LOCAL AND GLOBAL STABILITY OF FUSION FRAMES

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1 LOCAL AND GLOBAL STABILITY OF FUSION FRAMES Jerry Emidih Norbert Wiener Center Department of Mathematics University of Maryland, College Park November

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4 DEFINITION A frame F = {f i } i I in a Hilbert space H is a countable sequence {f i } H for which there exist A, B > 0 such that f H, A f 2 i I f, f i 2 B f 2. Developed by Richard Duffin and Albert Schaeffer in 1952 Generalization of orthonormal bases which allow for possibly redundant decompositions Frames have varied use in applications including image processing, wireless communication, and digital signal quantization because they are naturally robust to erasures

5 DEFINITION A frame F = {f i } i I in a Hilbert space H is a countable sequence {f i } H for which there exist A, B > 0 such that f H, A f 2 i I f, f i 2 B f 2. Developed by Richard Duffin and Albert Schaeffer in 1952 Generalization of orthonormal bases which allow for possibly redundant decompositions Frames have varied use in applications including image processing, wireless communication, and digital signal quantization because they are naturally robust to erasures

6 DEFINITION A frame F = {f i } i I in a Hilbert space H is a countable sequence {f i } H for which there exist A, B > 0 such that f H, A f 2 i I f, f i 2 B f 2. Developed by Richard Duffin and Albert Schaeffer in 1952 Generalization of orthonormal bases which allow for possibly redundant decompositions Frames have varied use in applications including image processing, wireless communication, and digital signal quantization because they are naturally robust to erasures

7 FRAME BASICS Frame operator S F : H H by S F (f ) = i I f, f i f i, A dual frame { f i } i I for F is a frame which satisfies f = f, f i f i =, i I i I f f i f i for all f H. By invertibility of the frame operator S F, {S 1 F f i} is a dual frame, known as the canonical dual frame.

8 In 2004, Peter Casazza and Shidong Li developed frames of subspaces for simple constructions of frames DEFINITION A fusion frame W = {(W i, c i )} i I in a separable Hilbert space H is a countable sequence of closed subspaces W i H and a sequence of weights {c i } i I R, c i > 0 for all i I for which there exist C, D > 0 such that f H, C f 2 i I c 2 i P Wi (f ) 2 D f 2, where P Wi is the orthogonal projection onto W i.

9 In 2004, Peter Casazza and Shidong Li developed frames of subspaces for simple constructions of frames DEFINITION A fusion frame W = {(W i, c i )} i I in a separable Hilbert space H is a countable sequence of closed subspaces W i H and a sequence of weights {c i } i I R, c i > 0 for all i I for which there exist C, D > 0 such that f H, C f 2 i I c 2 i P Wi (f ) 2 D f 2, where P Wi is the orthogonal projection onto W i.

10 DISTRIBUTED SENSING Casazza, Li, and Gitta Kutyniok give an example of a sensor network spread over a forest to measure temperature, where the sensors are divided into smaller sub-networks for processing. Within the fusion frame paradigm, the sub-networks form a set of redundant subspaces, so the signals can be processed globally, at one central processing center, or locally, at stations for each network. This example suggests that it s useful to consider fusion frames as a special case of frames rather than a generalization.

11 Tight fusion frames are defined to be fusion frames for which the bounds C and D can be chosen to be equal. A fusion frame system {(W i, c i, {f ij } j Ji )} i I is a fusion frame {(W i, c i )} i I for which F i = {f ij } j Ji is a frame for W i for each i I, known as a local frame.

12 Tight fusion frames are defined to be fusion frames for which the bounds C and D can be chosen to be equal. A fusion frame system {(W i, c i, {f ij } j Ji )} i I is a fusion frame {(W i, c i )} i I for which F i = {f ij } j Ji is a frame for W i for each i I, known as a local frame.

13 For a fusion frame W, define the analysis operator by T W (f ) := {c i P Wi f } i I and its adjoint, the synthesis operator, by TW ({v i} i I ) := i I c iv i, where f H and v i W i for each i I. Similar to the case of frames, the fusion frame operator S W : H H given by S W (f ) = TW T W(f ) = i I c2 i P Wi (f ) is a positive, self-adjoint, and invertible operator.

