Preconditioning techniques in frame theory and probabilistic frames

Transcription

1 Preconditioning techniques in frame theory and probabilistic frames Department of Mathematics & Norbert Wiener Center University of Maryland, College Park AMS Short Course on Finite Frame Theory: A Complete Introduction to Overcompleteness San Antonio, TX Friday January 9, 2015

2 Outline 1 Preconditioning of finite frames: Scalable frames 2 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential

3 Definition Preconditioning of finite frames: Scalable frames Definition Φ = {ϕ k } M k=1 RN is a frame for R N if A, B > 0 such that x R N, A x 2 M x, ϕ k 2 B x 2. k=1 If, in addition, ϕ k = 1 for each k, we say that Φ is a unit-norm frame. The set of frames for R N with M elements will be denoted by F. In addition, we let F u the the subset of unit-norm frames.

4 Analysis and Synthesis with frame Let Φ = {ϕ k } M k=1 RN. 1 The analysis operator, is defined by R N x Φ T x = { x, ϕ k } M k=1 R M. 2 The synthesis operator is defined by R M c = (c k ) M k=1 Φc = 3 The frame operator S = ΦΦ T is given by R N x Sx = M c k ϕ k R N. k=1 M x, ϕ k ϕ k R N. k=1 4 The Gramian (operator) G = Φ T Φ of the frame is the M M matrix whose (i, j) th entry is ϕ j, ϕ i.

5 Tight frames and FUNTFs 1 A frame Φ is a tight frame if we can choose A = B. 2 If Φ = {ϕ k } M k=1 RM is a frame then {ϕ k }M k=1 = {S 1/2 ϕ k } M k=1 R N is a tight frame and for every x R N, x = M x, ϕ k ϕ k. (1) k=1 3 If Φ is a tight frame of unit-norm vectors, we say that Φ is a finite unit-norm tight frame (FUNTF). In this case, the reconstruction formula (??) reduces to x R N, M x = N M x, ϕ k ϕ k. (2) k=1

6 The frame potential Theorem (Benedetto and Fickus, 2003) For each Φ = {ϕ k } M k=1 RN, such that ϕ k = 1 for each k, we have M M FP(Φ) = ϕ j, ϕ k 2 M N j=1 k=1 max(m, N). (3) Furthermore, If M N, the minimum of FP is M and is achieved by orthonormal systems for R N with M elements. If M N, the minimum of FP is M 2 N and is achieved by FUNTFs. FP(Φ) is the frame potential.

7 Why frames and FUNTFs Remark 1 Geometry of FUNTFs: N. Strawn. 2 Constructing all FUNTFs: D. Mixon. 3 Applications of FUNTFs and frames: P. Casazza; R. Balan; G. Chen and D. Needell; A. Powell and O. Yilmaz.

8 Optimally conditioned frames Remark 1 FUNTFs can be considered optimally conditioned frames. In particular the condition number of the frame operator is 1. 2 There are many preconditioning methods to improve the condition number of a matrix, e.g., Matrix Scaling. 3 A matrix A is (row/column) scalable if there exit diagonal matrices D 1, D 2 with positive diagonal entries such that D 1 A, AD 2, or D 1 AD 2 have constant row/column sum.

9 Goals of this section Remark 1 How to transform a (non) tight frame into a tight one? 2 Give theoretical guarantees and algorithms. 3 What transformations are allowed? 4 For a given transformation, what happens if a frame cannot be transformed exactly? In this part of the lecture we will only consider one transform and mostly answer the first two questions.

10 Main question Preconditioning of finite frames: 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 {S 1/2 ϕ k } M k=1. Involves the inverse frame operator. 2 What transformations are allowed?

11 Choosing a transformation 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?

12 Definition Preconditioning of finite frames: Scalable frames Definition A frame Φ = {ϕ k } M k=1 in RN is scalable, if {c k } M k=1 [0, ) such that {c k ϕ k } M k=1 is a tight frame for RN. The set of scalable frames is denoted by SC(M, N). In addition, if {c k } M k=1 (0, ), the frame is called strictly scalable and the set of strictly scalable frames is denoted by SC + (M, N).

