Spatio–spectral limiting on Boolean cubes Jubilee of Fourier Analysis and Applications, NWC at UMD, 2019 joint work with Jeff Hogan Spatio–spectral limiting
Overview 1. Review: Time and band limiting: on R , Z and Z N 2. Spatio-spectral limiting on graphs: definitions 3. Hypercube graphs 4. Results 5. Adjacency maps and invariant subspaces 6. Matrix reduction of spatio–spectral limiting operator 7. Numerical aspects 8. Potential extensions Spatio–spectral limiting
Time and band limiting on R : The 1960s Bell Labs Theory � R f ( t ) e − 2 π it ξ dt Fourier transform: � f ( ξ ) = Bandlimiting: P Ω f ( x ) = ( � f ✶ [ − Ω / 2 , Ω / 2] ) ∨ ( x ) Time limiting: ( Q T f )( x ) = ✶ [ − T , T ] ( x ) f ( x ) Spatio–spectral limiting
Bell Labs theory: basic questions 1. What are the eigenfunctions of P Ω Q T ? 2. What is the distribution of eigenvalues of P Ω Q T Spatio–spectral limiting
Eigenvalue distribution: Approximately 2Ω T − O (log(2Ω T )) eigenvalues close to one Plunge region of width proportional to 2Ω T Exponential decay of remaining eigenvalues Eigenvalues for 1025 points, normalized area of 64 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 0 20 40 60 80 100 120 Spatio–spectral limiting
Eigenfunctions: The lucky accident 1 P Ω Q T commutes with (4 T 2 − t 2 ) d 2 dt 2 − 2 t d dt − Ω 2 t 2 . ( PDO ) Eigenfunctions: Prolate Spheroidal Wave Functions (PSWFs) Methods to compute PSWFs based on PDO 1 S. Slepian, Some comments on Fourier analysis, uncertainty and modeling, SIAM Review, 25, 379–393 1983 Spatio–spectral limiting
2.5 2 1.5 1 0.5 0 −0.5 −1 −1.5 −2 −2.5 −1 −0.8 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 Figure: ϕ n , n = 0 , 3 , 10 , c = π T Ω / 2 = 5 Spatio–spectral limiting
� Finite dimensional analogue: cycle � ◦ � ◦ · · · ◦ � ◦ � ◦ ◦ ◦ ◦ Discrete setting Z ↔ T : Slepian, (1978) DPSS Finite Z N setting: Gr¨ unbaum (1981), others Results analogous to continuous setting Zhu et al 2018: Non-asymptotic bound on plunge region 2 Many other developments in time and band limiting since 2000 2 Z. Zhu, S. Karnik, M. A. Davenport, J. Romberg, and M. B. Wakin, The Eigenvalue Distribution of Discrete Periodic Time-Frequency Limiting Operators, IEEE Signal Process. Lett,, 25 , 95–99, 2018. Spatio–spectral limiting
� Hypercubes: N = 5 00000 10000 01000 00100 00010 00001 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 11110 11101 11011 10111 01111 11111 VS � ◦ � ◦ · · · ◦ � ◦ � ◦ ◦ ◦ ◦ Spatio–spectral limiting
Graphs and Spatio–spectral limiting Unnormalized Graph Laplacian and Fourier transform G = ( V , E ) � f : V → R , Lf ( v ) = f ( v ) − f ( w ) w ∼ v L = D − A D : degree of each vertex A : adjacency map (undirected) Graph Fourier transform ϕ n : eigenvectors of L . ˆ f ( λ ℓ ) = � f , ϕ ℓ � Analogue of Q T : truncation to path neighborhood of a vertex Analogues of P Ω : truncation to span { ϕ ℓ : λ ℓ small } Spatio–spectral limiting
Motivation for GFT (e.g, Sardellitti Barbarossa Di Lorenzo [2016]): identify smooth clusters in vertex data that varies across clusters Other time–frequency analysis on graphs: Shuman, Ricaud and Vandergheynst [e.g., ACHA 2016], Stankovi´ c, Dakovi´ c and Sedji´ c [IEEE SP Magazine, 2017] Our thesis: particular graphs admit concrete analytical expressions Spatio–spectral limiting
Very particular graphs: Boolean hypercubes B N = Z N 2 B N : unweighted metric Cayley graph v = v S = ( ǫ 1 , . . . , ǫ N ), S ⊂ { 1 , . . . , N } : i ∈ S ⇔ ǫ i = 1 L = D − A D = N I N A RS = 1 if R ∆ S is a singleton Spatio–spectral limiting
Figure: Adjacency matrix for N = 8 in dyadic lexicographic order. Spatio–spectral limiting
B 5 following dyadic lexicographic order Σ r : Hamming sphere of radius r : vertices with r one-bits 00000 10000 01000 00100 00010 00001 11000 10100 10010 10001 01100 01010 01001 00110 00101 00011 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 11110 11101 11011 10111 01111 11111 Spatio–spectral limiting
Why hypercubes Historical use: Sampling Known Fourier transform Non-Euclidean geometry Our thesis: particular graphs admit concrete analytical expressions Accessible generalizations and restrictions: generalized hypercubes, partial cubes Spatio–spectral limiting
