EE 290A: Generalized Principal Component Analysis Lecture 5: Generalized Principal Component...

EE 290A: Generalized Principal Component Analysis

Lecture 5: Generalized Principal Component Analysis

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GPCA: Problem definition Segmentation of multiple hyperplanes

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Recover subspaces from vanishing polynomial

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This Lecture

Segmentation of general subspace arrangements knowing the number of subspaces

Subspace segmentation without knowing the number of subspaces

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An Introductory Example

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Make use of the vanishing polynomials

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Recover Mixture Subspace Models

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Question: How to choose one representative point per subspace? (some loose answers)1. In noise-free case, randomly pick one.2. In noisy case, choose one close to the zero

set of vanishing polynomials. (How?)

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Summary

Using the vanishing polynomials, GPCA converts a CAE problem to a closed-form solution.

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Step 1: Fitting Polynomials

In general, when the dimensions of subspaces are mixed, the set of all K-th degree polynomials that vanish on A becomes more complicated.

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Polynomials may be dependent!

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With the closed-form solution, even when the sample data are noisy, if K and subspace dimensions are known, a complete list of linearly independent vanishing polynomials can be recovered from the (null space of) embedded data matrix!

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Step 2: Polynomial Differentiation

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Step 3: Sample Point Selection Given n sample points from K

subspaces, how to choose one point per subspace to evaluate the orthonormal basis for each subspace?

What is the notion of optimality in choosing the best sample when a set of vanishing polynomials is given (for any algebraic set)?

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In the case of segmenting hyperplanes?

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Draw a random line that does not pass the origin

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Lemma 3.9: For general arrangements We shall choose samples as close to the

zero set as possible (in the presence of noise)1. One shall avoid choosing points based on

P(x), as it is merely an algebraic error, not the geometric distance.

2. One shall discourage choosing points close to the intersection of two ore more subspaces, even when P(x)=0.

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Estimate the Rest (K-1) Subspaces Polynomial division

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GPCA without knowing K or d’s Determining K and d’s is

straightforward when subspaces are of equal dimension1. If d is known, project samples to (d+1)-

dim space. The problem becomes hyperplane segmentation.

2. If K is known, project samples to l-dim spaces, while l=1, 2, …, compute k-th order Veronese map until it drops rank.

3. If both K and d are unknown, try all the combinations

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GPCA without knowing K or d’s Determine arrangements of different

dimensions1. If data are noise-free, check the Hilbert

function table.

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2. When the data are noisy, apply GPCA recursively

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Please read Section 3.5 for the definition of Effective Dimension