Convergence of Eigenspaces in Kernel Principal Component Analysis
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1 Convergence of Eigenspaces in Kernel Principal Component Analysis Shixin Wang Advanced machine learning April 19, 2016 Shixin Wang Convergence of Eigenspaces April 19, / 18
2 Outline 1 Motivation 2 Result 3 Summary Shixin Wang Convergence of Eigenspaces April 19, / 18
3 Motivation Outline 1 Motivation 2 Result 3 Summary Shixin Wang Convergence of Eigenspaces April 19, / 18
4 Motivation Kernel Principal Component Analysis PCA: To find the most relevant lower-dimension projection of data. KPCA: Extend PCA to data mapped in a kernel feature space. Assumption: Target dimensionality of the projected data is fixed: D. Objective: To find the span S D of the first D eigenvectors of the covariance matrix. Shixin Wang Convergence of Eigenspaces April 19, / 18
5 Motivation How reliable is our estimation? Problem: The true covariance matrix is no known and has to be estimated from the available data. Question: How reliable is the D-eigenspace ŜD of the empirical covariance matrix compared to the D-eigenspace S D of the true covariance matrix? 1 The average reconstruction error of ŜD converges to the reconstruction error of S D. (Blanchard et al 2004, Shawe-Taylor 2005) 2 But this does not mean ŜD converges to S D! since different subspaces can have a very similar reconstruction error. Shixin Wang Convergence of Eigenspaces April 19, / 18
6 Motivation Why analyze the behavior of ŜD? PCA or KPCA is often used merely as a preprocessing step, so the behavior of ŜD is more important than just reconstruction error. We want to use u 1 in the future, so we need to show that ˆx i converges to the true ones, rather than only norm of x i ˆx i. Shixin Wang Convergence of Eigenspaces April 19, / 18
7 Outline 1 Motivation 2 Result 3 Summary Shixin Wang Convergence of Eigenspaces April 19, / 18
8 Notation X: interest variable taking values in measurable space X, with distribution P. φ(x) = k(x, ): feature mapping of x X into a reproduction kernel Hilbert space H D: target dimensionality of projected data C: covariance matrix of variable φ(x) λ 1 > λ 2 >...: simple nonzero eigenvalues of C ϕ 1, ϕ 2...: associated eigenvectors C n : empirical covariance matrix Shixin Wang Convergence of Eigenspaces April 19, / 18
9 Notation S D = span{ϕ 1,...ϕ D }: D-dimensional subspace of H such that the projection of φ(x) on S D has maximum average squared norm Ŝ D : subspace spanned by the first D eigenvectors of C n. P SD : the orthogonal projectors of X on S D P ŜD : the orthogonal projectors of X on ŜD Shixin Wang Convergence of Eigenspaces April 19, / 18
10 First Bound Tow steps to obtain the first bound: 1 A non-asympotic bound on the (Hilbert-Schmidt) norm of the difference between the empirical and the true covariance operators 2 An operator perturbation result bounding the difference between spectral projectors of two operators by the norm of their difference. Shixin Wang Convergence of Eigenspaces April 19, / 18
11 Lemma 1 Lemma Supposing that sup x X k(x, x) M, with probability greater than 1 e ξ, C n C 2M (1 + n ξ 2 ) Shixin Wang Convergence of Eigenspaces April 19, / 18
12 Proof of Lemma 1 Proof. Theorem (Bounded Differences Inequality) Suppose that X 1,..., X n X are independent, and f : X n R, Let c 1,..., c n satisfy sup f (x 1,..., x n ) f (x 1,..., x i 1, x i, x i+1,..., x n ) c i, x 1,...,x n,x i i = 1,..., n Then, P(f E[f ] t) exp( 2t2 n i=1 c2 i ) Shixin Wang Convergence of Eigenspaces April 19, / 18
13 Proof of Lemma 1 Proof. C n C = 1 n n C Xi E[C X ] i=1 C X = φ(x) φ(x) = k(x, X) M Here c i = 2M n, t = 2M ξ, then we get 2n P C n C E[ C n C ] 2M exp( 2t2 n i=1 c2 i ξ 2n ) = exp( 4M2 ξ n 4M 2 /n ) = e ξ Shixin Wang Convergence of Eigenspaces April 19, / 18
14 Theorem 2 Shixin Wang Convergence of Eigenspaces April 19, / 18
15 Improved bound Shixin Wang Convergence of Eigenspaces April 19, / 18
16 Improved bound Shixin Wang Convergence of Eigenspaces April 19, / 18
17 Summary Outline 1 Motivation 2 Result 3 Summary Shixin Wang Convergence of Eigenspaces April 19, / 18
18 Summary Summary Provide finite sample size confidence bounds of the eigenspaces of Kernel-PCA Prove a bound in which the complexity factor for estimating the eigenspace S D by its empirical counterpart depends only on the inverse gap between the D-th and D + 1-th eigenvalues Restriction: Assume that the eigenvalues to be simple. Shixin Wang Convergence of Eigenspaces April 19, / 18
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