Prony s method under an almost sharp multivariate Ingham inequality

Transcription

1 Prony s method under an almost sharp multivariate Ingham ineuality Stefan Kunis H. Michael Möller Thomas Peter Ulrich von der Ohe The parameter reconstruction problem in a sum of Dirac measures from its low freuency trigonometric moments is well understood in the univariate case and has a sharp transition of identifiability with respect to the ratio of the separation distance of the parameters and the order of moments. Towards a similar statement in the multivariate case, we present an Ingham ineuality which improves the previously best known dimension-dependent constant from suare-root growth to a logarithmic one. Secondly, we refine an argument that an Ingham ineuality implies identifiability in multivariate Prony methods to the case of commonly used max-degree by a short linear algebra argument, closely related to a flat extension principle and the stagnation of a generalized Hilbert function. Key words and phrases : freuency analysis, exponential sum, moment problem, super-resolution, Ingham ineuality. 1 AMS Mathematics Subject Classification : 65T4, 4C15, 3E5, 65F3 1 Introduction The reconstruction of a Dirac ensemble from its low freuency trigonometric moments or euivalently the parameter reconstruction problem in exponential sums has been studied since [13] as Prony s method, see e.g. [37, 35, 16, 41, 3, 15] and the survey [3]. Popular different approaches to this reconstruction problem include, without being exhaustive, convex optimization [8, 7, 4, 3], where the problem has been termed super-resolution and can be solved via a lifting techniue by semidefinite optimization, and maximum likelihood techniues [39, 5, 6, 1, 34, 7, 8, 4]. Multivariate variants of Prony-like methods have been considered e.g. in [18, 38, 1, 33, 31, 14, 1, 36, ] using generic arguments, projections to univariate problems, or Gröbner basis techniues. The previously best known condition for an Ingham ineuality to hold is n > d/, where n denotes the order of the used moments, the separation distance of the parameters, and d the dimension, respectively. Together with a Gröbner basis construction which asks for Osnabrück University, Institute of Mathematics {skunis,petert,uvonderohe}@uos.de TU Dortmund, Fakultät für Mathematik moeller@mathematik.tu-dortmund.de 1

2 a total-degree setting and thus introduces a factor d, this results in a deterministic a-priori bound n d 1) > d 3/ for the success of Prony s method, see also []. In this paper we refine an appropriate Ingham ineuality as well as the algebra techniues to improve this a-priori condition to n 1) > 3 + log d. After fixing notation, Section. improves the Ingham ineuality [17, 19, 33] by constructing a function with compact support in space domain and zero crossing at an l p -ball in freuency domain. In Section.3, we refine our previous analysis [] of the vanishing ideal of the parameter set and prove that an interpolation condition, euivalent to some full rank condition, implies that the parameters can be identified by polynomials of certain max-degree. In total, the multivariate Prony method correctly identifies the parameters under a deterministic condition which is sharp up to a logarithmic factor in the space dimension. Main results In what follows, we establish a deterministic condition for the success of Prony s method relying on two ingredients - a new Ingham ineuality and an analysis of the parameters vanishing ideal. Ingham s ineuality generalizes Parseval s identity in the sense that a norm euality between a space and a freuency representation of a vector is replaced by a certain norm euivalence. With respect to the identifiability in Prony s method, we are interested in establishing conditions on the parameters such that a non trivial lower bound in this ineuality exists. Theorem.4 and Corollary.5 give such a result under a separation condition on the parameters and improve upon [17, 19, 33] by a smaller space dimension dependent constant. Subseuently, Theorem.8 and Corollary.1 apply the established Ingham ineuality to prove success of Prony s method by adapting standard arguments from algebraic geometry [4, App. B] to our so-called max-degree setting..1 Preliminaries Throughout the paper, d N always denotes the dimension and T d := {z C : z = 1} d the torus with parameterization T d z = e πit for a uniue t [, 1) d. Now let M N, pairwise distinct parameters t j [, 1) d, j = 1,..., M, be given, and define their wrap-around) separation distance by := min r Z d, j l t j t l + r. For given coefficients ˆf j C \ {}, the trigonometric moment seuence of the complex Dirac ensemble τ : P[, 1) d ) C, τ = M ˆf j δ tj, is the exponential sum f : Z d C, M k e πikt dτt) = ˆf j zj k, [,1) d with parameters z j := e πit j = e πit j,1,..., e πit j,d) T d. A convenient choice for the truncation of this seuence is k := max{ k 1,..., k d } n and our aim is to reconstruct the parameters t j [, 1) d and coefficients ˆf j C\{} from the trigonometric moments fk), k Z d, k n. Prony s method tries to realize the parameters z j as common zeros of certain polynomials as follows. We identify C n+1)d with the space of polynomials of max-degree n, i.e., Π n :=