14 For a fusion frame W, define the analysis operator by T W (f ) := {c i P Wi f } i I and its adjoint, the synthesis operator, by TW ({v i} i I ) := i I c iv i, where f H and v i W i for each i I. Similar to the case of frames, the fusion frame operator S W : H H given by S W (f ) = TW T W(f ) = i I c2 i P Wi (f ) is a positive, self-adjoint, and invertible operator.

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16 PERTURBATION THEORY Let T : X X be a bounded linear map on a Banach space X. (I T )x λ( x + Tx ) = 1 λ 1+λ 1+λ Tx x 1 λ Tx for λ < 1. Paley and Wiener (1934) showed that if Sx Tx λ 1 Sx + λ 2 Tx then their codimensions are equal.

17 PERTURBATION THEORY Let T : X X be a bounded linear map on a Banach space X. (I T )x λ( x + Tx ) = 1 λ 1+λ 1+λ Tx x 1 λ Tx for λ < 1. Paley and Wiener (1934) showed that if Sx Tx λ 1 Sx + λ 2 Tx then their codimensions are equal.

18 PERTURBATION THEORY Let T : X X be a bounded linear map on a Banach space X. (I T )x λ( x + Tx ) = 1 λ 1+λ 1+λ Tx x 1 λ Tx for λ < 1. Paley and Wiener (1934) showed that if Sx Tx λ 1 Sx + λ 2 Tx then their codimensions are equal.

19 FUSION FRAME PERTURBATIONS If P and Q are orthogonal projections on H with 0 λ 1, λ 2 < 1 such that Pf Qf λ 1 Pf + λ 2 Qf for all f H, then P = Q. DEFINITION Let {W i } i I and {V i } i I be collections of closed subspaces in H with {c i } a positive real sequence, and let 0 λ 1, λ 2 < 1 and ɛ > 0. {(W i, c i )} is a (λ 1, λ 2, ɛ) perturbation of {(V i, c i )} if (P Wi P Vi )f λ 1 P Wi f + λ 2 P Vi f + ɛ f for all f H. Only the sub-stations are changed, not the weights.

20 FUSION FRAME PERTURBATIONS If P and Q are orthogonal projections on H with 0 λ 1, λ 2 < 1 such that Pf Qf λ 1 Pf + λ 2 Qf for all f H, then P = Q. DEFINITION Let {W i } i I and {V i } i I be collections of closed subspaces in H with {c i } a positive real sequence, and let 0 λ 1, λ 2 < 1 and ɛ > 0. {(W i, c i )} is a (λ 1, λ 2, ɛ) perturbation of {(V i, c i )} if (P Wi P Vi )f λ 1 P Wi f + λ 2 P Vi f + ɛ f for all f H. Only the sub-stations are changed, not the weights.

21 FUSION FRAME PERTURBATIONS If P and Q are orthogonal projections on H with 0 λ 1, λ 2 < 1 such that Pf Qf λ 1 Pf + λ 2 Qf for all f H, then P = Q. DEFINITION Let {W i } i I and {V i } i I be collections of closed subspaces in H with {c i } a positive real sequence, and let 0 λ 1, λ 2 < 1 and ɛ > 0. {(W i, c i )} is a (λ 1, λ 2, ɛ) perturbation of {(V i, c i )} if (P Wi P Vi )f λ 1 P Wi f + λ 2 P Vi f + ɛ f for all f H. Only the sub-stations are changed, not the weights.

22 PERTURBATION FUSION FRAME BOUNDS PROPOSITION (CASAZZA, LI, KUTYNIOK) Let {(W i, c i )} i I be a fusion frame for H with bounds C, D. Let {(V i, c i )} i I be a (λ 1, λ 2, ɛ) perturbation of {(W i, c i )} i I where 0 λ 1, λ 2 < 1, ɛ > 0, and (1 λ 1 ) C ɛ( i I c2 i ) 1/2 > 0. Then {(V i, c i )} i I is a fusion frame of H with fusion frame bounds [ (1 λ1 ) C ɛ( i I c2 i ) 1/2 ] 2 [ (1 + λ1 ) D + ɛ( and i I c2 i ) 1/2 ] λ 2 1 λ 2

23 In the case of a fusion frame system, we can also perturb the local frames: For {f i } i I and {g i } i I are sequences in H and 0 λ 1, λ 2 < 1, {g i } i I is a (λ 1, λ 2 )-perturbation of {f i } i I if i I a i (f i g i ) λ 1 i I a i f i + λ 2 i I a i g i for all {a i } i I l 2 (I). Useful lemma: Let W = span i I {f i } and V = span i I {g i }, then P W P V (f ) ( 1 λ1 1+λ 2 λ 1 1+λ 2 1 λ 1 λ 2 ) P V (f ) for all f H.

