Approximating scalable frames

Transcription

1 Kasso Okoudjou joint with X. Chen, G. Kutyniok, F. Philipp, R. Wang Department of Mathematics & Norbert Wiener Center University of Maryland, College Park 5 th International Conference on Computational Harmonic Analysis Vanderbilt University, Nashville, TN Monday May 19, 2014

2 Outline Frames and scalable frames 1 Frames and scalable frames Background Scalable frames Characterization of scalable frames 2 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames 3 4

3 Definition Frames and scalable frames Background Scalable frames Characterization of scalable frames Definition Let K = R or K = C. A set of vectors {ϕ i } M i=1 KN is called a finite frame for K N if there are two constants 0 < A B such that A x 2 M x, ϕ i 2 B x 2, for all x K N. (1) i=1 If the frame bounds A and B are equal, the frame {ϕ i } M i=1 KN is called a finite tight frame for K N.

4 Frames in applications Background Scalable frames Characterization of scalable frames Example Quantum computing: construction of POVMs Spherical t-designs Classification of hyper-spectral data Quantization Phase-less reconstruction Compressed sensing.

5 Main question Frames and scalable frames Background Scalable frames Characterization of scalable frames Question Given a (non-tight) frame Φ = {ϕ k } M k=1 RN can one transform Φ into a tight frame? If yes can this be done algorithmically and can the class of all frames that allow such transformations be described? Solution 1 If Φ denotes again the N M synthesis matrix, a solution to the above problem is the associated canonical tight frame {(ΦΦ) 1/2 ϕ k } M k=1. Involves the inverse frame operator. 2 What transformations are allowed?

6 Choosing a transformation Background Scalable frames Characterization of scalable frames Question Given a (non-tight) frame Φ = {ϕ k } M k=1 RN can one find nonnegative numbers {c k } M k=1 [0, ) such that Φ = {c k ϕ k } M k=1 becomes a tight frame?

7 Definition Frames and scalable frames Background Scalable frames Characterization of scalable frames Definition Let M, N be given such that N M. A frame Φ = {ϕ k } M k=1 in RN is scalable if there exists nonnegative scalars {x k } M k=1 such that the system Φ I = {x k ϕ k } M k=1 is a tight frame for RN. If all the coefficients can be chosen to be positive, then we say that the frame is strictly scalable.

8 An extension of scalable frame Background Scalable frames Characterization of scalable frames Definition Let M, m, N be given such that N m M. A frame Φ = {ϕ k } M k=1 in R N is m scalable if there exist a subset Φ I = {ϕ k } k I with #I = m, and nonnegative scalars {x k } k I such that the system Φ I = {x k ϕ k } k I is a tight frame for R N.

9 Background Scalable frames Characterization of scalable frames A geometric characterization of scalable frames Theorem (G. Kutyniok, F. Philipp, K. Tuley, K.O. (2012)) Let Φ = {ϕ k } M k=1 RN \ {0} be a frame for R N. Then the following statements are equivalent. (i) Φ is not scalable. (ii) There exists a symmetric M M matrix Y with trace(y ) < 0 such that ϕ j, Y ϕ j 0 for all j = 1,..., M. (iii) There exists a symmetric M M matrix Y with trace(y ) = 0 such that ϕ j, Y ϕ j > 0 for all j = 1,..., M.

10 Scalable frames in R 2 and R 3 Background Scalable frames Characterization of scalable frames Figures show sample regions of vectors of a non-scalable frame in R 2 and R 3. (a) (b) (c) Figure : (a) shows a sample region of vectors of a non-scalable frame in R 2. (b) and (c) show examples of sets in C 3 which determine sample regions in R 3.

11 Scalable frames and Farkas s lemma Background Scalable frames Characterization of scalable frames Setting Let F : R N R d, d := (N 1)(N + 2)/2, defined by F 0 (x) F 1 (x) F (x) =. F N 1 (x) x 2 1 x 2 2 x k x k+1 x 2 1 x 2 3 F 0 (x) =.,..., F x k x k+2 k(x) =. x 2 1 x 2 N x k x N and F 0 (x) R N 1, F k (x) R N k, k = 1, 2,..., N 1.

12 Scalable frames and Farkas s lemma Background Scalable frames Characterization of scalable frames Theorem (G. Kutyniok, F. Philipp, K.O. (2013)) Φ = {ϕ k } M k=1 RN is scalable if and only if F (Φ)u = 0 has a nonnegative non trivial solution, where F (Φ) is the d M matrix whose k th row is F (ϕ k ). This is equivalent to 0 being in the relative interior of the convex polytope whose extreme points are {F (ϕ k )} M k=1.