13 A more general definition Definition Given, N m M, a frame Φ = {ϕ k } M k=1 is said to be m-scalable, respectively, strictly m scalable, if Φ I = {ϕ k } k I with I {1, 2,..., M}, #I = m, such that Φ I = {ϕ k } k I is scalable, respectively, strictly scalable. We denote the set of m-scalable frames, respectively, strictly m-scalable frames in F(M, N) by SC(M, N, m), respectively, SC + (M, N, m).

14 Some basic examples Example 1 When M = N, a frame Φ = {ϕ k } N k=1 RN is scalable if and only if Φ is an orthogonal set. 2 When M N, if Φ contains an orthogonal basis, then it is clearly N scalable. 3 Thus, given M N, the set SC(M, N, N) consists exactly of frames that contains an orthogonal basis for R N.

15 Useful remarks Preconditioning of finite frames: Scalable frames Remark We note that a frame Φ = {ϕ k } M k=1 RN with ϕ k 0 for each k = 1,..., M is scalable if and only if Φ = { ϕ k ϕ k }M k=1 is scalable.

16 Useful remarks Preconditioning of finite frames: Scalable frames Remark Given a frame Φ R N, assume that Φ = Φ 1 Φ 2 where and Let Φ 1 = {ϕ (1) k Φ : ϕ (1) k (N) 0} Φ 2 = {ϕ (2) k Φ : ϕ (2) k (N) < 0}. Φ = Φ 1 ( Φ 2 ). Φ is scalable if and only if Φ is scalable. We shall assume that all the frame vectors are in the upper-half space, i.e., Φ R N 1 R +,0 where R +,0 = [0, ).

17 Elementary properties of scalable frames Proposition Let M N, and m 1 be integers. (i) If Φ F is m-scalable then m N. (ii) For any integers m, m such that N m m M we have that and SC(M, N, m) SC(M, N, m ), SC(M, N) = M m=n SC(M, N, m). (iii) Φ SC(M, N) if and only if T (Φ) SC(M, N) for one (and hence for all) orthogonal transformation(s) T on R N. (iv) Let Φ = {ϕ k } N+1 k=1 F(N + 1, N) \ {0} with ϕ k ±ϕ l for k l. If Φ SC + (N + 1, N), then Φ / SC + (N + 1, N + 1).

18 Scalable frames: When and How? Question 1 When is a frame Φ = {ϕ k } M k=1 RN scalable? 2 If Φ = {ϕ k } M k=1 RN is scalable, how to find the coefficients? 3 If Φ is not scalable, how close to scalable is it? 4 What are the topological properties of SC(M, N)?

19 A reformulation Preconditioning of finite frames: Scalable frames Fact Φ is (m-) scalable {x k } k I [0, ) with #I = m N such that Φ = ΦX satisfies where X = diag(x k ). (4) is equivalent to solving for Y = 1 Ã X2. Φ Φ T = ΦX 2 Φ T = ÃI N = k I x2 k ϕ k 2 N I N (4) ΦY Φ T = I N (5)

20 Scalable frame in R 2 Question When is Φ = {ϕ k } M k=1 S1 is a scalable frame in R 2? Solution Assume that Φ = {ϕ k } M k=1 R R +,0, ϕ k = 1, and ϕ l ϕ k for l k. Let 0 = θ 1 < θ 2 < θ 3 <... < θ M < π, then ( ) cos θk ϕ k = S 1. sin θ k Let Y = (y k ) M k=1 [0, ), then (5) becomes ( M k=1 y k cos 2 θ k M k=1 y k sin θ k cos θ k M k=1 y k sin θ k cos θ k M k=1 y k sin 2 θ k ) = ( ) 1 0. (6) 0 1

