ECE Department, U. of Minnesota, Minneapolis, MN. 3SPICE-ECE-UMN N. Sidiropoulos / SIAM50

Transcription

1 ( *;: <* < 4 7 #$ % %'&( +*), -/. # #0 98 =5 2> 4! " ECE Department, U. of Minnesota, Minneapolis, MN N. Sidiropoulos

2 Roadmap 2

3 Outline Motivation Carathéodory / Pisarenko Low-rank decomposition of N-way arrays (aka tensors) Kruskal Main results, #: Generalizing Kruskal to N-D Harmonic prelude: Carathéodory from N-D Kruskal Main results, #2: Generalizing Carathéodory to N-D Conclusions 3

4 . Displacement estimation in N dimensions; image modeling;... Multicarrier COM; fluorescence excitation-emission matrices; Joint AZ-EL-Delay-Doppler in SAP for COM and Radar; Seismic; Speech; Medicine (EEG, ECG);... Communications: Synchronization; Radar; Periodicities in time series: econometrics, astronomy; Harmonics everywhere 4

5 := ( ) 4 < : ( Underpinning technique: uniqueness of harmonic parameterization T, Windows, ilterbanks, Pisarenko, Capon, MUSIC, ESPRIT,... find ω f π π, f xi c f e jω f f ni i i I measurements * # % * - < ; & *,; Given a finite number of Spectral analysis 5

6 D (diag) contains positive reals. π ω f π f distinct v f e jω f e jω f Ï T, f V : I T VDVH Any psd I I Toeplitz T of rank I can be uniquely decomposed as Carathéodory / Pisarenko 6

7 are unique. > if I, then ω f π π and c f 2, f xi c f f 2 e jω f i i I *&( * (Carathéodory s Uniqueness Result, c. 9) Given Carathéodory / Pisarenko, Cont. 7

8 PARAAC/CANDECOMP X X = b a + + a 2 b 2 b 3 a 3 c c 2 c 3 = b + b b a a 2 a 3 act : Low-rank matrix (2-way array) decomposition not unique act 2: Low-rank 3- and higher-way array decomposition (PARAAC) is unique :-) 8

9 k-rank: ka maximum : r such that every r columns of A are linearly independent N Sidiropoulos & Bro J. Chemo 2000: any N, N n k ranks 2 : Kruskal 977, N 3 IR: ka kb kc 2 2 X = b + b b a a 2 a 3 c c 2 c 3 b a a 2 X = + + b 2 b 3 a 3 Backbone 9

10 ck k M f h k k M f : g M f key : C G H Define ML matrix C J i I j k ML yi j k : x i j k M k k M M Unfolding into 3-D (matricizing [Kiers]) xi j l m ai f b j f f gl f hm f Given quadrilinear model of rank Proof of Theorem (sketch) 0

11 kb A 0 whereas if ka 0 and/or kb 0 kb A min ka kb I = * (k-rank of KRP - Sidiropoulos & Liu, 999) A a, B J. If k A and kb, then it holds that b b a Instrumental Lemma

12 kb Induction for N 4 ka kb kgkh 2 3 else if kg kh then ka kb min kg kh WLOG WMA ka If kgkh 2, and hence ka kb then ka kb kg kh kg kh ka min kb kg kh 2 2 Apply 3-way result & Lemma: Proof of Theorem (sketch) 2

13 Higher N, better ID for given rank In N dimensions, decomposition of rank- arrays a.s. unique 2 n min In 2 N N Stronger than Kruskal applied to 3-D slices: matching issue Even disregarding the matching issue, consider: ka kb kg kh 2, 3, and 3 Kruskal: :-( New result: :-) ull k-rank: Discussion 3

14 unique. > with c fand a f, if I 2 then a f c f, f xi c f a i f f i I *&( * Given Harmonic Prelude: -D harmonics (including damping and phase) 4

15 im IM i... I IM I M I for xi im : x i im M f f c f a i im M f c f a i f aim f Define the M-way array Proof 5

16 Multidimensional embedding: a x y ay ax -D exponential M-D rank-one array! 6

17 Multidimensional embedding, Cont. 7

18 and thus the proof is complete 2 I or, equivalently, 2 I 2 I 2 M m min becomes Im for all ). Then the identifiability condition Pick M I and Im 2 for all m (this choice actually maximizes m min Im 2 M M Unique, provided Proof, Cont. 8

19 So we get samples of the original model from 2 ; from the previous result, unique if I I (since integer) - Carathéodory i i 0 to i2i i 2I 2I 2 2 zi : xi c f e jω f f I ï i I yi : x Ï i c f e jω f f i i c f : I c f 2 e jω f I ; xi c f f 2 e jω f i i I Given Recovering Carathéodory: Conjugation & folding 9

20 > a f then there exist unique c f n, n N that give rise to xi in. n In N 2 N a f n a f2 n, f f2 and all n, if for in In 2, n N, with c fand a f nsuch that xi in c f f n N a i n f n *&( * (Sidiropoulos, IEEE Trans. IT, 200) Given a sum of exponentials in N-dimensions result possible (also for N 2) or N 3 uniqueness by means of N-way Theorem, however, better N-D 20

21 N and c f, f are unique provided that the rank-one factors c f N n 2 m full k-rank, n I n ai n m f m n and hence a f rom N-way Theorem, and the fact that Vandermonde matrices have c f f n m N a i n f n in m In m n c f f n N a i n f n in In c f f In N n a i n f n f n ain In xi i I in in I N : x i i I I 2 in in I N IN 2 Define the extended multi-way array Proof n, 2

22 which completes the proof n In N 2 N Note that the sum on the right hand side is the total number of effective dimensions Equivalently, uniqueness holds provided n m N 2 2 In n In N Proof, Cont. 22

23 &;; ( - > ra B ka B min PL IJ I J a s 2 &( or a pair of matrices A I J and B, ra B ka B min PL IJ 2 a s A (Jiang, ten Berge, Sidiropoulos, IEEE Trans. SP, 200) or I a pair of Vandermonde matrices and J B *&( * P-a.s. Uniqueness: Key Result 23

24 b f ) is continuous with respect to the k l > and PL (the distribution used to draw the 2 complex exponential parameters, f 2 Lebesgue measure in, then the parameter triples c f, 2 f almost surely unique. are PL a f a f b f 2 K 2 L 2 if xk l k c f a f f K 4 L 4 f b l *&( * (Jiang, ten Berge, Sidiropoulos, IEEE Trans. SP, 200) Given P-a.s. Uniqueness: Damped Exponentials 24

25 ν f k l > Trick: 2-D conjugate folding then the parameter triples unique. ω f c f, f are PL Π2 -a.s. f ) is continuous with respect to the Lebesgue measure in Π2, and PL Π2 ω f ν f (the distribution used to draw the 2 frequencies, K 2 L 2 if xk l c f e jω f f K 3 L 3 k e jν f l *&( * (Liu, Sidiropoulos, IEEE Trans. SP, 2002) Given P-a.s. Uniqueness: Undamped Exponentials 25

26 P-a.s. Uniqueness Can be extended to N-D harmonic decomposition, see Sidiropoulos et al, IEEE Trans. SP Conditions close to eqns unknowns in 2-D 26

27 Concluding remarks retrieval algorithms (EVD) that work under ID only ML iterative algorithms that perform close to CRB even at moderate sample sizes and SNRs Necessity of best known ID conditions? ast algorithms that perform close to ML? 27