13 Some questions Frames and scalable frames Background Scalable frames Characterization of scalable frames Question Let Φ = {ϕ k } M k=1 RN be a frame. 1 If Φ is scalable, how to find the corresponding weights? 2 Can one find intrinsic measures of scalability? 3 In particular, if a frame is not scalable, how far it is to be so?

14 Some questions Frames and scalable frames Background Scalable frames Characterization of scalable frames Question Let Φ = {ϕ k } M k=1 RN be a frame. 1 If Φ is scalable, how to find the corresponding weights? 2 Can one find intrinsic measures of scalability? 3 In particular, if a frame is not scalable, how far it is to be so?

15 Some questions Frames and scalable frames Background Scalable frames Characterization of scalable frames Question Let Φ = {ϕ k } M k=1 RN be a frame. 1 If Φ is scalable, how to find the corresponding weights? 2 Can one find intrinsic measures of scalability? 3 In particular, if a frame is not scalable, how far it is to be so?

16 Fritz John s Theorem Frames and scalable frames Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Theorem (F. John (1948)) Let K B = B(0, 1) be a convex body with nonempty interior. There exits a unique ellipsoid E min of minimal volume containing K. Moreover, E min = B if and only if there exist {λ k } m k=1 (0, ) and {u k } m k=1 K SN 1, m N + 1 such that (i) m k=1 λ ku k = 0 (ii) x = m k=1 λ k x, u k u k, x R N where K is the boundary of K and S N 1 is the unit sphere in R N. In particular, the points u k are contact points of K and S N 1.

17 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames F. John s characterization of scalable frames Setting Let Φ = {ϕ k } M k=1 SN 1 be a frame for R N. We apply F. John s theorem to the convex body K = P Φ = conv({±ϕ k } M k=1 ). Let E Φ denote the ellipsoid of minimal volume containing P Φ, and V Φ = Vol(E Φ )/ω N where ω N is the volume of the euclidean unit ball. Theorem (X. Chen, R. Wang, K.O. (2014)) Let Φ = {ϕ k } M k=1 SN 1 be a frame. Then Φ is scalable if and only if V Φ = 1. In this case, the ellipsoid E Φ of minimal volume containing P Φ = conv({±ϕ k } M k=1 ) is the euclidean unit ball B.

18 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames F. John s characterization of scalable frames Setting Let Φ = {ϕ k } M k=1 SN 1 be a frame for R N. We apply F. John s theorem to the convex body K = P Φ = conv({±ϕ k } M k=1 ). Let E Φ denote the ellipsoid of minimal volume containing P Φ, and V Φ = Vol(E Φ )/ω N where ω N is the volume of the euclidean unit ball. Theorem (X. Chen, R. Wang, K.O. (2014)) Let Φ = {ϕ k } M k=1 SN 1 be a frame. Then Φ is scalable if and only if V Φ = 1. In this case, the ellipsoid E Φ of minimal volume containing P Φ = conv({±ϕ k } M k=1 ) is the euclidean unit ball B.

19 Reformulating the scalability problem Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Setting Φ = {ϕ i } M i=1 is scalable {c i } M i=1 [0, ) : ΦCΦ T = I, where C = diag(c i ). M C Φ = {ΦCΦ T = c i ϕ i ϕ T i : c i 0} i=1 is the (closed) cone generated by {ϕ i ϕ T i }M i=1. Φ = {ϕ i } M i=1 is scalable I C Φ. D Φ := min C 0 diagonal ΦCΦ T I F

20 Reformulating the scalability problem Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Setting Φ = {ϕ i } M i=1 is scalable {c i } M i=1 [0, ) : ΦCΦ T = I, where C = diag(c i ). M C Φ = {ΦCΦ T = c i ϕ i ϕ T i : c i 0} i=1 is the (closed) cone generated by {ϕ i ϕ T i }M i=1. Φ = {ϕ i } M i=1 is scalable I C Φ. D Φ := min C 0 diagonal ΦCΦ T I F

21 Reformulating the scalability problem Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Setting Φ = {ϕ i } M i=1 is scalable {c i } M i=1 [0, ) : ΦCΦ T = I, where C = diag(c i ). M C Φ = {ΦCΦ T = c i ϕ i ϕ T i : c i 0} i=1 is the (closed) cone generated by {ϕ i ϕ T i }M i=1. Φ = {ϕ i } M i=1 is scalable I C Φ. D Φ := min C 0 diagonal ΦCΦ T I F

22 Comparing D Φ to the frame potential Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Proposition (X. Chen, R. Wang, K.O. (2014)) (a) Φ is scalable if and only if D Φ = 0. (b) If Φ = {ϕ k } M k=1 RN is a unit norm frame we have D 2 Φ N M 2 FP(Φ), where FP(Φ) is the frame potential of Φ.