21 Scalable frame in R 2 Solution (6) is equivalent to M k=1 y k sin 2 θ k = 1 M k=1 y k cos 2θ k = 0 M k=1 y k sin 2θ k = 0. Consequently, for Φ to be scalable we must find a nonnegative( vector ) cos Y = (y k ) M k=1 in the kernel of the matrix whose 2θk kth column is. sin 2θ k

22 Scalable frame in R 2 Solution (6) is equivalent to M k=1 y k sin 2 θ k = 1 M k=1 y k cos 2θ k = 0 M k=1 y k sin 2θ k = 0. Consequently, for Φ to be scalable we must find a nonnegative( vector ) cos Y = (y k ) M k=1 in the kernel of the matrix whose 2θk kth column is. sin 2θ k

23 Scalable frame in R 2 Solution The problem is equivalent to finding non-trivial nonnegative vectors in the nullspace of ( ) 1 cos 2θ2... cos 2θ M. (7) 0 sin 2θ 2... sin 2θ M

24 Describing SC(3, 2) Example We first consider the case M = 3. In this case, we have 0 = θ 1 < θ 2 < θ 3 < π, and the (7) becomes ( ) 1 cos 2θ2 cos 2θ 3. (8) 0 sin 2θ 2 sin 2θ 3

25 Describing SC(3, 2) Example If θ k0 = π/2 for k 0 {2, 3}, then the corresponding frame contains an ONB and, hence is scalable. For example, when k 0 = 2, then 0 = θ 1 < θ 2 = π/2 < θ 3 < π. In this case, the fame is 2 scalable but not 3 scalable. Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

26 Describing SC(3, 2) Example If θ k0 = π/2 for k 0 {2, 3}, then the corresponding frame contains an ONB and, hence is scalable. For example, when k 0 = 2, then 0 = θ 1 < θ 2 = π/2 < θ 3 < π. In this case, the fame is 2 scalable but not 3 scalable. Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

27 Describing SC(3, 2) Example Suppose θ k π/2 for k = 2, 3. If θ 3 < π/2, then the frame cannot be scalable. Indeed, u = (z 1, z 2, z 3 ) belongs to the kernel of (8) if and only if { sin 2(θ3 θ2) z 1 = sin 2θ 2 z 3, sin (9) 2θ3 z 2 = sin 2θ 2 z 3, where z 3 R. The choice of the angles implies that z 2 z 3 < 0, unless z 3 = 0.

28 Describing SC(3, 2) Example This is illustrated by Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

29 Describing SC(3, 2) Example Suppose that 0 = θ 1 < θ 2 < π/2 < θ 3 < π. From (9) z 2 > 0 for all z 3 > 0 and z 1 > 0 for all z 3 > 0 if and only if θ 3 θ 2 < π/2. Consequently, when 0 = θ 1 < θ 2 < π/2 < θ 3 < π the frame Φ SC + (3, 2, 3) if and only if 0 < θ 3 θ 2 < π/2.

30 Describing SC(3, 2) Example Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

31 Describing SC(4, 2) Example When M = 4 we are lead to seek nonnegative non-trivial vectors in the null space of ( ) 1 cos 2θ2 cos 2θ 3 cos 2θ 4. 0 sin 2θ 2 sin 2θ 3 sin 2θ 4

32 Describing SC(4, 2) Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

33 Describing SC(4, 2) Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

34 Describing SC(4, 2) Figure : Blue=original frame; Red=the frames obtained by scaling; Green=associated canonical tight frame.

35 A more general reformulation 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.

36 The map F when N = 2 Example When N = 2 the map F reduces to ( ( ) x x F = y) 2 y 2. xy Note that in the examples given above we consider ( ( ) x x F = y) 2 y 2. 2xy

37 When is a frame scalable: A generic solution Question When is Φ = {ϕ k } M k=1 RN scalable? Proposition A frame Φ for R N is m-scalable, respectively, strictly m-scalable, if and only if there exists a nonnegative u ker F (Φ) \ {0} with u 0 m, respectively, u 0 = m, and where F (Φ) is the d M matrix whose k th column is F (ϕ k ).

