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Fast Krylov Methods for N-Body Learning

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x i. x i. Xj’s. Xj’s. O( N 2 x Number of Iterations). O( NlogN x Number of Iterations). O( N 3 ). O( N 2 x Number of Iterations). True Manifold. Sampled data. Embedding of SNE. Embedding of SNE+IFGT. Fast Krylov Methods for N-Body Learning.

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O(N2 x Number of Iterations)

O(NlogN x Number of Iterations)

O(N3 )

O(N2 x Number of Iterations)

True Manifold

Sampled data

Embedding of SNE

Embedding of SNE+IFGT

Fast Krylov Methods for N-Body Learning

Maryam Mahdaviani, Nando de Freitas, Yang Wang and Dustin Lang, University of British Columbia

Experimental results

Step 2: Fast N-Body Methods

Gaussian Processes with Large Dimensional Features: Using GPs to predict the labels of 128-dimensional SIFT features for object class detection and localization. Thousands of features per image => BIG covariance matrix

Fast Multipole Methods (Greengard et al):

We present a family of low storage, fast algorithms for:

Spectral clustering

Dimensionality reduction

Gaussian processes

Kernel methods


Work in low dimensions but O(N)

Fast Gauss Transform (FGT)

Improved FGT (IFGT)

Dual Trees (Gray & Moore)

Spectral Clustering and Image Segmentation: A generalized eigenvalue problem.

The inverse problem (e.g. in GPs) and the eigenvalue problem (e.g. in spectral clustering) have a computational cost of O(N3) and storage requirement O(N2) in the number of data points N.

Step 1: Krylov subspace iteration

And the storage is now O(N) instead of O(N2)

Does it still work with step 2?


GMRES for solving Ax = b

But how do the errors accumulate over successive iterations?

Original IFGT Dual Tree Nystrom

Stochastic neighbor embedding: Projecting to low dimensions (Roweis and Hinton)

In the paper we prove that:

  • The deviation in residuals is upper bounded

Other similar Krylov Methods: Lanczos, Conjugate Gradients, MINRES

2) The orthogonality of the Krylov subspace can be preserved

The upper-bounds in terms of the measured residuals will enable us to design adaptive algorithms

Expensive Step:

Requiring solving two kernel estimates

Embedding on S-curve and Swiss-roll datasets