Transcript Slides

Spectral Approaches to
Nearest Neighbor Search
arXiv:1408.0751
Robert Krauthgamer (Weizmann Institute)
Joint with: Amirali Abdullah, Alexandr Andoni,
Ravi Kannan
Les Houches, January 2015
Nearest Neighbor Search (NNS)
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Preprocess: a set 𝑃 of 𝑛 points in ℝ𝑑
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Query: given a query point 𝑞, report a
point 𝑝∗ ∈ 𝑃 with the smallest distance
to 𝑞
𝑝∗
𝑞
Motivation
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Generic setup:
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Application areas:
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Points model objects (e.g. images)
Distance models (dis)similarity measure
machine learning: k-NN rule
signal processing, vector quantization,
bioinformatics, etc…
Distance can be:
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Hamming, Euclidean,
edit distance, earth-mover distance, …
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𝑝∗
𝑞
Curse of Dimensionality
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All exact algorithms degrade rapidly with the
dimension 𝑑
Algorithm
Query time
Space
Full indexing
𝑂(log 𝑛 ⋅ 𝑑)
No indexing –
linear scan
𝑂(𝑛 ⋅ 𝑑)
𝑛𝑂(𝑑) (Voronoi diagram size)
𝑂(𝑛 ⋅ 𝑑)
Approximate NNS
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Given a query point 𝑞, report 𝑝′ ∈ 𝑃 s.t.
∗
𝑝′ − 𝑞 ≤ 𝑐 min
𝑝
−𝑞
∗
𝑝
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𝑐 ≥ 1 : approximation factor
randomized: return such 𝑝′ with probability
≥ 90%
Heuristic perspective: gives a set of
candidates (hopefully small)
𝑝∗
𝑞
𝑝′
NNS algorithms
It’s all about space partitions !
Low-dimensional
[Arya-Mount’93], [Clarkson’94], [Arya-MountNetanyahu-Silverman-We’98], [Kleinberg’97],
[HarPeled’02],[Arya-Fonseca-Mount’11],…
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High-dimensional
[Indyk-Motwani’98], [Kushilevitz-OstrovskyRabani’98], [Indyk’98, ‘01], [Gionis-IndykMotwani’99], [Charikar’02], [Datar-ImmorlicaIndyk-Mirrokni’04], [Chakrabarti-Regev’04],
[Panigrahy’06], [Ailon-Chazelle’06], [AndoniIndyk’06], [Andoni-Indyk-NguyenRazenshteyn’14], [Andoni-Razenshteyn’15]
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Low-dimensional
kd-trees,…
𝑐 =1+𝜖
runtime: 𝜖 −𝑂(𝑑) ⋅ log 𝑛
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High-dimensional
Locality-Sensitive Hashing
Crucial use of random projections
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Johnson-Lindenstrauss Lemma: project to random subspace of
dimension 𝑂(𝜖 −2 log 𝑛) for 1 + 𝜖 approximation
Runtime: 𝑛1/𝑐 for 𝑐-approximation
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Practice
Data-aware partitions
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optimize the partition to your dataset
PCA-tree [Sproull’91, McNames’01, Verma-Kpotufe-Dasgupta’09]
randomized kd-trees [SilpaAnan-Hartley’08, Muja-Lowe’09]
spectral/PCA/semantic/WTA hashing [Weiss-Torralba-Fergus’08,
Wang-Kumar-Chang’09, Salakhutdinov-Hinton’09, Yagnik-Strelow-RossLin’11]
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Practice vs Theory
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Data-aware projections often outperform (vanilla)
random-projection methods
But no guarantees (correctness or performance)
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JL generally optimal [Alon’03, Jayram-Woodruff’11]
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Even for some NNS setups! [Andoni-Indyk-Patrascu’06]
Why do data-aware projections outperform random projections ?
Algorithmic framework to study phenomenon?