38 A key tool: The Farkas Lemma Lemma For every real N M-matrix A exactly one of the following cases occurs: (i) The system of linear equations Ax = 0 has a nontrivial nonnegative solution x R M, i.e., all components of x are nonnegative and at least one of them is strictly positive. (ii) There exists y R N such that y T A is a vector with all entries strictly positive.

39 Farkas lemma with N = 2, M = 4 Figure : Bleu=original frame; Green=image by the map F. Both of these examples result in non scalable frames.

40 Farkas lemma with N = 2, M = 4 Figure : Bleu=original frame; Green=image by the map F. Both of these examples result in scalable frames.

41 Some convex geometry notions Fact Let X = {x i } M k=1 RN. 1 The polytope generated by X is the convex hull of X, denoted by P X (or co(x)). 2 The affine hull generated by X is denoted by aff(x). 3 The relative interior of the polytope co(x) denoted by ri co(x), is the interior of co(x) in the topology induced by aff(x). 4 It is true that ri co(x) whenever #X 2, and { M } M ri co(x) = α k x k : α k > 0, α k = 1, k=1 k=1

42 Scalable frames and Farkas s lemma Theorem Let M N 2, and let m be such that N m M. Assume that Φ = {ϕ k } M k=1 F (M, N) is such that ϕ k ±ϕ l when k l. Then the following statements are equivalent: (i) Φ is m scalable, respectively, strictly m scalable, (ii) There exists a subset I {1, 2,..., M} with #I = m such that 0 co(f (Φ I )), respectively, 0 ri co(f (Φ I )). (iii) There exists a subset I {1, 2,..., M} with #I = m for which there is no h R d with F (ϕ k ), h > 0 for all k I, respectively, with F (ϕ k ), h 0 for all k I, with at least one of the inequalities being strict.

43 A useful property of F For x = (x k ) N k=1 RN and h = (h k ) d k=1 Rd, we have that F (x), h = N l=2 N 1 h l 1 (x 2 1 x 2 l) + N k=1 l=k+1 h k(n 1 (k 1)/2)+l 1 x k x l. (10) Consequently, fixing h R d, F (x), h is a homogeneous polynomial of degree 2 in x 1, x 2,..., x N. The set of all polynomials of this form can be identified with the subspace of real symmetric N N matrices whose trace is 0.

44 A useful property of F Remark F (x), h = Q h x, x = 0 defines a quadratic surface in R N, and condition (iii) in the last Theorem stipulates that for Φ to be scalable, one cannot find such a quadratic surface such that the frame vectors (with index in I) all lie on (only) one side of this surface.

45 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.

46 Scalable frames in R 2 and R 3 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.

47 Fritz John s Theorem 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.

48 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 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.

49 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 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.

50 A measure of scalability Remark Let Φ S N 1 be a frame. Then V Φ is a measure of scalability : the closer it is to 1 the more scalable is the frame.

51 A quadratic programing approach to scalability 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 ΦCΦ T I F C 0 diagonal

52 A second measure of scalability Remark Let Φ S N 1 be a frame. Then D Φ is a measure of scalability : the closer it is to 0 the more scalable is the frame.

53 Comparing the measures of scalability 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 = 6, 11. The black line indicates the upper bound in the last theorem, while the red dash line indicates the lower bound.

54 Comparing the measures of scalability 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 = 15, 20. The black line indicates the upper bound in the last theorem, while the red dash line indicates the lower bound.

55 Concluding remarks on scalable frames 1 The problem can be reformulated as a linear programing one leading to numerical solutions. 2 When frame not scalable, one can define how close or far to being scalable it is: Notion of almost scalable. 3 Role of redundancy. 4 Size of SC(M, N). 5 Other methods of frame preconditioning

56 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Goals of this section Remark 1 Standard tools used in frame theory include: Functional and Harmonic Analysis, Operator Theory, Linear Algebra, Differential Geometry, Differential Equations. 2 Identifying frames with probability measures leads analyzing frames in the setting of the Wasserstein metric spaces. 3 For example, gradient flow methods from optimal transport theory can be used to minimize certain common potentials in frame theory.

