On the Recursions of Robust COMET Algorithm for Convexly Structured

On the Recursions of Robust COMET Algorithm for Convexly Structured Shape Matrix Bruno Meriaux, Chengfang Ren, Arnaud Breloy, Mohammed Nabil El Korso, P Forster, Jean-Philippe Ovarlez

To cite this version:

Bruno Meriaux, Chengfang Ren, Arnaud Breloy, Mohammed Nabil El Korso, P Forster, et al.. On the Recursions of Robust COMET Algorithm for Convexly Structured Shape Matrix. 27th European Signal Processing Conference (EUSIPCO 2019), Sep 2019, A Coruña, Spain. hal-02155905

HAL Id: hal-02155905 https://hal.archives-ouvertes.fr/hal-02155905 Submitted on 14 Jun 2019

HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. On the Recursions of Robust COMET Algorithm for Convexly Structured Shape Matrix

B. Meriaux´ ∗, C. Ren∗, A. Breloy†, M.N. El Korso†, P. Forster‡ and J.-P. Ovarlez∗§

∗SONDRA, CentraleSupelec,´ Universite´ Paris-Saclay, F-91190 Gif-sur-Yvette, France †LEME EA 4416, Universite´ Paris-Nanterre, F-92410 Ville d’Avray, France ‡SATIE, Universite´ Paris-Nanterre, F-94230 Cachan, France §DEMR, ONERA, Universite´ Paris-Saclay, F-91123 Palaiseau, France

Abstract—This paper addresses robust estimation of structured the common Complex Angular Elliptical (CAE) distribution. shape (normalized covariance) matrices. Shape matrices most Several robust methods have been proposed to leverage Tyler’s often own a particular structure depending on the application estimator formulation [7] in the context of structured shape of interest and taking this structure into account improves estimation accuracy. In the framework of robust estimation, we matrices [8]–[13]. A COnvexly ConstrAined (COCA) shape introduce a recursive robust shape matrix estimation technique matrix estimator has been recently proposed in [10]. Iterative based on Tyler’s M-estimate for convexly structured shape Majorization-Minimization algorithms for the computation of matrices. We prove that the proposed estimator is consistent, structured CM estimates are developped in [11] and a robust asymptotically efficient and Gaussian distributed and we notice extension of COMET, named RCOMET, has been derived in that it reaches its asymptotic regime faster as the number of recursions increases. Finally, in the particular wide spreaded case [13]. The references [8], [9], [12] considered the problem of of Hermitian persymmetric structure, we study the convergence robust shape matrix estimation with symmetric structures. of the recursions of the proposed algorithm. In this paper, we propose a Recursive version of RCOMET Index Terms—Robust shape matrix estimation, elliptical dis- (R-RCOMET) based on Tyler’s M-estimate and COMET tributions, Tyler’s M-estimator, structured estimation. criterion for convexly structured shape matrices. We conduct a theoretical analysis of the asymptotic performance of the pro- I.INTRODUCTION posed estimator. We also compare the non-asymptotic behavior Most of the adaptive signal processing algorithms require a with the RCOMET method. Finally, we analyse theoretically Covariance Matrix (CM) estimation. In addition to its Hermi- the convergence of the recursions in the particular case of the tian symmetry and positive definiteness, the CM may exhibit Hermitian persymmetric structure. a particular structure related to the application of interest. For In the following, convergence in distribution and in proba- example, a linear array that is symmetrically spaced w.r.t. the bility are respectively denoted by →L and →P . AT (respectively phase center leads to the Hermitian persymmetric structure