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Plan for the rest
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Model
Two spectral algorithms
Conclusion
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Our model
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“low-dimensional signal + large noise”
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inside high dimensional space
Signal: 𝑃 ⊂ 𝑈 for subspace 𝑈 ⊂ ℝ𝑑 of dimension 𝑘 ≪ 𝑑
Data: each point is perturbed by a full-dimensional
Gaussian noise 𝑁𝑑 (0, 𝜎 2 𝐼𝑑 )
𝑈
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Model properties
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Data 𝑃 = 𝑃 + 𝐺
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Query 𝑞 = 𝑞 + 𝑔𝑞 s.t.:
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points in P have at least unit norm
||𝑞 − 𝑝∗ || ≤ 1 for “nearest neighbor” 𝑝∗
||𝑞 − 𝑝|| ≥ 1 + 𝜖 for everybody else
Noise entries 𝑁(0, 𝜎 2 )
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up to factor poly(𝜖 −1 𝑘 log 𝑛)
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𝜎≈
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Claim: exact nearest neighbor is still the same
𝑑 1/4
Noise is large:
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has magnitude 𝜎 𝑑 ≈ 𝑑1/4 ≫ 1
top 𝑘 dimensions of 𝑃 capture sub-constant mass
JL would not work: after noise, gap very close to 1
Algorithms via PCA
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Find the “signal subspace” 𝑈 ?
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Use Principal Component Analysis (PCA)?
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then can project everything to 𝑈 and solve NNS there
≈ extract top direction(s) from SVD
e.g., 𝑘-dimensional space 𝑆 that minimizes
𝑝∈𝑃 𝑑
2 (𝑝, 𝑆)
If PCA removes noise “perfectly”, we are done:
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𝑆=𝑈
Can reduce to 𝑘-dimensional NNS
NNS performance as if we are in 𝑘 dimensions for full model?
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Best we can hope for
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dataset contains a “worst-case” 𝑘-dimensional instance
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Reduction from dimension 𝑑 to 𝑘
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Spoiler: Yes
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PCA under noise fails
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Does PCA find “signal subspace” 𝑈 under noise ?
No 
2 (𝑝, 𝑆)
𝑑
𝑝∈𝑃
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PCA minimizes
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good only on “average”, not “worst-case”
weak signal directions overpowered by noise directions
typical noise direction contributes 𝑛𝑖=1 𝑔𝑖2 𝜎 2 = Θ(𝑛𝜎 2 )
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𝑝∗
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1st Algorithm: intuition
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Extract “well-captured points”
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points with signal mostly inside top PCA space
should work for large fraction of points
Iterate on the rest
𝑝∗
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Iterative PCA
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Find top PCA subspace 𝑆
𝐶=points well-captured by 𝑆
Build NNS d.s. on {𝐶 projected onto 𝑆}
Iterate on the remaining points, 𝑃 ∖ 𝐶
Query: query each NNS d.s. separately
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To make this work:
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Nearly no noise in 𝑆: ensuring 𝑆 close to 𝑈
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Capture only points whose signal fully in 𝑆
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𝑆 determined by heavy-enough spectral directions (dimension may be
less than 𝑘)
well-captured: distance to 𝑆 explained by noise only
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Simpler model
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Assume: small noise
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𝑝𝑖 = 𝑝𝑖 + 𝛼𝑖 ,
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well-captured if 𝑑 𝑝, 𝑆 ≤ 2𝛼
Claim 1: if 𝑝∗ captured by 𝐶, will find it in NNS
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Query: query each NNS separately
can be even adversarial
Algorithm:
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where ||𝛼𝑖 || ≪ 𝜖
Find top-k PCA subspace 𝑆
𝐶=points well-captured by 𝑆
Build NNS on {𝐶 projected onto 𝑆}
Iterate on remaining points, 𝑃 ∖ 𝐶
for any captured 𝑝:
||𝑝𝑆 − 𝑞𝑆 || = || 𝑝 − 𝑞|| ± 4𝛼 = ||𝑝 − 𝑞|| ± 5𝛼
Claim 2: number of iterations is 𝑂(log 𝑛)
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𝑝∈𝑃 𝑑
2
(𝑝, 𝑆) ≤
𝑝∈𝑃 𝑑
2
𝑝, 𝑈 ≤ 𝑛 ⋅ 𝛼 2
for at most 1/4-fraction of points, 𝑑 2 𝑝, 𝑆 ≥ 4𝛼 2
hence constant fraction captured in each iteration
Analysis of general model
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Need to use randomness of the noise
Want to say that “signal” is stronger than “noise” (on
average)
Use random matrix theory
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𝑃 =𝑃+𝐺
𝐺 is random 𝑛 × 𝑑 with entries 𝑁(0, 𝜎 2 )
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𝑃 has rank ≤ 𝑘 and (Frobenius-norm)2 ≥ 𝑛
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All singular values 𝜆2 ≤ 𝜎 2 𝑛 ≈ 𝑛/ 𝑑
important directions have 𝜆2 ≥ Ω(𝑛/𝑘)
can ignore directions with 𝜆2 ≪ 𝜖𝑛/𝑘
Important signal directions stronger than noise!