57 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Motivation: The Welch bound Theorem For any frame Φ = {ϕ k } M k=1 SN 1 we have max ϕ M N k, ϕ l k l N(M 1), (11) and equality hold if and only if Φ is an ETF. Furthermore, equality can hold only when M N(N+1) 2.

58 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Definition of the p th frame potential Definition Let M be a positive integer, and 0 < p <. Given a collection of unit vectors Φ = {ϕ k } M k=1 SN 1, the p-frame potential is the functional FP p,m (Φ) = M k,l=1 When, p =, the definition reduces to FP,M (Φ) = max k l ϕ k, ϕ l. ϕ k, ϕ l p. (12)

59 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Special cases 1 p = 2 corresponds to the frame potential whose minimizers are the FUNTFs 2 For p = and fixed M, the minimizers of FP,M are called Grassmanian frames. 3 The potential FP,M always has a minimum but constructing these minimizers is challenging. Question What are the minimizers of FP p,m?

60 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Example: M = 3, N = 2 Question Find the minimizers of 3 FP p,3 (Φ) = ϕ k, ϕ l p k,l=1 when p (0, ] and Φ = {ϕ k } 3 k=1 S1.

61 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Solution for p = 2 and p = Solution 1 When p = 2, FP 2,3 (Φ) = 3 k,l=1 ϕ k, ϕ l 2 9/2 with equality if and only if Φ = {ϕ k } 3 k=1 S1 is a FUNTF. A minimizer of FP 2,3 is the MB-frame, see next slide. 2 When p =, FP,3 (Φ) = max k l ϕ k, ϕ l 1/ 2 with equality if and only if Φ = {ϕ k } 3 k=1 S1 is an ETF. Hence a solution is also given by the MB frame

62 the MB-frame Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Figure : An example of Equiangular FUNTF: the MB-frame.

63 Minimizers of FP p,3 for p (0, ] The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Proposition Let p 0 = log(3) log(2). Then FP p 0,3(Φ) 5, with equality holding if and only if Φ = {ϕ k } 3 k=1 is an orthonormal basis plus one repeated vector or an ETF. Furthermore, (1) for 0 < p < p 0, and Φ = {ϕ k } 3 k=1 S1, we have FP p,3 (Φ) 5, and equality holds if and only if Φ = {ϕ k } 3 k=1 is an orthonormal basis plus one repeated vector, (2) for p > p 0, and Φ = {ϕ k } 3 k=1 S1, we have FP p,3 (Φ) 2 p p 0 (6) 1 p p 0 + 3, and equality holds if and only if Φ = {ϕ k } 3 k=1 is an ETF.

64 Minimizers of FP p,3 for p (0, ) The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Remark µ p,3,2 = min{fp p,2 (Φ) : Φ = {ϕ k } 3 k=1 S 1 }

65 Minimizers of FP p,n+1 for p (0, ) The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Theorem Let p (0, ], and N be positive integer. Let Φ = {ϕ k } N+1 k=1 SN 1. N(N+1) log( 2 ) Set p 0 =. Assume that FP p0,n+1(φ) N + 3, with equality log(n) holding if and only if Φ = {ϕ k } N+1 k=1 is an orthonormal basis plus one repeated vector or an ETF. Then, (1) for 0 < p < p 0, and Φ = {ϕ k } N+1 k=1 SN 1, we have FP p,n+1 (Φ) N + 3, and equality holds if and only if Φ = {ϕ k } N+1 k=1 is an orthonormal basis plus one repeated vector, (2) for p 0 < p < 2, and Φ = {ϕ k } N+1 k=1 SN 1, we have FP p,n+1 (Φ) 2 p p 0 (N(N + 1)) 1 p p 0 + N + 1, and equality holds if and only if Φ = {ϕ k } N+1 k=1 is an ETF.