Spatio–spectral limiting: Tsitsvero, Barbarossa, Di Lorenzo [2016]: relate properties of compositions QP and PQ on graphs to (sub)-sampling strategies for recovery of sparse vertex functions. Sampling of bandlimited vertex functions was developed in the setting of hypercubes by Mansour et al in early 1990s in context of learning (sparse) Boolean functions. Spatio–spectral limiting
Fourier transform on B N : Hadamard matrix Lemma (Boolean Fourier transform) Let H S ( R ) = ( − 1) | R ∩ S | and L = L B N as above. Then H S is an eigenvector of L with eigenvalue 2 | S | . Spatio–spectral limiting
Figure: Hadamard (Fourier) matrix for N = 8 in dyadic lexicographic order. Spatio–spectral limiting
Spatial and spectral limiting on B N � 1 , R = S & | S | ≤ K Space-limiting matrix Q = Q K : Q R , S = 0 , else Spectrum-limiting matrix P = P K by P = ¯ HQ ¯ H Spatio–spectral limiting
Results and approach Results: identify eigenvectors of spatio–spectral limiting PQP Approach: ◮ Work in spectral domain : QPQ = ¯ HPQP ¯ H ◮ Identify salient invariant subspaces of QPQ ◮ These subspaces factor ◮ Reduce to small matrix problem on one of the factors ◮ Numerical computation via almost commuting operator and power method with a weight Spatio–spectral limiting
Eigenspaces of spatio-spectral limiting on B N A : adjacency matrix of B N (dyadic lexicographic order) A = A + + A − : A − = A T + ; A + : lower triangular A + maps data on Σ r to data on Σ r +1 : outer adjacency A − maps data on Σ r to data on Σ r − 1 : inner adjacency Figure: Highlighted: A − , Σ 3 → Σ 2 Spatio–spectral limiting
ℓ 2 (Σ r ) = A + ℓ 2 (Σ r − 1 ) ⊕ W r W r : the orthogonal complement of A + ℓ 2 (Σ r − 1 ) inside ℓ 2 (Σ r ). + W 0 ⊕ A r − 1 ℓ 2 (Σ r ) = A + ℓ 2 (Σ r − 1 ) ⊕ W r = · · · = A r + W 1 ⊕ · · · ⊕ W r Spatio–spectral limiting
Projection Matrix onto W r : columns form a Parseval frame Figure: Matrix of projection onto W r , N = 8, r = 3. Spatio–spectral limiting
Theorem Let V ∈ W r and k such that r + k < N. Then A − A k +1 [( N − 2 r ) + · · · + ( N − 2( r + k ))] A k V = + V + ( k + 1)( N − 2 r − k ) A k = + V m ( r , k ) A k ≡ + V Spatio–spectral limiting
� � � � � � � � � Base case ( k = 0, r = 2) 00000 10000 01000 00100 00010 00001 11000 10100 10010 01100 01010 01001 00110 00101 00011 10001 11100 11010 11001 10110 10101 10011 01110 01101 01011 00111 11110 11101 11011 10111 01111 11111 Spatio–spectral limiting
Commutators of A + and A − C = [ A − , A + ] = A − A + − A + A − : commutator of A + and A − . Proposition C is diagonal with C RR = N − 2 | R | . Theorem follows from induction on k Spatio–spectral limiting
Adjacency invariant subspaces V ∈ V r if V = � N − r k =0 c k A k + W , W ∈ W r Lemma A + and A − map V r to itself. Corollary A maps V r to itself. Polynomials p ( A ) preserve V r . Spatio–spectral limiting
Proposition The spectrum-limiting operator P = P K can be expressed as a polynomial p ( A ) of degree N. Proof. N K � � x − ( N − 2 j ) p k = ; p ( x ) = p k 2( j − k ) j =0 , j � = k k =0 Then P = p ( A ) as verified on Hadamard basis. Spatio–spectral limiting
P K factors through V r ≃ W r ⊗ R N − r +1 Matrix of Spectral limiting P K on V r M P ( N , K , r ) of size ( N − r + 1): represents P K on V r N − r � P ( A k M P ( N , K , r ) ( k , ℓ ) A ℓ + W ) = + W , ( W ∈ W r ) ℓ =0 N − r N − r N − r � � � d k A k M P ( N , K , r ) ( k , ℓ ) c ℓ A k PV = + W = + W ( W ∈ W r ) k =0 k =0 ℓ =0 Spatio–spectral limiting
Matrix of QPQ on V r M QPQ ( N , K , r ) : ( K − r + 1)-principal minor of M P ( N , K , r ) . K − r K − r K − r � � � d k A k M P ( N , K , r ) ( k , ℓ ) c ℓ A k QPQV = + W = + W , ( W ∈ W r ) k =0 k =0 ℓ =0 Spatio–spectral limiting
Corollary (Coefficient eigenvectors of QPQ ) If c = [ c 0 , . . . c K − r ] T is a λ -eigenvector of the principal minor M QPQ ( N , K , r ) of size ( K − r + 1) of the matrix M P ( N , K , r ) then V = � K − r k =0 c k A k + W , any W ∈ W r , is a λ -eigenvector of QPQ and ¯ HV is a λ -eigenvector of PQP. Remark (Completeness) Any eigenvector of QPQ is attached to one of the spaces V r Spatio–spectral limiting
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