Rate-Optimal Perturbation Bounds for Singular Subspaces with Applications to High-Dimensional Statistics Anru Zhang Department of Statistics University of Wisconsin – Madison
Introduction Introduction • Focus: singular value decomposition (SVD) X = U · Σ 1 · V ⊤ + U ⊥ · Σ 2 · V ⊤ ⊥ • Due to perturbation, ˆ X = X + Z , SVD is altered to V ⊤ + ˆ U · ˆ U ⊥ · ˆ V ⊤ X = ˆ ˆ Σ 1 · ˆ Σ 2 · ˆ ⊥ . Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 2
Introduction Introduction close ˆ V to V (or ˆ small perturbation + large signal → U and U ) • Problem: Perturbation Bounds on Singular Subspaces ◮ How to quantify the difference between ˆ V and V (or ˆ U and U )? ◮ Is there any upper bounds for the difference? ◮ Are U and ˆ U , V and ˆ V equally different? • Motivation : spectral method , which has been used in a wide range of modern high-dimensional statistical problems, utilize this property. Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 3
Introduction Application 1: Low-rank Matrix Denoising ˆ X = X + Z , Z iid ∼ sub-Gaussian (0 , σ 2 ) X is approximately rank-r , • Target: X , U or V . • Specific applications ◮ Magnetic Resonance Imaging (MRI) (Cand` es, Sing-Long and Trzasko, 2012); ◮ Relaxometry (Bydder and Du, 2006) U , ˆ ˆ V , the first r singular vectors of ˆ • Natural estimators for U , V : X . • Q: How do ˆ U , ˆ V perform, respectively? Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 4
Introduction Application 2: High-dimensional Clustering • Observe n points X 1 , . . . , X n ∈ R p , p ≥ n . • Each point belongs to one of two classes (Jin, Ke and Wang, 2015) iid X i = µ l i + ε i ∈ R p , ∼ sub-Gaussian (0 , σ 2 I p ) , i = 1 , . . . , n , ε i µ ∈ R p is the mean . l i ∈ {− 1 , 1 } are labels ; • Goal: recover labels l . Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 5
Introduction Other Applications • In addition, spectral method is often applied to find a “warm start” for more delicate iterative algorithms. ◮ phase retrieval (Cai, Li and Ma, 2016) ◮ matrix completion (Sun and Luo, 2015) ◮ community detection (Jin, 2015) Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 6
Introduction Other Applications Other applications of spectral methods include • community detection • matrix completion • principle component analysis • canonical correlation analysis • ... Specific practices include • collaborative filtering (the Netflix problem) • multi-task learning • system identification • sensor localization • ... Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 7
Perturbation Bounds for Singular Subspaces Problem Formulation X = U · Σ 1 · V ⊤ + U ⊥ · Σ 2 · V ⊤ ⊥ V ⊤ + ˆ U · ˆ U ⊥ · ˆ V ⊤ ˆ X = ˆ ˆ Σ 1 · ˆ Σ 2 · ˆ X = X + Z , ⊥ • Target: Measure the difference between ˆ V and V ( ˆ U and U ) Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 8
Perturbation Bounds for Singular Subspaces sin Θ Distance of Singular Sub-spaces Definition of sin Θ distances : • Suppose V ⊤ ˆ V have singular values σ 1 ≥ σ 2 ≥ · · · ≥ σ r ≥ 0 . • Define the sine principle angles as � � sin Θ ( V , ˆ 1 − σ 2 1 − σ 2 V ) = diag ( r ) . 1 , . . . , • Quantitative measure of distance: � sin Θ ( ˆ V , V ) � and � sin Θ ( ˆ V , V ) � F . Good properties : • Triangular inequality → indeed a distance; • Many other distances are equivalent → convenient to use. Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 9
Perturbation Bounds for Singular Subspaces Classic Results of Perturbation Bounds • The Perturbation bounds: develop the upper bound for � sin Θ ( V , ˆ � sin Θ ( U , ˆ � sin Θ ( V , ˆ � sin Θ ( U , ˆ V ) � , U ) � , V ) � F , U ) � F . • This problem has been widely studied in the literature (Davis and Kahan, 1970; Wedin, 1972; Weyl, 1912; Stewart, 1991, 2006; Yu et al., 2015; Fan, Wang and Zhong, 2016). • Classical tools: ◮ Davis and Kahan (1970): eigenvectors of symmetric matrices; ◮ Wedin (1972): singular vectors for asymmetric matrices. Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 10