66 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Remarks on the Theorem Remark 1 The hypothesis of the last theorem can be verified when N = 2. But for N 3 it is not known if this hypothesis is true. 2 There seems to be some universality of the minimizers of these potentials. With p 0 given above, any orthonormal basis plus one repeated vector minimizes FP p,n+1 for 0 < p p 0 and any ETF minimizes FP p,n+1 for p 0 p.

67 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Partial results on minimizing FP p,m for p (0, ) Proposition Let p (0, ], M, N be positive integers. Let Φ = {ϕ k } M k=1 SN 1 we have: (a) If M N and 2 < p <, then FP p,m (Φ) M(M 1) ( M N N(M 1)) p/2 + N, and equality holds if and only if Φ is an ETF. (b) Let 0 < p < 2 and assume that M = kn for some positive integer k. Then the minimizers of the p-frame potential are exactly the k copies of any orthonormal basis modulo multiplications by ±1. The minimum of (12) over all sets of M = kn unit norm vectors is k 2 N.

68 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Numerical simulations for N = 2 Remark We let µ p,m,2 = min{fp p,2 (Φ) : Φ = {ϕ k } M k=1 S 1 }

69 The p th frame potential and t-design The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Definition Let t be a positive integer. A spherical t-design is a finite subset {x i } M i=1 of the unit sphere S N 1 in R N, such that, 1 M M h(x i ) = h(x)dσ(x), S N 1 i=1 for all homogeneous polynomials h of total degree equals or less than t in N variables and where σ denotes the uniform surface measure on S N 1 normalized to have mass one.

70 FUNTFS and 2-design The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Proposition Φ = {ϕ k } M k=1 SN 1 is a spherical 2-design if and only if Φ is a FUNTF and M k=1 ϕ k = 0.

71 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential t-designs as minimizers of p th frame potentials Theorem Let p = 2k be an even integer and {x i } M i=1 = { x i} M i=1 SN 1, then FP p,m ({x i } M i=1) (p 1) N(N + 2) (N + p 2) M 2, and equality holds if and only if {x i } M i=1 is a spherical p-design.

72 Motivations Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Let Φ = {ϕ i } M i=1 be a frame in RN with bounds 0 < A B <. Define µ Φ := 1 M M i=1 δ ϕi then R N x, y 2 dµ Φ (y) = 1 M M x, ϕ k 2. k=1 For each x R N : A/M x 2 R N x, y 2 dµ Φ (y) B/M x 2 µ Φ is an example of probabilistic frames. P is the set of probability measures on R N, and { } P 2 = µ P : M2 2 (µ) = y 2 dµ(y) < R N

73 Definition Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Definition A Borel probability measure µ P is a probabilistic frame if there exist 0 < A B < such that A x 2 x, y 2 dµ(y) B x 2, for all x R N. (13) R N When A = B, µ is called a tight probabilistic frame.

74 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential When is a probability measure a probabilistic frame? Theorem A Borel probability measure µ P is a probabilistic frame if and only if µ P 2 and E µ = R N, where E µ denotes the linear span of supp(µ) in R N. Moreover, if µ is a tight probabilistic frame, then the frame bound is given by A = 1 N M 2 2 (µ) = 1 N y 2 dµ(y). R N

75 Examples Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Example (a) Let a = {a k } M k=1 (0, ) with M k=1 a k = 1. A set Φ = {ϕ k } M k=1 RN is a frame if and only if the probability measure µ Φ,a = M k=1 a kδ ϕk supported by the set Φ is a probabilistic frame. (c) The uniform distribution on the unit sphere S N 1 in R N is a tight probabilistic frame. That is, denoting the probability measure on S N 1 by dσ we have that for all x R N, x 2 N = R N x, y 2 dσ(y).