AH and A∗) stands for the transpose (respectively conjugate of the CM [1]. Another example is the Toeplitz structure for transpose and conjugate) matrix. The vec-operator vec(A) uniform linear arrays. Taking into account this structure in stacks all columns of A into a vector. The identity matrix the estimation scheme leads to a better estimation accuracy of size m is referred to as Im. The matrix Jm denotes since it decreases the degrees of freedom in the estimation the m-dimensional antidiagonal matrix, having 1 as non- problem [2]. In the Gaussian framework, this challenge has zero element. The operator ⊗ refers to the Kronecker matrix been extensively studied. Notably, the Covariance Matching product and finally, the subscript ”e” refers to the true value. Estimation Technique (COMET) has been proposed in [3]. This paper is organized as follows. In section II, a brief re- The latter is computationally less intensive than Maximum view on CAE distribution, Tyler’s M-estimate and RCOMET Likelihood (ML) estimation and still provides asymptotically procedure is presented. Section III focuses on the proposed efficient CM estimates. However, COMET is based on the algorithm and its performance analysis. We also analyse the Sample Covariance Matrix (SCM) estimate, thus it is sensi- convergence of the recursions in the case of the Hermitian tive to outliers. In a context of robust CM estimation, the persymmetric structure. Some simulations results in Section class of circular Complex Elliptically Symmetric distributions IV illustrate the theoretical analysis. (CES) turns out to be particularly suitable to model spiky radar clutter measurements [4]–[6]. Within this framework, II.BACKGROUNDAND PROBLEM SETUP a distribution free estimator of the scatter matrix is derived A. Complex Angular Elliptical Distribution in [7] and referred to as Tyler’s M-estimator. Furthermore, m the normalization of zero mean CES distributed data leads to Let x ∈ C be a circular centered CES distributed random vector [5] with scatter matrix M. If it exists, the covariance The work of B. Meriaux´ is partially funded by the Direction Gen´ erale´ matrix of x is proportional to M. The normalized vector de l’Armement (D.G.A). This work is also supported by the ANR ASTRID x y = , x 6= 0, follows a CAE distribution, denoted by referenced ANR-17-ASTR-0015. kxk y ∼ Um (M). The probability density function of the vector y Algorithm 1 R-RCOMET w.r.t. spherical measure [10] reads Require: N i.i.d. data, yn ∼ Um (Me) with N > m, any −m K ≥ 1 given p(y | M) ∝ |M|−1 yH M−1y (1) 1: Compute Mb FP from y1,..., yN with (2) where the matrix M is defined up to an arbitrary scale factor. 2: Initialize µb0 with (4) To avoid scaling ambiguity, M is normalized according to 3: for k = 1 to K do Tr (M) = m. We refer to M as the shape matrix of y. 4: Compute µbk from (6) 5: end for B. Tyler’s M-estimator 6: return µbK From a set of N i.i.d. CAE distributed data, yn ∼ Um (M), n = 1,...,N with N > m, Tyler’s M-estimate is the M (µ )) and asymptotically efficient and Gaussian distributed unstructured ML-estimate of the shape matrix, given by the e [13]. Specifically solution of the following fixed-point equation [7]: √ N (µ − µ ) →L N ( , ) N H b0 e 0 CRBCAE (5) m X ynyn Mb FP = −1 , H Mb FP (2) CRB ´ N H where CAE, denoting the Cramer-Rao Bound (CRB), is n=1 yn Mb FP yn detailled in [13]. Existence and uniqueness up to a scale factor of the above Although asymptotically efficient, RCOMET algorithm re- equation solution have been studied in [14]. In the follow- quires a substantial sample support