3 {p : pz) = k N d, k n p kz k }, and for K C n+1)d we let V K) := {z C d : pz) = for all p K}. Given the above moments, it turns out that an appropriate set of polynomials is the kernel of the multilevel Toeplitz matrix T := T n := fk l)) k,l N d, k, l n Cn+1)d n+1) d. Direct computation easily shows the factorization T = A DA with diagonal matrix D := diag ˆf j ) C M M and Vandermonde matrix ) A := A n := zj k,...,m C M n+1)d, z j := e πit j C d, k N d, k n which is our entry point for the subseuent analysis of the problem.. Ingham s ineuality The univariate case is well understood and the popular paper [6] used a so-called Beurling- Selberg majorant and minorant to prove a discrete Ingham ineuality which has been generalized to the unit disk recently []). Tensorizing the majorant easily gives an upper bound in a multivariate discrete Ingham ineuality but simple attempts to provide a minorant and thus a lower bound failed, see e.g. [5]. A construction of a valid minorant by linear combinations of univariate majorants and minorants can be found e.g. in [9], these seem to become effective only if n > C d, i.e., show a linear dependence in the space dimension. A similar bound follows from [] using localized trigonometric polynomials and a packing argument. Moreover note that the Fourier analytic approach to sphere packing problems [11, 4] asks for a similar but not uite the same construction of a function. The classical and currently best known construction [17, 19, 33] uses an eigenfunction of the Laplace operator on the cube and becomes effective if n > 1 d. Subseuently, we replace the Laplace operator by the sum of higher order derivatives, improve the previous bound for d > 5, and show a logarithmic growth of the space dependent constant. Lemma.1. Let d, r N, p := r, n, >, and the functions ϕ : R R, ) ) r 1 x, x < ϕx) :=,, otherwise, and ψ : R d R, ψ = πn) p 1) r d p x p s s=1 be given, see Figures.1 and. for examples, then ) d ϕ ϕ, i) the Fourier transform ˆψv) := R ψx)e πivx dx is bounded and obeys d {, v p n ˆψv), v p n, l=1 3

4 ii) suppψ = [, ] d, iii) and ψ) > if n > C p p d with C p p + 3)/eπ). Proof. We have the weak r-th derivative { ) 1) ϕ r) r r 4 r r!p x r, x < x) =,, otherwise, where P r denotes the r-th Legendre polynomial with normalization P r 1) = 1. This derivative is of bounded variation and thus, the Fourier transform of the function ϕ obeys ˆϕv) C1 + v ) r 1. Now the first assertion follows from ˆψv) = πn) p ) d d πv s ) p ˆϕv l )). l=1 The second claim easily follows from suppϕ = [, ]. Moreover, we have ϕ ϕ) = 1 1 and noting ϕ r) being odd for r odd, we get s=1 4 ϕ r) ϕ r) r ) r! 1) r ) = r 1 x ) p dx = πp! Γp + 3 ) 1 1 P r x)) dx = 4p r!) 1) r p + 1) p 1. Finally, this implies ) ψ) = ϕ ϕ)) d 1 πn) p ϕ ϕ) 1) r d ϕ r) ϕ r) ) = ϕ ϕ)) d 1 πn) p ) πp! Γp + 3 ) d4p r!) p + 1) p 1 >, provided the term in brackets is positive. Using the Legendre duplication formula for Γp+), this is euivalent to n > C p p d with a constant C p := Γ p + 1) Γ p + 3 ) ) π p Γ p+3 1 p )) 1 Γ p + 3 p π p + 3 eπ,.1) where the first ineuality follows from monotonicity and the second by Stirling s approximation. Remark.. The construction of Lemma.1 for p = reads as ) 1 x ϕx) :=, x <,, otherwise, 4

