![Page 9: Cobinatorial Algorithms for Nearest Neighbors, Near-Duplicates and Small World Design - Yury Lifshits - SODA 2009](https://reader034.vdocuments.net/reader034/viewer/2022052521/53ff97c88d7f724c088b46a7/html5/thumbnails/9.jpg)
Revision: Basic Assumptions
In theory:Triangle inequalityDoubling dimension is o(logn)
Typical web dataset has separation effect
For almost all i, j : 1/2 ≤ d(pi,pj) ≤ 1
Classic methods fail:Branch and bound algorithms visit every objectDoubling dimension is at least logn/2
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 5 / 29
![Page 10: Cobinatorial Algorithms for Nearest Neighbors, Near-Duplicates and Small World Design - Yury Lifshits - SODA 2009](https://reader034.vdocuments.net/reader034/viewer/2022052521/53ff97c88d7f724c088b46a7/html5/thumbnails/10.jpg)
Contribution
Navin Goyal, YL, Hinrich Schütze, WSDM 2008:
Combinatorial framework: new approach to datamining problems that does not require triangleinequality
Nearest neighbor algorithm
This work:
Better nearest neighbor search
Detecting near-duplicates
Navigability design for small worlds
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 6 / 29
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Contribution
Navin Goyal, YL, Hinrich Schütze, WSDM 2008:
Combinatorial framework: new approach to datamining problems that does not require triangleinequality
Nearest neighbor algorithm
This work:
Better nearest neighbor search
Detecting near-duplicates
Navigability design for small worlds
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 6 / 29
![Page 12: Cobinatorial Algorithms for Nearest Neighbors, Near-Duplicates and Small World Design - Yury Lifshits - SODA 2009](https://reader034.vdocuments.net/reader034/viewer/2022052521/53ff97c88d7f724c088b46a7/html5/thumbnails/12.jpg)
Outline
1 Combinatorial Framework
2 New Algorithms
3 Combinatorial Nets
4 Directions for Further Research
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 7 / 29
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1Combinatorial Framework
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Comparison Oracle
Dataset p1, . . . ,pn
Objects and distance (or similarity)function are NOT given
Instead, there is a comparison oracleanswering queries of the form:
Who is closer to A: B or C?
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Disorder InequalitySort all objects by their similarity to p:
p r s
rankp(r)
rankp(s)
Then by similarity to r:
r s
rankr(s)
Dataset has disorder D if∀p, r, s : rankr(s) ≤ D(rankp(r) + rankp(s))
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 10 / 29
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Disorder InequalitySort all objects by their similarity to p:
p r s
rankp(r)
rankp(s)
Then by similarity to r:
r s
rankr(s)
Dataset has disorder D if∀p, r, s : rankr(s) ≤ D(rankp(r) + rankp(s))
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 10 / 29
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Disorder InequalitySort all objects by their similarity to p:
p r s
rankp(r)
rankp(s)
Then by similarity to r:
r s
rankr(s)
Dataset has disorder D if∀p, r, s : rankr(s) ≤ D(rankp(r) + rankp(s))
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 10 / 29
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Combinatorial Framework
=
Comparison oracleWho is closer to A: B or C?
+
Disorder inequalityrankr(s) ≤ D(rankp(r) + rankp(s))
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Combinatorial Framework: FAQ
Disorder of a metric space? Disorder ofRk?
In what cases disorder is relatively small?
Experimental values of D for somepractical datasets?