to reach its asymptotic ing, the scale ambiguity is removed by fixing in the latter regime. In this paper, we propose a recursive version of solution the same constraint as for the shape matrix, i.e. RCOMET, for which we conduct a theoretical analysis of h i Tr Mb FP = m. The solution Mb FP is obtained by an iterative its asymptotic performance; we also notice that the latter are achieved at lower sample support. Furthermore, we analyse the algorithm, Mk+1 = H (Mk) with the normalization on the convergence behavior for a particular strucuture: the Hermitian trace, which converges to Mb FP, for any initialization point [7], persymmetric one. [15]. Furthermore, Mb FP is a consistent, unbiased estimator of M and its asymptotic distribution is given by [5], [15]: III.RECURSIVE RCOMETPROCEDURE √ d A. Algorithm Nvec Mb FP − M −→ CN (0, Σ, ΣKm) (3) M  In the RCOMET objective (4), b FP plays both the role of a m + 1 T H −1  Σ = ΓM M ⊗ M ΓM target together with a metric specification through Mb FP . Split- with m ting these roles can lead naturally to a recursive formulation 1 H  ΓM = Im2 − vec (M) vec (Im) where the weighting is refined at each step. For a finite and m given number of steps, K, we obtain the R-RCOMET estimate for µ, denoted by µK and achieved at the k-th iteration by where Km is the commutation matrix, which satisfies b T solving for k ∈ [[1,K]] Kmvec (A) = vec A [16]. n −1o2 µk = arg min Tr Mb FP − αM (µ) M µk−1 (6) C. Problem Setup and RCOMET Algorithm b α,µ b Let us consider N i.i.d. CAE distributed observations, with µb0 given by (4) and such that Tr [M (µbK )] = m. The R-RCOMET algorithm is recapped in the box Algorithm 1. In yn ∼ Um (Me), n = 1,...,N with N > m. The shape matrix belongs to the convex subset of Hermitian positive-definite practice, we can use a more elaborated stopping rule, e.g., a S matrices and there exists a one-to-one differentiable mapping combination of k ≤ Kmax and µbk+1 − µbk ≤ εtol kµbkk. p µ 7→ M (µ) from R to S . The vector µ is the unknown B. Asymptotic Analysis parameter of interest, with exact value µ and M = M (µ ). e e e First, we introduce an intermediary estimator, µ, of µ , for We recall that Tr [M ] = m. The RCOMET estimate, µ , of b e e b0 which its asymptotic performance is studied. Then, we deal µ is obtained by [13] e with the R-RCOMET asymptotic performance. n −1o2 µ0 = arg min Tr Mb FP − αM (µ) Mb (4) Lemma 1. Let µ be the solution obtained by b α,µ FP b 2 2 n −1o −1/2 −1/2 µ = arg min Tr Mb FP − αM (µ) Mb (7) = arg min Im − Mb [αM (µ)] Mb b α,µ α,µ FP FP F | {z } d(Mb FP,Mb ,αM(µ)) with α > 0 and satisfying Tr [M (µb0)] = m. The minimization of the strictly convex criterion in (4) w.r.t αM (µ) over such that Tr [M (µb)] = m and where Mb refers to any is a convex problem that admits a unique solution. Finally, P S consistent estimator of M up to a scale factor, i.e., M → κM , the one-to-one mapping and the constraint on the trace yield e b e κ > 0. Then, a unique solution for µ. The RCOMET estimator µ0 (respec- b P √ L tively M (µb0)) is a consistent estimator of µe (respectively µb → µe and N (µb − µe) → N (0, CRBCAE) Proof. The estimate µb is a function of Mb FP and Mb that we C. Convergence for Hermitian Persymmetric Structure denote by: In this subsection, we consider the particular case of the µb = g Mb FP, Mb Hermitian persymmetric structure, i.e., matrices which belong n m×m H T o to the set A ∈ C | A = A and A = JmA Jm . Let us Function g(·) satisfies g (Me, κMe) = µe since denote M the (r, s) element of the matrix M (µ). The d (Me, κMe, M(µe)) = 0 for any κ > 0. Moreover, r,s