Perturbation Bounds for Singular Subspaces Classic Result: Wedin’s sin Θ Theorem X = U · Σ 1 · V ⊤ + U ⊥ · Σ 2 · V ⊤ ⊥ V ⊤ + ˆ X = ˆ ˆ U · ˆ Σ 1 · ˆ U ⊥ · ˆ Σ 2 · ˆ V ⊤ ⊥ Wedin’s sin Θ Theorem (1972) states that if σ min (ˆ Σ 1 ) − σ max ( Σ 2 ) = δ > 0 , � � � Z ˆ V � , � ˆ U ⊤ Z � max � � � sin Θ ( V , ˆ V ) � , � sin Θ ( U , ˆ U ) � ≤ max . δ • joint upper bound for both ˆ U and ˆ V ; • may be sub-optimal. Figure: Intuitively, estimating V is more difficult than U for the matrix above. Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 11
Perturbation Bounds for Singular Subspaces Unilateral Perturbation Bound • Decompose � � V ⊤ � � Z 11 � Z 12 � Z = U U ⊥ . V ⊤ Z 21 Z 22 ⊥ Z 11 = U ⊤ ZV , Z 21 = U ⊥ ZV ⊤ , Z 12 = U ⊤ ZV ⊥ , Z 22 = U ⊥ ZV ⊥ . Define z ij : = � Z ij � for i , j = 1 , 2 . Theorem (Unilateral Perturbation Bound (Cai & Z. 2016)) Denote α : = σ min ( U ⊤ ˆ XV ⊥ ) . If α 2 > β 2 + z 2 ⊥ ˆ XV ) , β : = σ max ( U ⊤ 12 ∧ z 2 21 , then α z 12 + β z 21 � sin Θ ( V , ˆ V ) � ≤ ∧ 1 , α 2 − β 2 − z 2 21 ∧ z 2 12 α z 21 + β z 12 � sin Θ ( U , ˆ U ) � ≤ ∧ 1 . α 2 − β 2 − z 2 21 ∧ z 2 12 Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 12
Perturbation Bounds for Singular Subspaces Remark • Since α > β , α z 12 + β z 21 α z 21 + β z 12 if z 12 > z 21 , > , α 2 − β 2 − z 2 α 2 − β 2 − z 2 21 ∧ z 2 21 ∧ z 2 12 12 vice versa. • When α ≫ max( β, � Z � ) , the upper bound is approximately V ) � ≤ z 12 U ) � ≤ z 21 � sin Θ ( V , ˆ � sin Θ ( U , ˆ α , α . In contrast, Wedin’s sin Θ law only leads to V ) � ≤ � Z � U ) � ≤ � Z � � sin Θ ( V , ˆ � sin Θ ( U , ˆ α , α . • The upper bound in Frobenius norm sin Θ norm can be derived similarly. Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 13
Perturbation Bounds for Singular Subspaces Idea Behind � I r � � I r � . Let us take a look at ˆ Assume U = , V = X . 0 0 • When estimating U , z 12 becomes “signal” while z 21 becomes “noise.” • When estimating V , z 12 becomes “noise” while z 21 becomes “signal.” Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 14
Perturbation Bounds for Singular Subspaces Lower Bound Theorem (Perturbation Lower Bound) Define the class of p 1 × p 2 rank- r matrices and perturbations, � F r ,α,β, z 21 , z 12 = ( X , Z ) : rank ( X ) = r , � σ min ( U ⊺ ˆ XV ) ≥ α, � Z 22 � ≤ β, � Z 12 � ≤ z 12 , � Z 21 � ≤ z 21 . Provided that α 2 > β 2 + z 2 21 , r < p 1 ∧ p 2 12 + z 2 , 2 1 α z 12 + β z 21 � � � sin Θ ( V , ˜ � ≥ inf sup V ) ∧ 1 , √ � � α 2 − β 2 − z 2 12 ∧ z 2 ˜ 2 10 V ( X , Z ) ∈F α,β, z 21 , z 12 21 1 α z 21 + β z 12 � � � sin Θ ( U , ˜ � ≥ inf sup U ) √ ∧ 1 . � � α 2 − β 2 − z 2 12 ∧ z 2 ˜ 2 10 U ( X , Z ) ∈F α,β, z 21 , z 12 21 Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 15
Applications Matrix Denoising Application: Matrix Denoising ˆ X = X + Z , Z iid X is rank-r , ∼ sub-Gaussian (0 , 1) • Target: U or V . • Natural estimators for U , V : ˆ U , ˆ V , the first r singular vectors of ˆ X . • Q: How do ˆ U , ˆ V perform, respectively? Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 16
Applications Matrix Denoising • The r -th singular value of X , σ r ( X ) , is a good characterization for the difficulty of this problem. • Applying the perturbation bound, we obtain Theorem Suppose X = U · Σ · V ⊤ ∈ R p 1 × p 2 is of rank- r . Then � 2 ≤ C ( p 2 σ 2 r ( X ) + p 1 p 2 ) � � � sin Θ ( V , ˆ E V ) ∧ 1 , � � σ 4 r ( X ) � 2 ≤ C ( p 1 σ 2 r ( X ) + p 1 p 2 ) � � � sin Θ ( U , ˆ ∧ 1 . E U ) � � σ 4 r ( X ) Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 17
Applications Matrix Denoising Define the following class of low-rank matrices F r , t = � X ∈ R p 1 × p 2 : rank ( X ) = r , σ r ( X ) ≥ t � . Theorem (Lower Bound) If r ≤ p 1 16 ∧ p 2 2 , then � p 2 t 2 + p 1 p 2 � V ) � 2 ≥ c E � sin Θ ( V , ˜ inf sup ∧ 1 , t 4 ˜ V X ∈F r , t � p 1 t 2 + p 1 p 2 � U ) � 2 ≥ c E � sin Θ ( U , ˜ inf sup ∧ 1 . t 4 ˜ V X ∈F r , t To sum up, � p 2 t 2 + p 1 p 2 � V ) � 2 ≍ E � sin Θ ( V , ˜ inf sup ∧ 1 , t 4 ˜ V X ∈F r , t � p 1 t 2 + p 1 p 2 � U ) � 2 ≍ E � sin Θ ( U , ˜ inf sup ∧ 1 . t 4 ˜ V X ∈F r , t Anru Zhang (UW-Madison) Perturbation Bounds for Singular Subspaces 18
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