5 leading to 1 Z ϕ ϕ) = 1 x ) dx = 1 8, 15 ϕ ϕ ) = 4 4, 3 5/ and we get ψ) > for n > π d.533 d. A minor improvement is possible by choosing the function ϕ : R R, cos πx, x <, ϕx) :=, otherwise, which has a distributional -nd derivative ϕ = π ϕ + Cδ± and thus π ϕ ϕ). Proceeding as in Lemma.1, we get ψ) > for n > 1 d, see also [19, 33]. ϕ ϕ ) = ϕ ϕ ) = a) Contour plot of ψ, compactly supported in the b) Contour plot of ψ, positive inside and nondashed suare, positive at origin. positive outside the dashed ` -ball. Figure.1: p =, d =, =.1, n = 1, positive values blue hatched, negative values red. Remark.3. The construction of Lemma.1 for p = 4 reads as 1 x, x <, ϕx) :=, otherwise, leading to ϕ ϕ) = Z x )4 dx = 18, 315 ϕ ϕ ) = 64 16, 5 4 5

6 and we get ψ) > for n > 4 63/ π 4 d d. An alternative choice is the function ϕ : R R, { 1 + cos πx ϕx) :=, x <,, otherwise, which has a weak -nd derivative of bounded variation ϕ x) = 4π χ, ) x) ϕx) ) and the Fourier transforms obey the bounds ˆϕv) C1 + v ) 3 and ˆψv) C. Moreover, we have ϕ ϕ = 16π4 ) 4 ϕ ϕ ϕ χ, ) + χ, ) χ, ), ϕ ϕ) = 3, and ϕ χ, ) ) =, which yields ψ) = ϕ ϕ)) d 1 16π 4 n 4 ϕ ϕ)) dϕ ϕ )) ) = 3 8π 4 ϕ ϕ)) d 1 n 4 d3 ) 4 > if n > 4 d/ d. Finally, a minor improvement is possible as follows. Let σ.365 be the first positive root of cos t sinh t + cosh t sin t and the function ϕ : R R, ϕx) := { cosh σ cosh σ cos σ cosσx/) cos σ cosh σ cos σ coshσx/), x <,, otherwise, be given, then ϕ solves the biharmonic eigenvalue problem d 4 16σ4 ϕx) = dx4 4 ϕx), x <, ϕ± ) = ϕ ± ) =. Globally, we have a 1-st derivative, a weak -nd derivative, and distributional 3-rd and 4-th derivatives. In particular, we have ϕ 4) = 16σ 4 4 ϕ 4) + Cδ ± and thus ϕ ϕ ) = ϕ ϕ 4) ) = 16σ4 4 ϕ ϕ). Proceeding as in Lemma.1, we get ψ) > for n > σ π 4 d d. Theorem.4. Let n >, d, p, M N, p even, t j [, 1) d, j = 1,..., M, with and n > p+3 eπ min t j t l + r > r Z d, j l p d, then there exists a constant c >, depending only on d, p, n, such that k Z d k p n M ˆf j e πikt j c M ˆf j, for all choices ˆf j C, j = 1,..., M. 6

7 a) Contour plot of ψ, compactly supported in the b) Contour plot of ψ, positive inside and nondashed suare, positive at origin. positive outside the dashed `4 -ball. Figure.: p = 4, d =, =.1, n = 1, positive values blue hatched, negative values red. Proof. Let the function ψ be as in Lemma.1, then max ψ v) v Rd M fˆj eπiktj k Zd kkkp n ψ k) M M fˆj fˆ` `=1 = ψ) fˆj eπiktj k Zd = M M ψtj t` + r) r Zd fˆj, where the ineuality is implied by Lemma.1 i), Poisson s summation formula yields the first euality, and Lemma.1 ii) and the separation of the nodes tj the last euality. Finally, the assertion follows since Lemma.1 iii) assures ψ) >. Corollary.5. With the notation of Section.1, the condition n > cd, see Table.1, or n > 3 + log d implies rank An = M. Proof. First note that An = zjk,...,m k Nd,kkk n n d e,...,d n e) = diag zj zjk,...,m k { d n e,...,b n c}d and thus An has rank M if and only if the second matrix on the right hand side has this rank. p Dropping the last column for odd n, this is assured by Theorem.4 if n > p+3 d, since eπ M k Zd kkk n fˆj eπiktj M k Zd fˆj eπiktj > kkkp n 7