for a smooth parameterization M(µ), g(·) is differentiable natural parameterization of a Hermitian persymmetric matrix, and thus continuous. Then, the consistency of Mb FP w.r.t Me with the minimal number of parameters, consists in stacking [15] (respectively M w.r.t. κM , for any κ > 0) and the the real and imaginary parts of the elements Mr,s satisfying b e continuity of g imply µ = g M , M →P g (M , κM ) = µ . s ≥ r and s ≤ m + 1 − r. The length of the vector µ is b b FP b e e e m(m + 1) P equal to p = . Hence, there exists a full column Consequently M (µ) → M (µ ) b e 2 2 We can rewrite (7) as the following concentrated function rank matrix J ∈ Cm ×p, which relates the vectorized matrix H −1/2Y⊥ −1/2 M (µ) to µ as µb = arg min ηbFPWb −1/2 Wb ηbFP (8) µ Wb η(µ) η (µ) = vec (M (µ)) = J µ (10) | {z } f (µ) † Mb ,Mb FP The full column rank matrix J admits a left inverse J = −1 −1/2 H −1/2 H H † Q⊥ Wb η (µ) η (µ) Wb J J J verifying J J = Ip [19]. where −1/2 = I − is W η(µ) H b η (µ) Wb η (µ) Proposition 1. Let W = AT ⊗ A, where A ∈ m×m −1/2 C the orthogonal projector onto Wb η (µ) and with η (µ) = is Hermitian persymmetric. Then the inverse of the matrix T H J H W−1J is J †WJ † . vec (M (µ)), Wb = Mb ⊗Mb and ηbFP = vec Mb FP . It follows from the Delta method [17, Chapter 3] that Proof. See Appendix √ N (µ − µ ) →L N (0, Γ ) (9) b e µ Corollary 1. Let µbK be the R-RCOMET estimate of µe. −1 −H Then, where Γµ = H(µ ) R∞H(µ ) in which [18] e e ∗ †  2 ∀ K ∈ N µbK = J ηbFP (11) −2 ∂ fMe,Me (µ)  H(µe) = κ  ∂µ∂µT Proof. For K = 1, the R-RCOMET estimate reads  µe   H  −1   H −1 H −1 ∂f (µ) ∂f (µ) µ1 ∝ J Wb 0 J J Wb 0 ηFP  −4 Me,Mb FP Me,Mb FP b b  R∞ = κ E       ∂µ ∂µ  T  = M (µ ) ⊗ M (µ ) µ  µe µe where Wb 0 b0 b0 and b0 is given by (4). According to Proposition 1, µ1 can be rewritten as Thus, Γµ is independent of κ. RCOMET estimator being a b H −1 M = M † † H particular case of the problem (7) (where b b FP is a µb1 ∝ J Wb 0J J Wb 0 ηbFP consistent estimator of Me), we finally obtain from (5) that 1 † −1 ∝ J Wb 0 [Im2 + Jm2 Km] Wb η (cf. Lemma 2) Γµ = CRBCAE, which concludes the proof. 2 0 bFP 1 † † Theorem 1. Let µK be the R-RCOMET estimator of µe based ∝ J [I 2 + J 2 K ] η = J η b 2 m m m bFP bFP on N i.i.d. observations, yn ∼ U (Me). µbK is consistent w.r.t. Hence, to verify Tr [M (µ )] = m, we have necessarily µe, asymptotically Gaussian distributed and efficient: b1 √ L † µ1 = J ηFP (12) N (µbK − µe) → N (0, CRBCAE) b b † Proof. By applying Lemma 1 at each iteration, the theorem By recurrence, we can show that µb1 = ... = µbK = J ηbFP P ∗ † for any K ∈ N . Indeed, let us assume µK = J ηFP, then follows immediately, since M (µbk) → Me for k ≥ 0. b b −1 −1 −1 K H H Finally, for a finite number of steps , R-RCOMET yields µbK+1 ∝ J Wb K J J Wb K ηbFP the same asymptotic performance as RCOMET. The practical T interest is that it empirically improves in most cases the where Wb K = M (µbK ) ⊗ M (µbK ). According to Propo- performance at low sample support, which can be intuited sition 1 and the trace constraint, we finally obtain µbK+1 = † by noting that the minimized norm is refined at each step. J ηbFP. Consequently, the R-RCOMET procedure for Hermi- Notice that the fixed-point iterations are heuristic as they do tian persymmetric matrices converges in only one step, i.e., not solve an underlying optimization problem when K → ∞. the resulting outcome is identical for any K ≥ 1. The estimator exists for a finite number of iterations but the † J ηbFP can be interpreted as the solution of the Euclidean convergence of R-RCOMET when K → ∞ requires a case projection of Tyler’s M-estimate onto the subspace of Hermi- by case study depending on the structure. In the following, tian persymmetric matrices. the convergence of R-RCOMET for Hermitian persymmetric † 2 J ηFP = arg min kηFP − αJ µk s.t. Tr [M (µ)] = m (13) structure is established. b α,µ b Consequently, surprisingly, the R-RCOMET procedure coin- where T ∈ Cm×m is a unitary matrix, defined in [8, Proposi- T cides with the classical Euclidean projection of Tyler’s M- tion 1] and verifying the relation T T = Jm. estimate in the case of Hermitian persymmetric matrices. Again, the R-RCOMET procedure converges in one step in this IV. NUMERICAL RESULTS particular case and it boils down to an Euclidean projection of Tyler’s M-estimate for Hermitian persymmetric matrices. In this section, we illustrate the results of the previous theoretical analysis and compare performance with state-of- B. Hermitian Toeplitz Structure the-art algorithms. Secondly, we examine the Hermitian Toeplitz structure. In this case, the minimal parameterization consists in stacking A. Hermitian Persymmetric Structure the real and imaginary parts of the first row of the matrix First, we consider the Hermitian persymmetric structure. M (µ). Furthermore, there exists a full column rank matrix m2×p For m = 8, we generate 5000 sets of N independent m- J 1 ∈ C with p = 2m − 1, which relates M (µ) to µ as dimensional CAE distributed samples. We compare the per- vec (M (µ)) = J 1µ. It is worth noting that the matrix J 1 for formance of RCOMET and R-RCOMET, the Persymmetric Hermitian Toeplitz matrices differs from J for Hermitian per- Fixed-Point (PFP) estimate derived in [8] and the Persymmet- symmetric matrices since the minimal number of parameters is ric Sample Covariance Matrix (PSCM) estimator. The latter different on these cases. For m = 8, we generate 5000 sets of is obtained by substituting Tyler’s M-estimate in the PFP N independent m-dimensional CAE distributed samples. We estimator with the SCM. Finally, the related CRB, CRBU , compare the performance of RCOMET and R-RCOMET for is drawn for the comparison [10], [20]. different numbers of recursions K, the Euclidean projection of Tyler’s M-estimate as well as the COCA shape estimator 5 introduced in [10]. The standard semi-definite program solver, CVX, is used to compute this estimator [21]. The related CRB, CRBU , is also drawn for the comparison [13]. 0 0 Tr (CRBU ) Euclid. proj. Tyler (dB)

} −5 COCA [10] ) b µ −5 RCOMET [13] ( R-RCOMET, K = 1 (dB) R-RCOMET, K = 5 MSE −10 } ) { Tr (CRBU ) −10

b R-RCOMET, K = 20 µ Tr PFP [8] ( RCOMET [13] −23 0 −24 −15 R-RCOMET, K = 1 MSE −15 { R-RCOMET, K = 2 −25 Tr −5 PSCM 102.8 102.9 103 −20 1 2 3 10 10 10 −10 Number of samples N −25 101 101.2 101.4 101 102 103 Fig. 1. Comparison on the MSE for Hermitian persymmetric structure Number of samples N From Fig. 1, we notice that the R-RCOMET estimates reach the CRB as well as the RCOMET estimate, which reflects their Fig. 2. Comparison on the MSE for Hermitian Toeplitz structure asymptotic efficiency. Furthermore, R-RCOMET estimates are From Fig. 2, we verify that the R-RCOMET estimates reach identical for 1 and 2 iterations and seems also to coincide with the CRB as well as the RCOMET estimate. As already stated, the PFP estimate. The PSCM estimator does not perform well we observe that the CRB is reached faster with R-RCOMET since the SCM computed from CAE distributed data is the than RCOMET, especially when the number of recursions, K normalized SCM and is biased [15]. increases. However, in this case R-RCOMET does