23 Comparing the measures of scalability Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Theorem (X. Chen, R. Wang, K.O. (2014)) Let Φ = {ϕ k } M k=1 RN is a unit norm frame, then N(1 D 2 Φ ) N D 2 Φ V 4/N Φ N(N 1 D2 Φ ) (N 1)(N D 2 Φ ) 1, where the leftmost inequality requires D Φ < 1. Consequently, V Φ 1 is equivalent to D Φ 0.

24 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Values of V Φ and D Φ for randomly generated frames of M vectors in R 4. 1 Frames of size Frames of size V Φ 0.5 V Φ D Φ D Φ Figure : Relation between V Φ and D Φ with M = 11, 20. The black line indicates the upper bound in the last theorem, while the red dash line indicates the lower bound.

25 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Projecting a frame onto the scalable frames Setting We denote the set of scalable frames of M vectors in R N by Sc(M, N). Given a unit norm frame Φ = {ϕ k } M k=1 RN, let d Φ := inf Φ Ψ F. Ψ Sc(M,N) Proposition (X. Chen, R. Wang, K.O. (2014)) If Φ = {ϕ k } M k=1 RN is a unit norm frame such that d Φ < 1 then there exists ˆΦ Sc(M, N) such that Φ ˆΦ F = d Φ.

26 Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Projecting a frame onto the scalable frames Setting We denote the set of scalable frames of M vectors in R N by Sc(M, N). Given a unit norm frame Φ = {ϕ k } M k=1 RN, let d Φ := inf Φ Ψ F. Ψ Sc(M,N) Proposition (X. Chen, R. Wang, K.O. (2014)) If Φ = {ϕ k } M k=1 RN is a unit norm frame such that d Φ < 1 then there exists ˆΦ Sc(M, N) such that Φ ˆΦ F = d Φ.

27 Comparing the measures of scalability Fritz John s ellipsoid theorem and scalable frames Distance to the cone of nonnegative diagonal matrices Distance to the set of scalable frames Theorem (X. Chen, R. Wang, K.O. (2014)) Let Φ = {ϕ k } M k=1 RN is a unit norm frame and assume that d Φ < 1. Then with K := min{m, N(N+1) 2 } and ω := D Φ + K we have D Φ ω + ω 2 D 2 Φ ( d Φ KN 1 V 2/N Φ Consequently, we can bound d Φ below and above by expressions of D Φ or expressions of V Φ. ).

28 Approximating with scalable frames Theorem (X. Chen, R. Wang, K.O. (2014)) Let Φ = {ϕ k } M k=1 RN is a unit norm frame and assume that d Φ 1 2 (1 + K) 1. Let ˆΦ be given by, ˆΦ = arg inf Ψ Sc(M,N) Φ Ψ F, and let E Φ = E(X) be the minimal ellipsoid of Φ, where X 1 = M i=1 ρ iϕ i ϕ T i. Then there exists a (constructible) scalable frame Φ = { ϕ i } M i=1 which is a good approximation to Φ in the following sense: where K = min{m, N(N+1) 2 }. Φ Φ F = K NO(d Φ ),

29 Probability of a frame to be scalable Theorem (X. Chen, R. Wang, K.O. (2013)) Given Φ = {ϕ i } M i=1 RN, be a frame such that each frame vector ϕ i is drawn independently and uniformly from S N 1, if P M,N indicates the probability of Φ being scalable, then (a) P M,N = 0, when M < N(N+1) 2, (b) P M,N > 0, when M N(N+1) 2, (c) 1 C(N) ( 1 A N 1 ) M ) M N α PM,N 1 (1 A N 1 arccos(1/ N), where C(N) is the number of caps with angular radius 1 2 arccos N 1 N needed to cover S N 1. Consequently, lim M P M,N = 1.

30 Frames and scalable frames J. Cahill and X. Chen, A note on scalable frames, Proceedings of the 10th International Conference on Sampling Theory and Applications, pp X. Chen, K. A. Okoudjou, and R. Wang,, preprint. M. S. Copenhaver, Y. H. Kim, C. Logan, K. Mayfield, S. K. Narayan, and J. Sheperd, Diagram vectors and tight frame scaling in finite dimensions, Oper. Matrices, 8, no.1 (2014), G. Kutyniok, K. A. Okoudjou, F. Philipp, and K. E. Tuley, Scalable frames, Linear Algebra Appl., 438 (2013), G. Kutyniok, K. A. Okoudjou, F. Philipp, Scalable frames and convex geometry, Contemp. Math., to appear. F. John, Extremum problems with inequalities as subsidiary conditions, Studies and Essays Presented to R. Courant on his 60 th Birthday, January 8, 1948, Interscience Publishers, Inc., New York, N. Y., 1948.

31 Thank You! okoudjou