Download presentation

Presentation is loading. Please wait.

Published byEdgar Surridge Modified over 2 years ago

1
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

2
Nearest Neighbor Search (NNS)

3
Motivation Generic setup: Points model objects (e.g. images) Distance models (dis)similarity measure Application areas: machine learning: k-NN rule signal processing, vector quantization, bioinformatics, etc… Distance can be: Hamming, Euclidean, edit distance, earth-mover distance, … 000000 011100 010100 000100 010100 011111 000000 001100 000100 110100 111111

4
Curse of Dimensionality AlgorithmQuery timeSpace Full indexing No indexing – linear scan

5
Approximate NNS

6
NNS algorithms 6 It’s all about space partitions ! Low-dimensional [Arya-Mount’93], [Clarkson’94], [Arya-Mount- Netanyahu-Silverman-We’98], [Kleinberg’97], [HarPeled’02],[Arya-Fonseca-Mount’11],… High-dimensional [Indyk-Motwani’98], [Kushilevitz-Ostrovsky- Rabani’98], [Indyk’98, ‘01], [Gionis-Indyk- Motwani’99], [Charikar’02], [Datar-Immorlica- Indyk-Mirrokni’04], [Chakrabarti-Regev’04], [Panigrahy’06], [Ailon-Chazelle’06], [Andoni- Indyk’06], [Andoni-Indyk-Nguyen- Razenshteyn’14], [Andoni-Razenshteyn’15]

7
Low-dimensional 7

8
High-dimensional 8

9
Practice Data-aware partitions 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-Ross- Lin’11] 9

10
Practice vs Theory 10 Data-aware projections often outperform (vanilla) random-projection methods But no guarantees (correctness or performance) JL generally optimal [Alon’03, Jayram-Woodruff’11] Even for some NNS setups! [Andoni-Indyk-Patrascu’06] Why do data-aware projections outperform random projections ? Algorithmic framework to study phenomenon?

11
Plan for the rest 11 Model Two spectral algorithms Conclusion

12
Our model 12

13
Model properties 13

14
Algorithms via PCA 14

15
15

16
PCA under noise fails 16

17
1 st Algorithm: intuition 17 Extract “well-captured points” points with signal mostly inside top PCA space should work for large fraction of points Iterate on the rest

18
Iterative PCA 18

19
Simpler model 19

20
Analysis of general model 20

21
Closeness of subspaces ? 21

22
Wedin’s sin-theta theorem 22

23
Additional issue: Conditioning 23 After an iteration, the noise is not random anymore! non-captured points might be “biased” by capturing criterion Fix: estimate top PCA subspace from a small sample of the data Might be purely due to analysis But does not sound like a bad idea in practice either

24
Performance of Iterative PCA 24

25
2 nd Algorithm: PCA-tree 25 Closer to algorithms used in practice

26
2 algorithmic modifications 26 Centering: Need to use centered PCA (subtract average) Otherwise errors from perturbations accumulate Sparsification: Need to sparsify the set of points in each node of the tree Otherwise can get a “dense” cluster: not enough variance in signal lots of noise

27
Analysis 27

28
Wrap-up 28 Here: Model: “low-dimensional signal + large noise” like NNS in low dimensional space via “right” adaptation of PCA Immediate questions: Other, less-structured signal/noise models? Algorithms with runtime dependent on spectrum? Broader Q: Analysis that explains empirical success? Why do data-aware projections outperform random projections ? Algorithmic framework to study phenomenon?

Similar presentations

OK

Efficient Image Search and Retrieval using Compact Binary Codes Rob Fergus (NYU) Jon Barron (NYU/UC Berkeley) Antonio Torralba (MIT) Yair Weiss (Hebrew.

Efficient Image Search and Retrieval using Compact Binary Codes Rob Fergus (NYU) Jon Barron (NYU/UC Berkeley) Antonio Torralba (MIT) Yair Weiss (Hebrew.

© 2017 SlidePlayer.com Inc.

All rights reserved.

Ads by Google

Ppt on how google search engine works Evs ppt on pollution Ppt on south african culture pictures Ppt on electricity from wastewater to drinking Ppt on energy crisis in india Ppt on poverty and unemployment in india Ppt on virtual file system Ppt on scope of wind farming in india Ppt on mammals and egg laying animals Ppt on sectors of indian economy