not coincide Remark: Actually, we can show theoretically that the with the Euclidean projection of Tyler’s M-estimate, which is R-RCOMET estimate coincides with the PFP estimate, since not asymptotically efficient. Finally, COCA estimator is consistent (as shown in [10]) but not asymptotically efficient. This µbPFP given in [8] can be expressed by estimator shows its interest at low sample support, however it † T H 1 h H ∗ T T i suffers from a heavy computational cost. R-RCOMET allows µPFP = J T ⊗ T vec TMb FPT + T Mb FPT b 2 for an interesting performance-computational cost trade-off in 1 † † this context. Indeed, with the simulations running in Matlab = J [I 2 + J 2 K ] η = J η (14) 2 m m m bFP bFP R2017a on E3-1270 v5 CPU, the average calculation time is 50.40s for COCA (respectively 0.17s for R-RCOMET) for REFERENCES N = 1000. [1] M. Haardt, M. Pesavento, F. Roemer,¨ and M. N. El Korso, “Subspace methods and exploitation of special array structures,” in Array and ONCLUSION V. C Statistical Signal Processing, ser. Academic Press Library in Signal In this paper, we have introduced a recursive version of Processing. Elsevier, Jan. 2014, vol. 3, ch. 15, pp. 651–717. [2] T. A. Barton and D. R. Fuhrmann, “Covariance structures for multi- RCOMET for convexly structured shape matrix estimation. dimensional data,” Multidimensional Systems and Signal Processing, We have shown that the proposed R-RCOMET method is vol. 4, no. 2, pp. 111–123, 1993. consistent, asymptotically unbiased, efficient and Gaussian [3] B. Ottersten, P. Stoica, and R. Roy, “Covariance matching estimation techniques for array signal processing applications,” ELSEVIER Digital distributed. In addition, the latter empirically performs better Signal Processing, vol. 8, no. 3, pp. 185–210, 1998. at lower sample support than RCOMET. In the particular [4] K. D. Ward, C. J. Baker, and S. Watts, “Maritime surveillance radar. case of the Hermitian persymmetric structure, we studied the part 1: Radar scattering from the ocean surface,” IEE Proceedings F (Communications, Radar and Signal Processing), vol. 137, no. 2, pp. convergence of the recursions and we related R-RCOMET to 51–62, Apr. 1990. the classical Euclidean projection of Tyler’s M-estimate. [5] E. Ollila, D. E. Tyler, V. Koivunen, and H. V. Poor, “Complex elliptically symmetric distributions: Survey, new results and applications,” IEEE VI.APPENDIX Transactions on Signal Processing, vol. 60, no. 11, pp. 5597–5625, Nov. This appendix presents the proof of Proposition 1, which 2012. [6] K. J. Sangston, F. Gini, and M. S. Greco, “Adaptive detection of requires the following lemma. radar targets in compound-Gaussian clutter,” in Proc. of IEEE Radar m2×p Conference, May, 2015, pp. 587–592. Lemma 2. Let J ∈ C be the matrix defined in (10). [7] D. E. Tyler, “A distribution-free M-estimator of multivariate scatter,” † 2 Its left inverse is the matrix J ∈ Cp×m such that J † = The Annals of Statistics, vol. 15, no. 1, pp. 234–251, 1987. H −1 H † [8] G. Pailloux, P. Forster, J.-P. Ovarlez, and F. Pascal, “Persymmetric adap- J J J and J J = Ip. Then, we have the equality tive radar detectors,” IEEE Transactions on Aerospace and Electronic † 1 Systems, vol. 47, no. 4, pp. 2376–2390, Oct. 2011. JJ = (I 2 + J 2 K ) (15) 2 m m m [9] Y. Gao, G. Liao, S. Zhu, X. Zhang, and D. Yang, “Persymmetric adaptive detectors in homogeneous and partially homogeneous environments,” † Proof. Let us introduce PJ = JJ and Q = IEEE Transactions on Signal Processing, vol. 62, no. 2, pp. 331–342, 1 2014. (I 2 + J 2 K ). On the one hand, P is a projection 2 m m