8 d c d Table.1: Explicit constants c d = min p N C p p d, see Euation.1), or its minor improvements via Remark. and.3. for ˆf C M \ {}. Choosing p := log d yields the assertion since p + 3 p p + 3 p d e p = p + 3 eπ eπ π e log d π. e Remark.6. Corollary.5 can also be found in [, Cor. 4.7] and [33, Lem. 3.1] under the stronger conditions n + 1) > d and n > d, respectively. Regarding a trivial lower bound of the constant c d, a d-fold Cartesian product of euispaced parameters in each dimension consists of M = d parameters in total and thus rank A n + 1) d < M if n + 1) < 1. Moreover, a non-trivial lower bound in [9, Section 5.7] indicates that functions as in Lemma.1 necessarily imply that the constant c d tends to infinity for d. Going beyond the rank of the Vandermonde matrix A n and considering its condition number or euivalently upper and lower bounds in the Ingham ineuality, [, Cor. 4.7] implies a uniform bound κa n W A n) 1 + d) d+1 n + 1)) d+1 d) d+1 for each -separated parameter set with n + 1) 4d and a certain diagonal preconditioner W. On the other hand, every weaker condition n + 1) c d 1 ε, c >, ε, 1), implies at least a subexponential growing condition number, since [, Cor. 4.11] leads to κa n A n) = n + 1) n + 1) ) d n + 1) ) d exp d ε c 1 log ) for euispaced nodes with n + 1) N. For a numerical example, let d = 56 and consider euispaced nodes with N n + 1) 4.98 c 56 - while Corollary.5 assures full rank of A n, we have κa n A n) = 5/4) Prony s method As outlined in Section.1, Prony s method tries to realize the unknown parameters z j as common roots of d-variate polynomials belonging to the kernel of the multilevel Toeplitz matrix T n. In [1], we provided the well-known a-priori condition n > M which however implies the need of OM d ) trigonometric moments for the reconstruction of M parameters. We improved this in [] to the a-priori condition n d 1) > d 3/ using Gröbner basis arguments and a variant of the flat extension principle [1, 3]. This allows for the reconstruction of M well distributed parameters from OM) trigonometric moments but the constant deteriorates with larger space dimension. Subseuently, we refine a variant of the flat extension principle to our max-degree setting and give a simple linear algebra proof. Together with the improved Ingham ineuality, Prony s method succeeds under the weakened condition n 1) > 3 + log d. 8

9 Lemma.7. With the notation of Section.1, if rank A n = rank A n+1 for some n N, then rank A l = M for all l n. Proof. Let r := rank A n and pick k 1,..., k r N d with k i n such that the matrix B := z k i j ),...,M C M r i=1,...,r has rank r. Subseuently, we let m N d, m / {k 1,..., k r }, m n + and B m := B, z m j ),...,M ) C M r+1. If m n+1, then B m is a submatrix of A n+1 and r = rank B rank B m rank A n+1 = r by assumption. If m = n +, then m = s + k, s = 1, k = n + 1, implies z m j = z s j z k j = r i=1 c i z s+k i j, c i C, s + k i n + 1, and thus z m j ),...,M is linear dependent on the columns of A n+1 and thus on the columns of B, i.e., rank B m = r. This shows rank A n+ = r and inductively rank A l = r for all l n. Finally, the space Π M of polynomials of max-degree at most M interpolates any set of M points and thus rank A M = M and r = M. Theorem.8. With the notation of Section.1, if rank A n = M, then V ker A n+1 ) = {z 1,..., z M }. Proof. Clearly, holds. To prove the reverse inclusion, assume that there is a For l N consider the matrix z V ker A n+1 ) \ {z 1,..., z M }. C l := zj k ) j=,...,m C M+1 l+1)d, k N d, k l i.e. A l extended by the row z k) k l. We show that ker C l = ker A l for all l n + 1. The inclusion is clear. Let p ker A l. Since l n + 1, we can consider p as an element of ker A n+1, which, by assumption, yields pz ) =. By the definition of C l we have C l p = considering p as an element of ker A l again). Thus ker C l = ker A l for all l n + 1, and therefore rank C l = rank A l. In particular, rank C n = rank A n = M and rank C n+1 = rank A n+1 = M, thus rank C n = rank C n+1. By Lemma.7 we have rank C n = {z,..., z M } = M + 1, a contradiction. Remark.9. Inspection of the proofs readily shows that the results of Lemma.7 and Theorem.8 hold for arbitrary pairwise different z j C d. The Vandermonde matrix A n is the representation matrix of the evaluation homomorphism which maps a polynomial to its values at the prescribed points z j. In algebraic geometry and with the max-degree replaced by the total degree, the rank of such maps is called Hilbert function. It is well known that such a function is strictly increasing with n up to stagnation at the value M, see e.g. [4, App. B]. The previous Lemma.7 shows by elementary arguments that this is true also for the max-degree. 9