76 Probabilistic frame operator The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Let µ P be a probability measure. 1 The probabilistic analysis operator is given by T µ : R N L 2 (R N, µ), x x,. 2 The probabilistic synthesis operator is defined by Tµ : L 2 (N d, µ) R N, f f(x)xdµ(x). R N 3 The probabilistic frame operator of µ is S µ = TµT µ. 4 The probabilistic Gram operator of µ, is defined on L 2 (R N, µ) by G µ f(x) = T µ Tµf(x) = x, y f(y)dµ(y). R N

77 Probabilistic frame operator The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Fact The probabilistic frame operator is given by S µ : R d R d, S µ (x) = x, y ydµ(y) R N and is the matrix of second moments of µ: If {e j } N j=1 is the canonical orthonormal basis for RN, then S µ e i = N m i,j (µ)e j, j=1 where m i,j (µ) = y (i) y (j) dµ(y). R N

78 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Probabilistic frame operator Proposition Let µ P, then S µ is well-defined (and hence bounded) if and only if M 2 (µ) <. Furthermore, µ is a probabilistic frame if and only if S µ is positive definite.

79 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Duality If µ is a probabilistic frame then S µ is positive definite. 1 The push-forward of µ through S 1 µ is given by µ(b) = µ((s 1 µ ) 1 B) = µ(s µ B). 2 µ is a probabilistic frame called the probabilistic canonical dual frame of µ. 3 The push-forward of µ through S 1/2 µ is given by µ (B) = µ(s 1/2 B).

80 Reconstruction formula The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Proposition Let µ P be a probabilistic frame with bounds 0 < A B <. Then: (a) µ is a probabilistic frame with frame bounds 1/B 1/A. (b) µ is a tight probabilistic frame. Consequently, for each x R N we have: x, y S µ y d µ(y) = R N Sµ 1 x, y y dµ(y) = x, R N (14) and x, y y dµ (y) = R N Sµ 1/2 x, y Sµ 1/2 y dµ(y) = x. R N (15)

81 Definition Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Question When is a probability measure µ a tight probabilistic frame? Definition The probabilistic frame potential is the nonnegative function defined on P and given by PFP(µ) = x, y 2 dµ(x) dµ(y), (16) R N R N for each µ P.

82 Definition Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Question When is a probability measure µ a tight probabilistic frame? Definition The probabilistic frame potential is the nonnegative function defined on P and given by PFP(µ) = x, y 2 dµ(x) dµ(y), (16) R N R N for each µ P.

83 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential The probabilistic frame potential and Gramian operator Proposition Let µ P, then PFP(µ) is the Hilbert-Schmidt norm of the probabilistic Gramian operator G µ, that is G µ 2 HS = x, y 2 dµ(x)dµ(y). R d R d Furthermore, if µ P 2, (which is the case when µ is a probabilistic frame) then we have PFP(µ) M 4 2 (µ) <.

84 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Probabilistic tight frames as minimizers of the PFP Theorem Let µ P 2 be such that M 2 (µ) = 1 and set E µ = span(supp(µ)), then the following estimate holds PFP(µ) 1/n (17) where n is the number of nonzero eigenvalues of S µ. Moreover, equality holds if and only if µ is a tight probabilistic frame for E µ. In particular, given any probabilistic frame µ P 2 with M 2 (µ) = 1, we have PFP(µ) 1/N and equality holds if and only if µ is a tight probabilistic frame. Remark When µ is a discrete measure, then PFP(µ) is the frame potential.

85 Definition Preconditioning of finite frames: Scalable frames The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential For p (0, ) set P p = { µ P : Mp p (µ) = y p dµ(y) < }. R N Definition For each p (0, ), the probabilistic p frame potential is given by PFP(µ, p) = x, y p dµ(x) dµ(y). (18) R N R N When supp(µ) = Φ = {ϕ k } M k=1 SN 1, PFP(µ, p) reduces to FP p,m.

