Holder Homeomorphisms and Approximate Nearest Neighbors Ilya RazenshteynMicrosoft
Holder Homeomorphisms and Approximate Nearest Neighbors Ilya Razenshteyn(Microsoft Research Redmond) joint with Alexandr Andoni, Assaf Naor, Aleksandar Nikolov, Erik Waingarten
How to measure distances? Metric spaces In some basis, the norm is symmetric wrt coordinate permutationsand sign flips Normed spaces Symmetric normed spaces edit(ELEPHANT, RELEVANT) = 3 Euclidean spaces
Near Neighbor Search (NNS) •
Possible solutions •
Approximate Near Neighbor Search (ANN) •
Efficient data structures
![In practice Glo. Ve word embeddings [Pennington, Socher, Manning 2014] • Nearest neighbor](http://slidetodoc.com/presentation_image_h/04996bcdb4124adf4c97182928e0985b/image-7.jpg "In practice Glo. Ve word embeddings [Pennington, Socher, Manning 2014] • Nearest neighbor")
In practice Glo. Ve word embeddings [Pennington, Socher, Manning 2014] • Nearest neighbor
![For more information, see… • Theses: [Andoni 2009], [R 2017] • Surveys: [Andoni, Indyk](http://slidetodoc.com/presentation_image_h/04996bcdb4124adf4c97182928e0985b/image-9.jpg "For more information, see… • Theses: [Andoni 2009], [R 2017] • Surveys: [Andoni, Indyk")
For more information, see… • Theses: [Andoni 2009], [R 2017] • Surveys: [Andoni, Indyk 2017], [Andoni, Indyk, R 2018] (accompanying Indyk’s ICM 2018 talk)
How does the geometry of a metric/normed space affect the complexity of the ANN problem?
Summary •
Our results Parameter Approximation Algorithm 1 Algorithm 2 Space Query time Preprocessing time This talk: Algorithm 1
Approach at a high level •
Reducing ANN to the bounded case •
Mazur map •
Uniform convexity • not uniformly convex
Complex interpolation •
Why is complex interpolation nice? •
![Complex interpolation in TCS • [Blais, Canonne, Gur 2017] • [Garg, Lee, Song, Shrivastava](http://slidetodoc.com/presentation_image_h/04996bcdb4124adf4c97182928e0985b/image-21.jpg "Complex interpolation in TCS • [Blais, Canonne, Gur 2017] • [Garg, Lee, Song, Shrivastava")
Complex interpolation in TCS • [Blais, Canonne, Gur 2017] • [Garg, Lee, Song, Shrivastava 2018]
Analytic form of John’s theorem •
Daher’s theorem •
“Construction” •
Making interpolation algorithmic •
How to use Daher’s theorem for ANN? •
Unconditional spaces •
Better approximation? •
![Overview of the algorithm Daher’s embedding [Matousek 1996] Fixed-point argument Sparse cuts in embedded](http://slidetodoc.com/presentation_image_h/04996bcdb4124adf4c97182928e0985b/image-29.jpg "Overview of the algorithm Daher’s embedding [Matousek 1996] Fixed-point argument Sparse cuts in embedded")
Overview of the algorithm Daher’s embedding [Matousek 1996] Fixed-point argument Sparse cuts in embedded graphs Minimax “Data-dependent” random space partitions Standard ANN data structures
Sparse cuts in embedded graph •
Nice cuts •
Conclusions and open problems • Questions?
- Slides: 32