Invariant Subspace Perturbations or: How I Learned to Stop Worrying and Love Eigenvectors

Transcription

1 Invariant Subspace Perturbations or: How I Learned to Stop Worrying and Love Eigenvectors Alexander Cloninger Norbert Wiener Center Department of Mathematics University of Maryland, College Park

2 Outline Introduction to Eigenvector Perturbation 1 Introduction to Eigenvector Perturbation 2 3

3 Outline Introduction to Eigenvector Perturbation 1 Introduction to Eigenvector Perturbation 2 3

4 Error in A Matrix Eigenvector decomposition is ubiquitous in mathematics Principle Component Analysis Quantum States Fourier Analysis Spectral Graph Theory Most Common Problem in Mathematics Find (λ, v) such that Av = λv One imagines computers made this problem trivial [U, S] = eig(a) Question: What happens to eigenpairs if matrix has tiny errors à = A + E

5 Error in A Matrix Answer for eigenvalues: You re fine and they behave rather continuously Weyl s Inequality Answer for eigenvectors: Problem!

6 Eigenvector Perturbations Eigenvectors under perturbation require careful treatment Dependent on separation of spectrum Example ( ) ( ) 1 ɛ Let A = = σ(a) = {1 ɛ, 1 + ɛ}, V = ɛ 0 1 ( ) ( ) 1 ɛ 1 1 Let à = = ɛ 1 σ(ã) = {1 ɛ, 1 + ɛ}, Ṽ =. 1 1 V and Ṽ are as far apart as possible

7 Eigenvector Perturbations Geometrically Problem is that image of A is rotationally symmetric

8 Stable Eigenvector Geometrically Lack of symmetry allows for robust perturbations

9 Eigenvector Perturbations Separation of spectrum creates stable perturbations Example ( ) 1 0 Let B = ( 0 2 ) Let B 1 ɛ = ɛ 2 = σ(b) = {1, 2}, V = = σ( B) = { ɛ 2 Ṽ = ( ( ) , ɛ 2 2 }, ) for ɛ =.1 Rotation between V and Ṽ is 5

10 Outline Introduction to Eigenvector Perturbation 1 Introduction to Eigenvector Perturbation 2 3

11 Eigenvalue Separation Consider symmetric matrices A, E R n n Eigenvectors of A + E depend on separation of spectrum σ(a) Definition (Separation of Spectrum) The separation of the spectrum σ(a) at λ is sep(λ, σ(a) \ λ) = min{ λ γ : γ σ(a) \ λ}

12 Invariant Subspace Perturbations I Theorem (Davis, 1963) Let A, E C n n be Hermetian. Let (λ, x) be an eigenpair of A such that Let sep(λ, σ(a) \ λ) = δ. P be a spectral projector of A such that Px = x P be the corresponding spectral projector of A + E, and P be the orthogonal complement P z = z P z. Then if E ɛ δ/2, P P ɛ δ ɛ.

13 Invariant Subspace Perturbations II Side Note on Advisers:

14 Similar Perturbation Theorems Davis, Kahan (1970): Clustered Subspaces are Preserved Stewart (1973): Take Direction of Error Into Account

15 Outline Introduction to Eigenvector Perturbation 1 Introduction to Eigenvector Perturbation 2 3

16 Concentration As Opposed to Angle Similarity Previous theory defined similarity by angle Θ[V, Ṽ ] Not only way to consider similarity Can also consider eigenvector localization Important when: σ(a) has high density in interval A is adjacency matrix for network graph Plot of Eigenvectors for A R λ 20 = λ 25 = λ 30 = λ 31 = λ 32 =

17 Concentration Pertrubation Plot of Eigenvectors for A, Ã R , Ã = A + E v 25 v 31 ev 25 ev 31

18 Eigenvector Localization Theorem (C., 2014) Let A R n n be symmetric with eigendecomposition A = V ΣV. Let (λ i, v i ) be an eigenpair of A. Assume Then Partition V = [V 1, V 2, v i, V 3, V 4 ] where V 2, V 3 R n s, ordered by λ 1... λ n C {1,..., n} such that supp(v i ) C and supp(v j ) C where v j is a column of V 2, V 3. Let ( λ, x) an eigenvector of the perturbed matrix à = A + E, where x = [x 1,..., x n ]. j C c x j 2 ( λ λ i )x Ex 2 2 min(λ i λ i s, λ i+s λ i ) 2.

19 Conclusions Eigenvector perturbation depends on inverse of spectrum separation Led to Nobel Prize in Physics for particle localization (Anderson 1977) Concentration / localization is lesser restriction on eigenvectors Depends on cluster of vectors concentrated in similar area Applications in eigenstate localization on data-dependent graphs