10 Corollary.1. With the notation of Section.1, the condition n 1) > c d, see Table.1, or n 1) > 3 + log d implies V ker T n ) = {z 1,..., z M }. Proof. Corollary.5 implies rank A n 1 = M and thus V ker A n ) = {z 1,..., z M } follows from Theorem.8. Finally note that always ker A n ker T n and euality follows from rank A n = M since p ker T n implies A n p kera nd) = imd A n ) = {} and thus p ker A n. 3 Summary We have shown that Prony s method identifies a Dirac ensemble from its first moments provided an associated Vandermonde matrix has full rank. This is indeed fulfilled if the number of given coefficients exceeds a constant divided by the separation distance of the unknown parameters. Explicit bounds for the involved constant grow logarithmically in the space dimension and improve over previous constructions also for small space dimensions. Acknowledgment. The authors thank the referees for their valuable suggestions and additional pointers to related literature. Moreover, we gratefully acknowledge support by the DFG within the research training group 1916: Combinatorial structures in geometry and by the DAAD-PRIME-program. References [1] F. Andersson, M. Carlsson, and M. V. de Hoop. Nonlinear approximation of functions in two dimensions by sums of exponential functions. Appl. Comput. Harmon. Anal., 9: , 1. [] C. Aubel and H. Bölcskei. Vandermonde matrices with nodes in the unit disk and the large sieve. Appl. Comput. Harmon. Anal., accepted. [3] T. Bendory, S. Dekel, and A. Feuer. Exact Recovery of Dirac Ensembles from the Projection Onto Spaces of Spherical Harmonics. Constr. Approx., 4):183 7, 15. [4] T. Bendory, S. Dekel, and A. Feuer. Super-resolution on the sphere using convex optimization. IEEE Trans. Signal Process., 64:53 6, 15. [5] Y. Bresler and A. Macovski. Exact maximum likelihood parameter estimation of superimposed exponential signals in noise. IEEE Trans. Acoust. Speech Signal Process., 345): , [6] J. A. Cadzow. Signal enhancement-a composite property mapping algorithm. IEEE Trans. Acoust. Speech Signal Process., 361):49 6, [7] E. J. Candès and C. Fernandez-Granda. Super-resolution from noisy data. J. Fourier Anal. Appl., 196):19 154, 13. [8] E. J. Candès and C. Fernandez-Granda. Towards a mathematical theory of superresolution. Comm. Pure Appl. Math., 676):96 956, 14. [9] J. Carruth, F. Goncalves, and M. Kelly. The Beurling-Selberg box minorant problem. Ariv e-prints, 17. 1