86 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Minimizers of the probabilistic p th frame potential Theorem Let 0 < p < 2, then the minimizers of (18) over all the probability measures supported on the unit sphere S N 1 are exactly those probability measures µ that satisfy (i) there is an orthonormal basis {e 1,..., e N } for R N such that {e 1,..., e N } supp(µ) {±e 1,..., ±e N } (ii) there is f : S N 1 R such that µ(x) = f(x)ν ±x1,...,±x N (x) and f(x i ) + f( x i ) = 1 N, where the measure ν ±x1,...,±x N (x) represent the counting measure of the set {±x i : i = 1,..., N}.

87 Probabilistic p frame The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Definition For 0 < p <, we call µ M(S N 1, B) a probabilistic p-frame for R N if and only if there are constants A, B > 0 such that A y p x, y p dµ(x) B y p, y R N. (19) S N 1 We call µ a tight probabilistic p-frame if and only if we can choose A = B.

88 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Examples Example By symmetry considerations, it is not difficult to show that the uniform surface measure σ on S N 1 is always a tight probabilistic p-frame, for each 0 < p <. Lemma If µ is probabilistic frame, then it is a probabilistic p-frame for all 1 p <. Conversely, if µ is a probabilistic p-frame for some 1 p <, then it is a probabilistic frame.

89 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Tight probabilistic p-frames and spherical t designs Theorem Let p be an even integer. For any probability measure µ on S N 1, PFP(µ, p) (p 1) N(N + 2) (N + p 2), and equality holds if and only if µ is a probabilistic tight p-frame.

90 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Tight probabilistic p-frames and spherical t designs Proposition Let p = 2k be an even positive integer. A set Φ = {ϕ k } M k=1 SN 1 is a spherical p design if and only if the probability measure µ Φ = 1 M N k=1 δ ϕ k is a probabilistic tight p frame.

91 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Concluding remarks on probabilistic frames The 2-Wasserstein metric given by { } W2 2 (µ, ν) := min x y 2 dγ(x, y), γ Γ(µ, ν), (20) R N R N where Γ(µ, ν) is the set of all Borel probability measures γ on R N R N whose marginals are µ and ν, respectively. (P 2, W 2 ) form a metric space. Construction of frame path with various constraint. Optimization of frame related functionals, e.g., the probabilistic p th frame potentials, in the context of the Wasserstein metrics.

92 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Concluding remarks on probabilistic frames The 2-Wasserstein metric given by { } W2 2 (µ, ν) := min x y 2 dγ(x, y), γ Γ(µ, ν), (20) R N R N where Γ(µ, ν) is the set of all Borel probability measures γ on R N R N whose marginals are µ and ν, respectively. (P 2, W 2 ) form a metric space. Construction of frame path with various constraint. Optimization of frame related functionals, e.g., the probabilistic p th frame potentials, in the context of the Wasserstein metrics.

93 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Concluding remarks on probabilistic frames The 2-Wasserstein metric given by { } W2 2 (µ, ν) := min x y 2 dγ(x, y), γ Γ(µ, ν), (20) R N R N where Γ(µ, ν) is the set of all Borel probability measures γ on R N R N whose marginals are µ and ν, respectively. (P 2, W 2 ) form a metric space. Construction of frame path with various constraint. Optimization of frame related functionals, e.g., the probabilistic p th frame potentials, in the context of the Wasserstein metrics.

94 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Concluding remarks on probabilistic frames The 2-Wasserstein metric given by { } W2 2 (µ, ν) := min x y 2 dγ(x, y), γ Γ(µ, ν), (20) R N R N where Γ(µ, ν) is the set of all Borel probability measures γ on R N R N whose marginals are µ and ν, respectively. (P 2, W 2 ) form a metric space. Construction of frame path with various constraint. Optimization of frame related functionals, e.g., the probabilistic p th frame potentials, in the context of the Wasserstein metrics.

95 The p th frame potentials : definition and basic properties Probabilistic frame potential Probabilistic p th frame potential Thank You! okoudjou