4.3 A Compact Formulation for ` 2,1 Minimization
4.3.1 Gridless SPARROW
The SPARROW formulation in (4.51) relies on grid-based frequency estimation. As discussed for the on-grid assumption (4.3) and for atomic norm minimization in Section 4.1.2, grid-based frequency estimation requires a fine frequency grid to to reduce the frequency estimation error, which, however, results in a high computational cost. The atomic norm minimization problem presented in Section 4.2.2 provides an approach for gridless frequency estimation in ULAs by means of the Vandermonde decomposition in Definition 4.1. In a similar fashion, the Vandermonde decomposition can be employed to derive a gridless version of the SPARROW formulation in (4.51).
Consider a ULA, where the sensing matrix A has a Vandermonde structure and the matrix product ASAH = HToep(w) forms a Hermitian Toeplitz matrix, as discussed in Section 4.1.2. Based on the uniqueness of the Vandermonde decomposition, problem (4.51) can be rewritten as the GridLess (GL-) SPARROW formulation
min w Tr (HToep(w) + λIM) −1R +ˆ 1 MTr HToep(w) (4.58a) s.t. HToep(w) 0, (4.58b)
where the identity Tr(S) = 1 MTr(A H AS) = 1 MTr(ASA H ) = 1 MTr HToep(w) (4.59)
was employed, with the factor 1/M resulting from ka(ν)k22 = M , for all ν ∈ [−1, 1). Us- ing the ideas of Corollaries 4.3 and 4.4, problem (4.58) is equivalent to the semidefinite program min w,WN 1 NTr WN + 1 MTr HToep(w) (4.60a) s.t. WN Y H Y HToep(w) + λIM 0 (4.60b) HToep(w) 0, (4.60c) or, alternatively, to min w,WM Tr WMR +ˆ 1 MTr HToep(w) (4.61a) s.t. WM IM IM HToep(w) + λIM 0 (4.61b) HToep(w) 0. (4.61c)
Given a minimizer ˆw of problem (4.60) or (4.61), the number of sources, i.e., the model order, can be estimated according to
ˆ
L = rank HToep( ˆw), (4.62)
while the frequencies {ˆµl} ˆ L
l=1 and corresponding magnitudes {ˆsl} ˆ L
l=1 can be estimated
by Vandermonde decomposition of the Hermitian Toeplitz matrix HToep( ˆw) according to Definition 4.1. With the estimated frequencies in {ˆµl}
ˆ L
l=1 and signal magnitudes in
{ˆsl} ˆ L
l=1, the corresponding signal matrix estimate ˆX can be reconstructed by appli-
cation of (4.52). As discussed in the context of the Vandermonde decomposition in Definition 4.1, different methods can be applied for frequency reconstruction from the Hermitian Toeplitz matrix HToep( ˆw), such as root-MUSIC [Bar83], ESPRIT [RK89], the matrix pencil method [HS90, HW91] and others, similar to the atomic norm mini- mization problem in Section 4.1.2. The specific relation to the root-MUSIC estimator will be investigated in more detail in the following section. In this context the gridless SPARROW can also be considered as a signal subspace estimator, represented by the Hermitian Toeplitz matrix HToep(w).
Unique Vandermonde decomposition requires that ˆL = rank HToep( ˆw) < M . The rank ˆL can be interpreted as the counterpart of the number of non-zero elements in the minimizer ˆS in the grid-based SPARROW formulation (4.51). Analogous to the grid- based SPARROW formulation (4.51), where the regularization parameter λ determines the sparsity level of ˆS, i.e., the number of its non-zero elements, there always exists a value λ which yields a minimizer ˆw of the gridless formulations (4.60) and (4.61) which fulfills ˆL = rank HToep( ˆw) < M , such that a unique Vandermonde decomposition (4.59) is obtained. Appropriate regularization parameter selection schemes are given by (4.40) and (4.41).
Comparing the GL-SPARROW formulation (4.60) and the ANM problem (4.49), a similar structure in the objective functions and semidefinite constraints can be observed. In fact, both problems are equivalent as given by the following theorem:
4.3 A Compact Formulation for `2,1 Minimization 47
Theorem 4.5. The atomic norm minimization problem (4.48) and the correspond- ing SDP implementation (4.49), with minimizer ˆwANM, are equivalent to the gridless
SPARROW formulation (4.60), with minimizer ˆwSPARROW, in the sense that the min-
imizers are related by
ˆ
wSPARROW= ˆwANM/
√
N . (4.63)
A proof of Theorem 4.5 is provided in Appendix D. For both problem formulations, GL-SPARROW (4.60) and ANM (4.49), the spatial frequencies ν are encoded in the vectors ˆwSPARROW and ˆwANM, as found by Vandermonde decomposition (4.23), such
that both formulations provide the same frequency estimates.
However, from a computational viewpoint, in contrast to the GL-SPARROW problem in (4.60), the ANM problem in (4.49) has additional M N complex-valued variables in the matrix Y0, which need to be matched to the multi snapshot matrix Y by
an additional quadratic term in the objective function. The dimensionality reduction techniques for ANM, discussed in Section 4.2.2, can similarly be applied to SPARROW. Due to the reduced number of variables, the GL-SPARROW formulations (4.60) and (4.61) are less demanding in the number of computational operations and memory requirements and thus admit significantly reduced computational cost as compared to the ANM formulation (4.49).
The case of a thinned linear array of M sensors, discussed in Section 2.1.2, can be treated in the GL-SPARROW formulation by application of a proper selection matrix
¨
J . To see this, let ¨J be the M0× M matrix selecting the sensors in the thinned linear
array from a virtual ULA of M0sensors. Application in the GL-SPARROW formulation
(4.61) yields min w,WM Tr WMR +ˆ 1 M0 Tr HToep(w) (4.64a) s.t. " WM IM IM J¨ T HToep(w) ¨J + λIM # 0 (4.64b) HToep(w) 0, (4.64c)
where w is an M0 element vector. A more detailed analysis of the gridless SPARROW
formulation for thinned linear arrays is provided in [SSSP17b].