11 [1] M. P. Clark and L. L. Scharf. On the complexity of iml algorithms. IEEE Trans. Signal Process., 47): , 199. [11] H. Cohn and N. Elkies. New upper bounds on sphere packings I. Annals of Math., 157: , 3. [1] R. Curto and L. A. Fialkow. The truncated complex K-moment problem. Trans. Amer. Math. Soc., 35:85 855,. [13] B. G. R. de Prony. Essai éxperimental et analytiue: sur les lois de la dilatabilité de fluides élastiue et sur celles de la force expansive de la vapeur de l alkool, a différentes températures. Journal de l école polytechniue, 1):4 76, [14] B. Diederichs and A. Iske. Parameter estimation for bivariate exponential sums. In Sampling Theory and Applications SampTA), 15 International Conference on, pages , 15. [15] F. Filbir, H. N. Mhaskar, and J. Prestin. On the problem of parameter estimation in exponential sums. Constr. Approx., 35:33 343, 1. [16] Y. Hua and T. K. Sarkar. Matrix pencil method for estimating parameters of exponentially damped/undamped sinusoids in noise. IEEE Trans. Acoust. Speech Signal Process., 385):814 84, 199. [17] A. E. Ingham. Some trigonometrical ineualities with applications to the theory of series. Math. Z., 411): , [18] T. Jiang, N. D. Sidiropoulos, and J. M. F. ten Berge. Almost-sure identifiability of multidimensional harmonic retrieval. IEEE Trans. Signal Process., 499): , 1. [19] V. Komornik and P. Loreti. Fourier Series in Control Theory. Springer-Verlag, New York, 4. [] S. Kunis, H. M. Möller, and U. von der Ohe. Prony s method on the sphere. Ariv e-prints, 16. [1] S. Kunis, T. Peter, T. Römer, and U. von der Ohe. A multivariate generalization of Prony s method. Linear Algebra Appl., 49:31 47, 16. [] S. Kunis and D. Potts. Stability results for scattered data interpolation by trigonometric polynomials. SIAM J. Sci. Comput., 9: , 7. [3] M. Laurent and B. Mourrain. A generalized flat extension theorem for moment matrices. Archiv der Mathematik, 931):87 98, 9. [4] P. Lemmerling, L. Vanhamme, S. V. Huffel, and B. D. Moor. IQML-like algorithms for solving structured total least suares problems: a unified view. Signal Process., 819): , 1. [5] W. Liao. MUSIC for multidimensional spectral estimation: stability and superresolution. IEEE Trans. Signal Process., 633): ,

12 [6] A. Moitra. Super-resolution, extremal functions and the condition number of Vandermonde matrices. In Proceedings of the Forty-Seventh Annual ACM on Symposium on Theory of Computing, STOC 15, pages 81 83, New York, NY, USA, 15. ACM. [7] B. D. Moor. Total least suares for affinely structured matrices and the noisy realization problem. IEEE Trans. Signal Process., 411): , [8] V. Nagesha and S. Kay. On freuency estimation with the IQML algorithm. IEEE Trans. Signal Process., 49):59 513, [9] A. Olevskii and A. Ulanovskii. On multi-dimensional sampling and interpolation. Anal. Math. Phys., ):149 17, 1. [3] G. Plonka and M. Tasche. Prony methods for recovery of structured functions. GAMM- Mitt., 37):39 58, 14. [31] G. Plonka and M. Wischerhoff. How many Fourier samples are needed for real function reconstruction? J. Appl. Math. Comput., 41-): , 13. [3] D. Potts and M. Tasche. Parameter estimation for exponential sums by approximate Prony method. Signal Processing, 95): , 1. [33] D. Potts and M. Tasche. Parameter estimation for multivariate exponential sums. Electron. Trans. Numer. Anal., 4:4 4, 13. [34] B. D. Rao and K. S. Arun. Model based processing of signals: a state space approach. Proc. IEEE, 8):83 39, 199. [35] R. Roy and T. Kailath. ESPRIT estimation of signal parameters via rotational invariance techniues. In Signal Processing, Part II, volume 3 of IMA Vol. Math. Appl., pages Springer, New York, 199. [36] T. Sauer. Prony s method in several variables. Numer. Math., 136): , 17. [37] R. O. Schmidt. Multiple emitter location and signal parameter estimation. Antennas and Propagation, IEEE Transactions on, 343):76 8, [38] P. Shukla and P. L. Dragotti. Sampling schemes for multidimensional signals with finite rate of innovation. IEEE Trans. Signal Process., 557, part ): , 7. [39] D. W. Tufts and R. Kumaresan. Estimation of freuencies of multiple sinusoids: Making linear prediction perform like maximum likelihood. Proc. IEEE, 79): , 198. [4] W. V. Vasconcelos. Computational methods in commutative algebra and algebraic geometry, volume of Algorithms and Computation in Mathematics. Springer-Verlag, Berlin, With chapters by David Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. [41] M. Vetterli, P. Marziliano, and T. Blu. Sampling signals with finite rate of innovation. IEEE Trans. Signal Process., 56): ,. [4] M. Viazovska. The sphere packing problem in dimension 8. Annals of Math., 1853): , 17. 1