Applications of Random Matrices in Spectral Computations and Machine Learning

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1 Applications of Random Matrices in Spectral Computations and Machine Learning Dimitris Achlioptas UC Santa Cruz

2 This talk Viewpoint: use randomness to transform the data

3 This talk Viewpoint: use randomness to transform the data Random Projections Fast Spectral Computations Sampling in Kernel PCA

4 The Setting

5 The Setting n d n d n d

6 The Setting n d n d n d

7 The Setting n d n d n d P Output: AP

8 The Johnson-Lindenstrauss lemma

9 The Johnson-Lindenstrauss lemma Algorithm: Projecting onto a random hyperplane (subspace) of dimension succeeds with probability

10 Applications Approximation algorithms [Charikar 02] Hardness of approximation [Trevisan 97] Learning mixtures of Gaussians [Arora, Kannan 01] Approximate nearest-neighbors [Kleinberg 97] Data-stream computations [Alon et al. 99, Indyk 00] Min-cost clustering [Schulman 00]. Information Retrieval (LSI) [Papadimitriou et al. 97]

11 How to pick a random hyperplane

12 How to pick a random hyperplane Take where the are independent random variables [Dasgupta Gupta 99] [Indyk Motwani 99] [Johnson Lindenstrauss 82]

13 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in

14 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in Each column is an unbiased, independent estimator of (via its squared inner product)

15 How to pick a random hyperplane Take where the are independent random variables Intuition: Each column of P points to a uniformly random direction in Each column is an unbiased, independent estimator of (via its squared inner product) is the average estimate (since we take the sum)

16 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated

17 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated Without orthonormalization:

18 How to pick a random hyperplane Take where the are independent random variables With orthonormalization: Estimators are equal Estimators are uncorrelated Without orthonormalization: Same thing!

19 Orthonormality: Take #1 Random vectors in high-dimensional Euclidean space are very nearly orthonormal.

20 Orthonormality: Take #1 Random vectors in high-dimensional Euclidean space are very nearly orthonormal. Do they have to be uniformly random? Is the Gaussian distribution magical?

21 JL with binary coins Take where the are independent random variables with

22 JL with binary coins Take where the are independent random variables with Benefits: Much faster in practice ± Only operations (no ) Fewer random bits Derandomization Slightly smaller(!) k

23 JL with binary coins Take where the are independent random variables with Preprocessing with a randomized FFT [Ailon, Chazelle 06]

24 Let s at least look at the data

25 The Setting n d n d n d P Output: AP

26 Spectral Norm: Low Rank Approximations

27 Low Rank Approximations Spectral Norm: Frobenius Norm:

28 Low Rank Approximations Spectral Norm: Frobenius Norm:

29 Low Rank Approximations Spectral Norm: Frobenius Norm:

30 Low Rank Approximations Spectral Norm: Frobenius Norm:

31 Start with a random How to compute A k

32 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x:

33 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x: Synthesize a new candidate by combining the rows of A according to their enthusiasm for x: (This is power iteration on. Also known as PCA.)

34 Start with a random Repeat until fixpoint How to compute A k Have each row in A vote for x: Synthesize a new candidate by combining the rows of A according to their enthusiasm for x: (This is power iteration on. Also known as PCA.) Project A on subspace orthogonal to x and repeat

35 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise

36 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A.

37 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A. Intuition: The perturbation vectors are nearly orthogonal

38 PCA for Denoising Assume that we perturb the entries of a matrix A by adding independent Gaussian noise Claim: If σ is not too big then the optimal projections for are close to those for A. Intuition: The perturbation vectors are nearly orthogonal No small subspace accommodates many of them

39 Rigorously Lemma: For any matrices A and

40 Rigorously Lemma: For any matrices A and Perspective: For any fixed x we have w.h.p.

41 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix

42 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix Computation-friendly noise:

43 Two new ideas A rigorous criterion for choosing k: Stop when A-A k has as much structure as a random matrix Computation-friendly noise: Inject data-dependent noise

44 Quantization

45 Quantization

46 Quantization

47 Quantization

48 Sparsification

49 Sparsification

50 Sparsification

51 Sparsification

52 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint

53 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data

54 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data L2 error can be quadratically better than Nystrom

55 Orthonormality: Take #2 Matrices with independent, 0-mean entries are white noise matrices

56 A scalar analogue

57 A scalar analogue Crude quantization at extremely high rate

58 A scalar analogue Crude quantization at extremely high rate + low-pass filter

59 A scalar analogue Crude quantization at extremely high rate + low-pass filter

60 A scalar analogue Crude quantization at extremely high rate + low-pass filter = 1-bit CD player ( Bitstream )

61 Accelerating spectral computations By injecting sparsification/quantization noise we can accelerate spectral computations: Fewer/simpler arithmetic operations Reduced memory footprint Amount of noise that can be tolerated increases with redundancy in data L2 error can be quadratically better than Nystrom Useful even for exact computations

62 Accelerating exact computations

63 Kernels

64 Kernels & Support Vector Machines Red and Blue pointclouds Which linear separator (hyperplane)? Maximum margin Optimal can be expressed by inner products with (a few) data points

65 Not always linearly separable

66 Population density

67 Kernel PCA

68 Kernel PCA We can also compute the SVD via the spectrum of n d n n d n

69 Kernel PCA We can also compute the SVD via the spectrum of n d n n d Each entry in AA T n is the inner product of two inputs

70 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function

71 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function Work implicitly in high-dimensional space

72 Kernel PCA We can also compute the SVD via the spectrum of n n d n d n Each entry in AA T is the inner product of two inputs Replace inner product with a kernel function Work implicitly in high-dimensional space Good linear separators in that space

73 From linear to non-linear PCA X-Y p kernel illustrates how the contours of the first 2 components change from straight lines for p=2 to non-linea for p=1.5, 1 and 0.5. From Schölkopf and Smola, Learning with kernels, MIT 2002

74 Kernel PCA with Gaussian Kernel KPCA with Gaussian kernels. The contours follow the cluster densities! First two kernel PCs separate the data nicely. Linear PCA has only 2 components, but kernel PCA has more, since the space dimension is usually large (in this case infinite)

75 Good News: KPCA in brief - Work directly with non-vectorial inputs

76 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03]

77 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03] Bad News: n 2 kernel evaluations are too many.

78 KPCA in brief Good News: - Work directly with non-vectorial inputs - Very powerful: e.g. LLE, Isomap, Laplacian Eigenmaps [Ham et al. 03] Bad News: n 2 kernel evaluations are too many. Good News: [Shaw-Taylor et al. 03] good generalization rapid spectral decay

79 So, it s enough to sample n n d n d n In practice, 1% of the data is more than enough In theory, we can go down to n polylog(n)

80 Important Features are Preserved

81 Open Problems How general is this stability under noise? For example, does it hold for Support Vector Machines? When can we prove such stability in a black-box fashion, i.e. as with matrices? Can we exploit if for data privacy?

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