Frequency Estimation from the Dual Problem
The gridless SPARROW formulation in (4.58) provides a simple approach of removing the grid-based sensing matrix A from the optimization problem to obtain high res- olution frequency estimates. To gain further insight to the estimation problem, this
section considers an alternative approach of frequency estimation. To this end, consider the Lagrange dual problem of the SPARROW formulation (4.57), which is given by
max Υ0,Υ1 − 2 ReTr(Υ1) − λ Tr(Υ0) (4.65a) s.t. ˆ R Υ1 ΥH1 Υ0 0 (4.65b) 1 − aHkΥ0ak≥ 0 for k = 1, . . . , K, (4.65c)
where Υ0 is an M × M Hermitian, positive semidefinite matrix, while Υ1 is an M × M
complex-valued matrix without specific structure (see Appendix E for details). Note that the primal problem in (4.57) as well as the dual problem in (4.65) are both strictly feasible, such that strong duality applies [VB96]. The constraint (4.65c) forms the dual to the nonnegativity constraint on the elements of S in the primal problem in (4.57), such that by complementary slackness any dual optimal variable ˆΥ0 must fulfill
1 − aHkΥˆ0ak
(
= 0 if ˆsk ≥ 0
≥ 0 if ˆsk= 0,
(4.66)
for k = 1, . . . , K. Equation (4.66) indicates that the spatial frequency estimates {ˆµl} ˆ L l=1,
as defined in (4.34), can equivalently be identified by the elements νk satisfying 1 −
aH kΥ0ak = 0, i.e., {ˆµl} ˆ L l=1 = { νk | 1 − aHkΥˆ0ak = 0, k = 1, . . . , K }. (4.67)
This leads to the conclusion that, instead of solving the primal problem (4.57) and perform frequency estimation according to equation (4.34), the dual problem (4.65) can be solved to obtain frequency estimates according (4.67). Note that similar observations have been made in the context of minimization of the total variation norm for gridless sparse recovery [CFG13, CFG14].
Consider the special case of a uniform linear array with sensor positions rm = (m−1) ∆,
for m = 1, . . . , M , and move from a finite set of K steering vectors a(ν1), . . . , a(νK) to
the array response vector
a(z) = [1, z, . . . , zM −1]T (4.68)
with z = e− j πµ∆, as discussed in Section 2.1.1. This transforms the finite set of constraints (4.65c) into an infinite-dimensional constraint, given as
1 − aH(z)Υ0a(z) = 1 − M −1
X
m=−M +1
cmz−m ≥ 0. (4.69)
Equation (4.69) constitutes a nonnegative trigonometric polynomial on the unit circle |z| = 1 with coefficients
4.3 A Compact Formulation for `2,1 Minimization 49
given as the sum of the main-diagonal and off-diagonal elements of Υ0, with Θm being
the elementary Toeplitz matrix with ones on the mth diagonal and zeros elsewhere, as introduced in Section 2.1.
Define a second polynomial equation
1 =
M −1
X
m=−M +1
dmz−m = aH(z)Ha(z) (4.71)
for some Hermitian positive semidefinite matrix H 0, with coefficients dm fulfilling
dm = Tr(ΘmH) =
(
1 if m = 0
0 otherwise, (4.72)
i.e., the main-diagonal elements of H must sum to 1 while the elements on the off- diagonals of H must sum to 0, such that (4.71) always holds true on the unit circle |z| = 1. The polynomial in (4.71) is employed to provide an upper bound on the polynomial in (4.69) as
1 − aH(z)Υ0a(z) = aH(z)(H − Υ0)a(z) ≥ 0, (4.73)
which holds true for H − Υ0 0. Equation (4.73) forms the infinite-dimensional
analogue to the finite-dimensional constraint (4.65c). Using (4.73) and the conditions on H in (4.72), the gridless version of the dual problem (4.65) can be formulated as
max Υ0,Υ1,H − 2 ReTr(Υ1) − λ Tr(Υ0) (4.74a) s.t. ˆ R Υ1 ΥH1 Υ0 0 (4.74b) H − Υ0 0 (4.74c) Tr(H) = 1, Tr(ΘmH) = 0 for m = 1, . . . , M. (4.74d)
Given a minimizer ˆΥ0 to the problem in (4.74), the spatial frequencies ˆµl can be
estimated from the ˆL roots {ˆzl} ˆ L
l=1 of the polynomial (4.69) on the unit circle, according
to (4.67).
Frequency estimation from the dual problem of the SPARROW formulation shows strong links to the root-MUSIC approach discussed in Section 2.1. However, while in root-MUSIC the L roots closest to the unit circle are selected, SPARROW only considers the roots on the unit circle for frequency estimation. For a proper selection of the regularization parameter λ, the existence of roots on the unit circle is ensured by complementary slackness according to (4.66) and its continuous analogue in (4.69), i.e., roots will be located on the unit circle in the grid-based case as well as in the gridless case. This is a fundamental difference to the MUSIC approach, where no such guarantee of roots on the unit circle exists, neither for the gridless root-MUSIC nor for grid-based MUSIC method. Instead, the grid-based MUSIC approach searches the
minima on the unit circle while root-MUSIC finds the roots in the entire complex plane, as discussed in Section 3.2. This leads to a degradation in resolution performance of MUSIC as compared to root-MUSIC. As will be shown by numerical experiments in Section 4.4, no such degradation occurs for grid-based SPARROW, as compared to GL-SPARROW.