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Disorder vs. Others
If expansion rate is c, disorder constant isat most c2
Doubling dimension and disorderdimension are incomparable
Disorder inequality implies combinatorialform of “doubling effect”
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 13 / 29
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Combinatorial Framework: Pro & Contra
Advantages:
Does not require triangle inequality for distances
Applicable to any data model and any similarityfunction
Require only comparative training information
Limitation: worst-case form of disorder inequality
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Combinatorial Framework: Pro & Contra
Advantages:
Does not require triangle inequality for distances
Applicable to any data model and any similarityfunction
Require only comparative training information
Limitation: worst-case form of disorder inequality
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 14 / 29
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2New Algorithms
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Nearest Neighbor Search
Assume S ∪ {q} has disorder constant D
TheoremThere is a deterministic and exact algorithmfor nearest neighbor search:
Preprocessing: O(D7n log2n)
Search: O(D4 logn)
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Near-Duplicates
Assume, comparison oracle can also tell uswhether σ(x,y) > T for some similaritythreshold T
TheoremAll pairs with over-T similarity can be founddeterministically in time
poly(D)(n log2n+ |Output|)
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 17 / 29
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Visibility Graph
TheoremFor any dataset S with disorder D there existsa visibility graph:
poly(D)n log2n construction time
O(D4 logn) out-degrees
Naïve greedy routingdeterministically reachesexact nearest neighbor of the given target qin at most logn steps
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qp1
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p3
p4
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p4
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p2
p3
p4
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p2
p3
p4
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qp1
p2
p3
p4
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qp1
p2
p3
p4
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3Combinatorial Nets
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Combinatorial Ball
B(x, r) = {y : rankx(y) < r}
In other words, it is a subset of dataset S: theobject x itself and r − 1 its nearest neighbors
xB(x,10)
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Combinatorial NetA subset R ⊆ S is called a combinatorialr-net iff the following two properties holds:Covering: ∀y ∈ S,∃x ∈ R, s.t. rankx(y) < r.Separation: ∀xi,xj ∈ R, rankxi(xj) ≥ r OR rankxj(xi) ≥ r
How to construct a combinatorial net?What upper bound on its size can we guarantee?
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Combinatorial NetA subset R ⊆ S is called a combinatorialr-net iff the following two properties holds:Covering: ∀y ∈ S,∃x ∈ R, s.t. rankx(y) < r.Separation: ∀xi,xj ∈ R, rankxi(xj) ≥ r OR rankxj(xi) ≥ r
How to construct a combinatorial net?What upper bound on its size can we guarantee?
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Basic Data Structure
Combinatorial nets:For every 0 ≤ i ≤ logn, construct a n
2i-net
Pointers, pointers, pointers:
Direct & inverted indices: links between centersand members of their balls
Cousin links: for every center keep pointers toclose centers on the same level
Navigation links: for every center keep pointers toclose centers on the next level
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Basic Data Structure
Combinatorial nets:For every 0 ≤ i ≤ logn, construct a n
2i-net
Pointers, pointers, pointers:
Direct & inverted indices: links between centersand members of their balls
Cousin links: for every center keep pointers toclose centers on the same level
Navigation links: for every center keep pointers toclose centers on the next level
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 23 / 29
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Fast Net Construction
TheoremCombinatorial nets can be constructed inO(D7n log2n) time
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Up’n’Down Trick
Assume your have 2r-net for the dataset
To compute an r-ball around some object p:
1 Take a center p′ of 2r ball that is covering p
2 Take all centers of 2r-balls nearby p′
3 For all of them write down all members of theirs2r-balls
4 Sort all written objects with respect to p and keep rmost similar ones.
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4Directions for Further Research
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Future of Combinatorial FrameworkOther problems in combinatorial framework:
Low-distortion embeddingsClosest pairsCommunity discoveryLinear arrangementDistance labellingDimensionality reduction
What if disorder inequality has exceptions?
Insertions, deletions, changing metric
Experiments & implementation
Unification challenge: disorder + doubling = ?
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Summary
Combinatorial framework:
comparison oracle + disorder inequality
New algorithms:
Nearest neighbor search
Deterministic detection of near-duplicates
Navigability design
Thanks for your attention!Questions?
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 28 / 29
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Summary
Combinatorial framework:
comparison oracle + disorder inequality
New algorithms:
Nearest neighbor search
Deterministic detection of near-duplicates
Navigability design
Thanks for your attention!Questions?
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 28 / 29
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Linkshttp://yury.name
http://simsearch.yury.nameTutorial, bibliography, people, links, open problems
Yury Lifshits and Shengyu Zhang
Combinatorial Algorithms for Nearest Neighbors, Near-Duplicates andSmall-World Design
http://yury.name/papers/lifshits2008similarity.pdf
Navin Goyal, Yury Lifshits, Hinrich Schütze
Disorder Inequality: A Combinatorial Approach to Nearest Neighbor Search
http://yury.name/papers/goyal2008disorder.pdf
Benjamin Hoffmann, Yury Lifshits, Dirk Novotka
Maximal Intersection Queries in Randomized Graph Models
http://yury.name/papers/hoffmann2007maximal.pdf
Yury Lifshits, Shengyu Zhang ()Combinatorial Nearest Neighbors 29 / 29