Affine-invariant Principal Components Charlie Brubaker and Santosh Vempala Georgia Tech School of...
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![Page 1: Affine-invariant Principal Components Charlie Brubaker and Santosh Vempala Georgia Tech School of Computer Science Algorithms and Randomness Center.](https://reader035.fdocuments.us/reader035/viewer/2022062421/56649d355503460f94a0cc3e/html5/thumbnails/1.jpg)
Affine-invariant Principal Components
Charlie Brubaker and Santosh Vempala
Georgia TechSchool of Computer Science
Algorithms and Randomness Center
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What is PCA?
“PCA is a mathematical tool for finding directions in which a distribution is stretched out.”
• Widely used in practice• Gives best-known results for some problems
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History• First discussed by Euler in a work on inertia of rigid bodies (1730).• Principal Axes identified as eigenvectors by Lagrange.• Power method for finding eigenvectors published in 1929, before computers• Ubiquitous in practice today:
Bioinformatics, Econometrics,
Data Mining, Computer Vision, ...
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Principal Components Analysis
For points a1…am in Rn, the principal components are orthogonal vectors v1…vn s.t. Vk = span{v1…vk} minimizes among all k-subspaces.
Like regression.
Computed via SVD.
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Singular Value Decomposition (SVD)
Real m x n matrix A can be decomposed as:
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PCA (continued)
• Example: for a Gaussian the principal components are the axes of the ellipsoidal level sets.
v1v2
• “top” principal components = where the data is “stretched out.”
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Why Use PCA?1. Reduces computation or space. Space goes from O(mn) to O(mk+nk).
--- Random Projection, Random Sampling
also reduce space requirement
2. Reveals interesting structure that is hidden in high dimension.
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Problem
• Learn a mixture of Gaussians
Classify unlabeled samples
Each component is a logconcave distribution (e.g., Gaussian).
Means, variances and mixing weights are unknown
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Distance-based Classification
“Points from the same component should be closer to each other than those from different components.”
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Mixture models
• Easy to unravel if components are far enough apart
• Impossible if components are too close
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Distance-based classificationHow far apart?
Thus, suffices to have
[Dasgupta ‘99][Dasgupta, Schulman ‘00][Arora, Kannan ‘01] (more general)
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PCA
• Project to span of top k principal components of the data
Replace A with
• Apply distance-based classification in this subspace
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Main idea
Subspace of top k principal components spans the means of all k Gaussians
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SVD in geometric terms
Rank 1 approximation is the projection to the line
through the origin that minimizes the sum of squared
distances.
Rank k approximation is projection k-dimensional subspace minimizing sum of squared distances.
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Why?
• Best line for 1 Gaussian?
- Line through the mean
• Best k-subspace for 1 Gaussian?
- Any k-subspace through the mean
• Best k-subspace for k Gaussians?
- The k-subspace through all k means!
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How general is this?
Theorem [V-Wang’02]. For any mixture of weakly isotropic distributions, the best k-subspace is the span of the means of the k components.
“weakly isotropic”: Covariance matrix = multiple of identity
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PCA
Projection to span of means gives
For spherical Gaussians,
Span(means) = PCA subspace of dim k.
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Sample SVD
• Sample SVD subspace is “close” to mixture’s SVD subspace.
• Doesn’t span means but is close to them.
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2 Gaussians in 20 Dimensions
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4 Gaussians in 49 Dimensions
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Mixtures of Logconcave Distributions
Theorem [Kannan-Salmasian-V ’04].
For any mixture of k distributions with SVD subspace V,
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Mixtures of Nonisotropic, Logconcave Distributions
Theorem [Kannan, Salmasian, V, ‘04].
The PCA subspace V is “close” to the span of the means, provided that means are well-separated.
where is the maximum directional variance.
Polynomial was improved by Achlioptas-McSherry.
Required separation:
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However,…
PCA collapses separable “pancakes”
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Limits of PCA
• Algorithm is not affine invariant.