24 In the case of a fusion frame system, we can also perturb the local frames: For {f i } i I and {g i } i I are sequences in H and 0 λ 1, λ 2 < 1, {g i } i I is a (λ 1, λ 2 )-perturbation of {f i } i I if i I a i (f i g i ) λ 1 i I a i f i + λ 2 i I a i g i for all {a i } i I l 2 (I). Useful lemma: Let W = span i I {f i } and V = span i I {g i }, then P W P V (f ) ( 1 λ1 1+λ 2 λ 1 1+λ 2 1 λ 1 λ 2 ) P V (f ) for all f H.

25 THEOREM 1 THEOREM (CASAZZA, LI, KUTYNIOK) Let {(W i, c i, {f ij } j Ji )} i I be a fusion frame system for H with fusion frame bounds C, D. Choose 0 λ 1, λ 2 < 1 and ɛ > 0 such that (1 λ 1 ) C ɛ( i I c2 i ) 1/2 > 0 and 1 ɛ2 2 = 1 λ 1 1+λ 1+λ 2 λ λ 1 λ 2. For every i, let {g ij } j Ji be a (λ 1, λ 2 )-perturbation of {f ij } j Ji and let V i = span{g ij } j Ji. Then {(V i, c i, )} i I is a fusion frame for H with fusion frame bounds [ C ɛ ( i I c 2 i ) 1/2 ] 2 and [ D + ɛ ( i I c 2 i ) 1/2 ] 2.

26 THEOREM 1 PROOF 1 Use the useful lemma to get two estimates: (I P Vi )(P Wi f ) 2 ɛ2 2 P W i f 2 and vice versa. 2 (P Wi P Vi )f 2 = (P Wi P Vi ) 2 f, f = (P Wi P Vi P Wi +P Vi P Wi P Vi )f, f (I P Vi )(P Wi f ) + (I P Wi )(P Vi f ) f ɛ2 2 P W i f f + ɛ2 2 P V i f f ɛ 2 f 2. 3 Hence, W is a (0, 0, ɛ)-perturbation of V.

27 THEOREM 1 PROOF 1 Use the useful lemma to get two estimates: (I P Vi )(P Wi f ) 2 ɛ2 2 P W i f 2 and vice versa. 2 (P Wi P Vi )f 2 = (P Wi P Vi ) 2 f, f = (P Wi P Vi P Wi +P Vi P Wi P Vi )f, f (I P Vi )(P Wi f ) + (I P Wi )(P Vi f ) f ɛ2 2 P W i f f + ɛ2 2 P V i f f ɛ 2 f 2. 3 Hence, W is a (0, 0, ɛ)-perturbation of V.

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29 FUSION FRAME DUAL Canonical fusion frame dual is W = {(S 1 W W i, c i S 1 W W i )} i I, where S 1 W W i is the inverse of the fusion frame operator restricted to W i.

30 WEIGHTED FUSION FRAME PERTURBATION DEFINITION Let µ > 0 and let W = {(W i, c i )} i I and V = {(V i, d i )} i I be fusion frames on H. V is said to be a µ perturbation of W if T W T V µ, where T W and T V are the analysis operators of W and V, respectively. This implies c i P Wi d i P Vi µ for every i I. In finite dimensions, when the sequences of weights are the same, both definitions are equivalent.

31 WEIGHTED FUSION FRAME PERTURBATION DEFINITION Let µ > 0 and let W = {(W i, c i )} i I and V = {(V i, d i )} i I be fusion frames on H. V is said to be a µ perturbation of W if T W T V µ, where T W and T V are the analysis operators of W and V, respectively. This implies c i P Wi d i P Vi µ for every i I. In finite dimensions, when the sequences of weights are the same, both definitions are equivalent.