m J [10] I. Soloveychik and A. Wiesel, “Tyler’s covariance matrix estimator in matrix onto the image of J and has a rank equal to p = elliptical models with convex structure,” IEEE Transactions on Signal m(m + 1) Processing, vol. 62, no. 20, pp. 5251–5259, Oct. 2014. . On the other hand, Q is also a projection matrix [11] Y. Sun, P. Babu, and D. P. Palomar, “Robust estimation of structured 2 covariance matrix for heavy-tailed elliptical distributions,” IEEE Trans- onto the image of J . Furthermore, we have QPJ = PJ . To actions on Signal Processing, vol. 14, no. 64, pp. 3576–3590, Jul. 2016. conclude that Q = PJ , we need to show that rank (Q) = p. [12] I. Soloveychik, D. Trushin, and A. Wiesel, “Group symmetric robust covariance estimation,” IEEE Transactions on Signal Processing, vol. 62, Since Km is full rank, we have no. 1, pp. 244–257, Jan. 2016. rank (Q) = rank (QKm) = rank (Km + Jm2 ) [13]B.M eriaux,´ C. Ren, M. N. El Korso, A. Breloy, and P. Forster, “Robust- COMET for covariance estimation in convex structures: algorithm and Let us analyse the columns of the matrix B = Km + Jm2 ∈ statistical properties,” in Proc. of IEEE International Workshop on Com- m2×m2 putational Advances in Multi-Sensor Adaptive Processing (CAMSAP), R . To this end, let be ei the i-th column unit vector 2 2 Dec. 2017, pp. 1–5. of order m . Thus, for (k, `) ∈ [[1; m]] , we have for i = [14] F. Pascal, Y. Chitour, J.-P. Ovarlez, P. Forster, and P. Larzabal, “Covari- k + (` − 1)m ∈ [[1; m2]] ance structure maximum-likelihood estimates in compound-Gaussian noise: Existence and algorithm analysis,” IEEE Transactions on Signal Bei = e`+(k−1)m + em2−i+1 = e`+(k−1)m + em(m−`)+m−k+1 Processing, vol. 56, no. 1, pp. 34–48, Jan. 2008. [15] F. Pascal, P. Forster, J.-P. Ovarlez, and P. Larzabal, “Performance The columns of B have either only one non-zero coordinate, analysis of covariance matrix estimates in impulsive noise,” IEEE equal to 2 (m different possibilities), or only two non-zero Transactions on Signal Processing, vol. 56, no. 6, pp. 2206–2217, Jun. m(m − 1) 2008. coordinates, equals to 1 different possibilities . [16] J. R. Magnus and H. Neudecker, “The commutation matrix: some 2 properties and applications,” The Annals of Statistics, vol. 7, no. 2, pp. m(m − 1) 381–394, Mar. 1979. Finally, we obtain rank (Q) = rank (B) = m+ = p, 2 [17] A. W. Van der Vaart, Asymptotic Statistics (Cambridge Series in Sta- hence the relation (15). In addition, by right multiplying (15) tistical and Probabilistic Mathematics). Cambridge University Press, Jun. 2000, vol. 3. with J , we obtain J = Jm2 KmJ . [18] J.-P. Delmas and H. Abeida, “Survey and some new results on per- Proof of Proposition 1. Let us introduce X = J H W−1J and formance analysis of complex-valued parameter estimators,” ELSEVIER Signal Processing † †H , vol. 111, pp. 210–221, 2015. Y = J WJ . We must show that XY = YX = Ip. First [19] R. Penrose, “A generalized inverse for matrices,” Mathematical Pro- according to Lemma 2, we have ceedings of the Cambridge Philosophical Society, vol. 51, no. 3, pp. 406–413, 1955. H −1 † †H 1 H −1 †H XY = J W JJ WJ = J Im2 + W Jm2 KmW J [20] P. Stoica and B. C. Ng, “On the Cramer-Rao´ bound under parametric 2 constraints,” IEEE Signal Processing Letters, vol. 5, no. 7, pp. 177–179, Then, it is easy to show that the matrix W−1 is persymmetric Jul. 1998. −1 [21] M. Grant and S. Boyd, “CVX: Matlab software for disciplined convex thus W Jm2 KmW = Jm2 Km. Finally, we obtain programming, version 2.1,” http://cvxr.com/cvx, Mar. 2014. 1 H †H XY = J [I 2 + J 2 K ] J = I = YX. 2 m m m p