• Any instance can be made bad by an affine transformation.
• Spherical Gaussians become parallel pancakes but remain separable.
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Parallel Pancakes
• Still separable, but previous algorithms don’t work.
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Separability
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Hyperplane Separability
• PCA is not affine-invariant.
• Is hyperplane separability sufficient to learn a mixture?
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Affine-invariant principal components?
• What is an affine-invariant property that distinguishes 1 Gaussian from 2 pancakes?
• Or a ball from a cylinder?
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Isotropic PCA
1. Make point set isotropic via an affine transformation.
2. Reweight points according to a spherically symmetric function f(|x|).
3. Return the 1st and 2nd moments of reweighted points.
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Isotropic PCA [BV’08]
• Goal: Go beyond 1st and 2nd moments to find “interesting” directions.
• Why? What if all 2nd moments are equal?
v?
v? v?v?
• This isotropy can always be achieved by an affine transformation.
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Ball vs Cylinder
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Algorithm
1. Make distribution isotropic.
2. Reweight points.
3. If mean shifts, partition along this direction. Recurse.
4. Otherwise, partition along top principle component. Recurse.
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Step1: Enforcing Isotropy
• Isotropy:a. Mean = 0 and b. Variance = 1 in every direction
• Step 1a: move the origin to the mean (translation).
• Step 1b: apply linear transformation
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Step 1: Enforcing Isotropy
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Step 1: Enforcing Isotropy
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Step 1: Enforcing Isotropy
• Turns every well-separated mixture into (almost) parallel pancakes, separable along the intermean direction.
• PCA no longer helps us!
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Algorithm
1. Make distribution isotropic.
2. Reweight points (using a Gaussian).
3. If mean shifts, partition along this direction. Recurse.
4. Otherwise, partition along top principle component. Recurse.
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Two parallel pancakes
• Isotropy pulls apart the components
• If one is heavier, then overall mean shifts along the separating direction
• If not, principal component is along the separating direction
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Steps 3 & 4: Illustrative Examples• Imbalanced Pancakes:
• Balanced Pancakes:Mean
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Step 3: Imbalanced Case
• Mean shifts toward heavier component
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Step 4: Balanced Case
• Mean doesn’t move by symmetry.• Top principle component is inter-mean direction
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Unraveling Gaussian Mixtures
Theorem [Brubaker-V. ’08]
The algorithm correctly classifies samples from two arbitrary Gaussians “separable by a hyperplane” with high probability.
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Original Data
• 40 dimensions, 8000 samples (subsampled for visualization)
• Means of (0,0) and (1,1).
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Random Projection
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PCA
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Isotropic PCA
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Results:k=2
• Theorem: For k=2, algorithm succeeds if there is some direction v such that:
(i.e., hyperplane separability.)
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Fisher Criterion• For a direction p,
intra-component variance along p
J(p) = ------------------------------------------------
total variance along p
• Overlap: Min J(p) over all directions p.(
small overlap => well-separated)
• Theorem: For k=2, algorithm suceeds if overlap is
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Results:k>2
• For k > 2, we need k-1 orthogonal directions with small overlap
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Fisher Criterion
J(S)= max intra-component variance within S
• Make F isotropic. For subspace S
• Overlap is affine-invariant.
• Overlap = Min J(S), S: k-1 dim subspace
• Theorem [BV ’08]: For k>2, the algorithm succeeds if the overlap is at most
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Original Data (k=3)
• 40 dimensions, 15000 samples (subsampled for visualization)
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Random Projection
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PCA
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Isotropic PCA
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Conclusion
• Most of this in a new book: “Spectral Algorithms,” with Ravi Kannan
• IsoPCA gives an affine-invariant clustering(independent of a model)
• What do Iso-PCA directions mean?
• Robust PCA (Brubaker 08; robust to small changes in point set); applied to noisy/best-fit mixtures.
• PCA for tensors?