32 THEOREM 2 THEOREM (KUTYNIOK, PATERNOSTRO, PHILIPP) Let W = {(W i, c i )} i I be a fusion frame for H with frame bounds A B and let V = {(V i, d i )} i I be a µ perturbation of W, where 0 < µ < A. If there exists τ > 0 such that τ is a lower bound for both {c i } i I and {d i } i I, then the canonical fusion frame dual Ṽ of V is a Cµ perturbation of the canonical fusion frame dual W of W, where C = α2 + β 2 [ 1 + (A 1 + B) 2 ( 2 + αβ 2) ] + β 2 (1 + α 2 β 2 ) A A τ with α := 2 B + µ and β := ( A µ) 1.

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34 Perhaps only one or a few subspaces are perturbed. What about when an entire subspace is erased? Major strength of frames is robustness to erasure. Robert Calderbank, Taotao Liu, Gitta Kutyniok, and Ali Pezeshki considered conditions for finite fusion frames that optimize recovery.

35 Perhaps only one or a few subspaces are perturbed. What about when an entire subspace is erased? Major strength of frames is robustness to erasure. Robert Calderbank, Taotao Liu, Gitta Kutyniok, and Ali Pezeshki considered conditions for finite fusion frames that optimize recovery.

36 FINITE FUSION FRAMES For the rest of the talk, all finite dimensional real Hilbert spaces H = R M. We also want to consider equally weighted spaces to simplify the problem. Let m i denote the dimension of subspace W i.

37 FINITE FUSION FRAMES For the rest of the talk, all finite dimensional real Hilbert spaces H = R M. We also want to consider equally weighted spaces to simplify the problem. Let m i denote the dimension of subspace W i.

38 MODEL OF VECTOR RECOVERY z i = Ui T x + n i, for i = 1,..., N is the fusion frame measurement at each subspace. x is a random zero-mean vector with variance σx 2, n i is a realization of an additive white noise vector with variance σn 2 U i is a left orthogonal N m i -matrix such that U T i U i = I mi and U i U T i = P Wi.

39 MEAN-SQUARED RECOVERY ERROR Want to minimize mean-squared error (MSE) in linearly estimating x from z i. Let MSE k denote the MSE when k subspaces are erased. Minimum is achieved when fusion frame is tight, given by MSE 0 = Mσ2 nσx 2. σn 2 + Lσ2 x M

40 MEAN-SQUARED RECOVERY ERROR Want to minimize mean-squared error (MSE) in linearly estimating x from z i. Let MSE k denote the MSE when k subspaces are erased. Minimum is achieved when fusion frame is tight, given by MSE 0 = Mσ2 nσx 2. σn 2 + Lσ2 x M

41 ONE ERASURE, MSE 1 We consider where m i = m is constant for all i, since MSE 1 naturally increases with m i. MSE 1 = MSE 0 + Erasure Term MSE 1 is then a function of m which has a maximum at m m min, if m max m or m = if m min m m max and MSE(m min ) MSE(m max ), m max, otherwise.

42 ONE ERASURE, MSE 1 We consider where m i = m is constant for all i, since MSE 1 naturally increases with m i. MSE 1 = MSE 0 + Erasure Term MSE 1 is then a function of m which has a maximum at m m min, if m max m or m = if m min m m max and MSE(m min ) MSE(m max ), m max, otherwise.

43 ONE ERASURE, MSE 1 We consider where m i = m is constant for all i, since MSE 1 naturally increases with m i. MSE 1 = MSE 0 + Erasure Term MSE 1 is then a function of m which has a maximum at m m min, if m max m or m = if m min m m max and MSE(m min ) MSE(m max ), m max, otherwise.

44 ONE ERASURE, MSE 1 We consider where m i = m is constant for all i, since MSE 1 naturally increases with m i. MSE 1 = MSE 0 + Erasure Term MSE 1 is then a function of m which has a maximum at m m min, if m max m or m = if m min m m max and MSE(m min ) MSE(m max ), m max, otherwise.