Closeness of subspaces ?
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Trickier than singular values
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Top singular vector not stable under perturbation!
Only stable if second singular value much smaller
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How to even define “closeness” of subspaces?
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To the rescue: Wedin’s sin-theta theorem
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sin 𝜃 𝑆, 𝑈 = max min ||𝑥 − 𝑦||
𝑥∈𝑆 𝑦∈𝑈
|𝑥|=1
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𝑆
𝜃
𝑈
Wedin’s sin-theta theorem
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Developed by [Davis-Kahan’70], [Wedin’72]
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Theorem:
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Consider 𝑃 = 𝑃 + 𝐺
𝑆 is top-𝑙 subspace of 𝑃
𝑈 is the 𝑘-space containing 𝑃
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Then: sin 𝜃 𝑆, 𝑈 ≤
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𝜃
||𝐺||
𝜆𝑙 (𝑃)
Another way to see why we need to take directions with
sufficiently heavy singular values
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Additional issue: Conditioning
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After an iteration, the noise is not random anymore!
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non-captured points might be “biased” by capturing criterion
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Fix: estimate top PCA subspace from a small sample of
the data
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Might be purely due to analysis
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But does not sound like a bad idea in practice either
Performance of Iterative PCA
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Can prove there are 𝑂
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In each, we have NNS in ≤ 𝑘 dimensional space
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Overall query time: 𝑂
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Reduced to 𝑂
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𝑑 log 𝑛 iterations
1
𝜖𝑂 𝑘
⋅ 𝑑 ⋅ log 3/2 𝑛
𝑑 log 𝑛 instances of 𝑘-dimension NNS!
2nd Algorithm: PCA-tree
Closer to algorithms used in practice
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Find top PCA direction 𝑣
Partition into slabs ⊥ 𝑣
Snap points to ⊥ hyperplane
Recurse on each slice
≈ 𝜖/𝑘
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Query:
• follow all tree paths that
may contain 𝑝∗
2 algorithmic modifications
Find top PCA direction 𝑣
Partition into slabs ⊥ 𝑣
Snap points to ⊥ hyperplanes
Recurse on each slice
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Centering:
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Query:
• follow all tree paths that
may contain 𝑝∗
Need to use centered PCA (subtract average)
Otherwise errors from perturbations accumulate
Sparsification:
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Need to sparsify the set of points in each node of the tree
Otherwise can get a “dense” cluster:
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not enough variance in signal
lots of noise
Analysis
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An “extreme” version of Iterative PCA Algorithm:
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just use the top PCA direction: guaranteed to have signal !
Main lemma: the tree depth is ≤ 2𝑘
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because each discovered direction close to 𝑈
snapping: like orthogonalizing with respect to each one
cannot have too many such directions
𝑘 2𝑘
𝜖
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Query runtime: 𝑂
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Overall performs like 𝑂(𝑘 ⋅ log 𝑘)-dimensional NNS!
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Wrap-up
Why do data-aware projections outperform random projections ?
Algorithmic framework to study phenomenon?
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Here:
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Immediate questions:
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Model: “low-dimensional signal + large noise”
like NNS in low dimensional space
via “right” adaptation of PCA
Other, less-structured signal/noise models?
Algorithms with runtime dependent on spectrum?
Broader Q: Analysis that explains empirical success?
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