45 TWO ERASURES, MSE 2 MSE 2 = MSE 1 + Cross Term from mixed projections P Wi P Wj Want to minimize the trace of P Wi P Wj minimizing eigenvalues, which is equivalent to Define principal angles θ k (i, j) for 1 k M as inverse cosines of eigenvalues and chordal distance between W i and W j, as ( M ) 1/2 d c (i, j) := k=1 sin2 θ k (i, j) MSE 2 is minimized when MSE 1 is minimized and chordal distance is maximized

46 TWO ERASURES, MSE 2 MSE 2 = MSE 1 + Cross Term from mixed projections P Wi P Wj Want to minimize the trace of P Wi P Wj minimizing eigenvalues, which is equivalent to Define principal angles θ k (i, j) for 1 k M as inverse cosines of eigenvalues and chordal distance between W i and W j, as ( M ) 1/2 d c (i, j) := k=1 sin2 θ k (i, j) MSE 2 is minimized when MSE 1 is minimized and chordal distance is maximized

47 TWO ERASURES, MSE 2 MSE 2 = MSE 1 + Cross Term from mixed projections P Wi P Wj Want to minimize the trace of P Wi P Wj minimizing eigenvalues, which is equivalent to Define principal angles θ k (i, j) for 1 k M as inverse cosines of eigenvalues and chordal distance between W i and W j, as ( M ) 1/2 d c (i, j) := k=1 sin2 θ k (i, j) MSE 2 is minimized when MSE 1 is minimized and chordal distance is maximized

48 TWO ERASURES, MSE 2 MSE 2 = MSE 1 + Cross Term from mixed projections P Wi P Wj Want to minimize the trace of P Wi P Wj minimizing eigenvalues, which is equivalent to Define principal angles θ k (i, j) for 1 k M as inverse cosines of eigenvalues and chordal distance between W i and W j, as ( M ) 1/2 d c (i, j) := k=1 sin2 θ k (i, j) MSE 2 is minimized when MSE 1 is minimized and chordal distance is maximized

49 MORE THAN TWO ERASURES For equidistant tight fusion frame with constant m dimension, MSE k = MSE 3 for k 3. It is not known if such a fusion frame exists. If it does, it s an optimal Grassmannian packing of m -dimensional subspaces in R N.

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51 HETEROGENEOUS DATA FUSION We want to combine heterogenous information, not homogeneous like in distributed processing example. How do we combine different information to make up for gaps in knowledge of one? Studying stability explains the applicability of fusion frames as a data fusion model.

52 HETEROGENEOUS DATA FUSION We want to combine heterogenous information, not homogeneous like in distributed processing example. How do we combine different information to make up for gaps in knowledge of one? Studying stability explains the applicability of fusion frames as a data fusion model.

53 HETEROGENEOUS DATA FUSION We want to combine heterogenous information, not homogeneous like in distributed processing example. How do we combine different information to make up for gaps in knowledge of one? Studying stability explains the applicability of fusion frames as a data fusion model.

54 PRIMARY REFERENCES Casazza, P. G., Kutyniok, G., & Li, S. (2008). Fusion frames and distributed processing. Applied and computational harmonic analysis, 25(1), Kutyniok, G., Paternostro, V., & Philipp, F. (2015). The Effect of Perturbations of Frame Sequences and Fusion Frames on Their Duals. arxiv preprint arxiv: Kutyniok, G., Pezeshki, A., Calderbank, R., & Liu, T. (2009). Robust dimension reduction, fusion frames, and Grassmannian packings. Applied and Computational Harmonic Analysis, 26(1),

55 ADDITIONAL REFERENCES Casazza, P. G., Fickus, M., Mixon, D. G., Wang, Y., & Zhou, Z. (2011). Constructing tight fusion frames. Applied and Computational Harmonic Analysis, 30(2), Casazza, P. G., & Kalton, N. J. (1999). Generalizing the Paley-Wiener perturbation theory for Banach spaces. Proceedings of the American Mathematical Society, Casazza, P. G., & Kutyniok, G. (2004). Frames of subspaces. Contemporary Mathematics, 345,

FUSION FRAMES AND DISTRIBUTED PROCESSING

FUSION FRAMES AND DISTRIBUTED PROCESSING PETER G. CASAZZA, GITTA KUTYNIOK, AND SHIDONG LI Abstract. Let { i } be a (redundant) sequence of subspaces of a Hilbert space each being endowed with a weight