A numerical algorithm for stable 2D autoregressive filter design

Woerdeman, Hugo J.; Geronimo, Jeffrey S.; Castro, Glaysar

doi:10.1016/s0165-1684(03)00057-4

Cited by 7 publications

(5 citation statements)

References 13 publications

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“…If one seeks the stable factorizations guaranteed in [29], one may worry that optimizing over polynomials which satsify the constraint given in Theorem II.4 may itself be difficult. That said, numerical work in [29,51,52] each support that this constraint leads to well-defined semi-definite programming problems that, while not fully characterized, appear empirically compatible with common classical optimization algorithms discussed above.…”

Section: B the Numerical Outlook For M-qspmentioning

confidence: 97%

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi,

Chuang

2022

Preprint

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Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, and say nothing about computing joint properties of two or more oracles; these can be far cheaper to compute given the ability to pit oracles against one another coherently.This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected.M-QSP is numerically stable and inherits the approximative efficiency of its single-variable analogue; however, its underlying theory is substantively more complex. M-QSP provides one bridge from quantum algorithms to a rich theory of algebraic geometry over multiple variables. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms.

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Section: B the Numerical Outlook For M-qspmentioning

confidence: 97%

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi,

Chuang

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…If one seeks the stable factorizations guaranteed in [29], one may worry that optimizing over polynomials which satsify the constraint given in Theorem 2.4 may itself be difficult. That said, numerical work in [29,53,54] each support that this constraint leads to well-defined semi-definite programming problems that, while not fully characterized, appear empirically compatible with common classical optimization algorithms discussed above.…”

Section: The Numerical Outlook For M-qspmentioning

confidence: 97%

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Rossi

Chuang

2022

Quantum

View full text Add to dashboard Cite

Recent work shows that quantum signal processing (QSP) and its multi-qubit lifted version, quantum singular value transformation (QSVT), unify and improve the presentation of most quantum algorithms. QSP/QSVT characterize the ability, by alternating ansätze, to obliviously transform the singular values of subsystems of unitary matrices by polynomial functions; these algorithms are numerically stable and analytically well-understood. That said, QSP/QSVT require consistent access to a single oracle, saying nothing about computing joint properties of two or more oracles; these can be far cheaper to determine given an ability to pit oracles against one another coherently. This work introduces a corresponding theory of QSP over multiple variables: M-QSP. Surprisingly, despite the non-existence of the fundamental theorem of algebra for multivariable polynomials, there exist necessary and sufficient conditions under which a desired stable multivariable polynomial transformation is possible. Moreover, the classical subroutines used by QSP protocols survive in the multivariable setting for non-obvious reasons, and remain numerically stable and efficient. Up to a well-defined conjecture, we give proof that the family of achievable multivariable transforms is as loosely constrained as could be expected. The unique ability of M-QSP to obliviously approximate joint functions of multiple variables coherently leads to novel speedups incommensurate with those of other quantum algorithms, and provides a bridge from quantum algorithms to algebraic geometry.

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“…completions. For any such that is invertible, let (18) Then the determinant-maximizing p.d. Toeplitz-block matrix completion is unique and satisfies (19) In other words, the ME completion has the property that all elements in the inverse, in positions corresponding to the same completed element in the direct matrix, sum to zero.…”

Section: A Step 1: Me-completion Ofmentioning

confidence: 99%

“…For clutter that is stationary both in slow time (i.e., a strictly periodic radar waveform) and space (i.e., a perfectly ULA), whose -variate covariance matrix is Toeplitz-block-Toeplitz in structure, the natural choice for STAP applications is the 2-D autoregressive model [16], [17]. For this model, the covariance matrix is uniquely specified by the -variate matrix [18] ( . If similarly to the ideal ULA case, it is possible to impose parametric (order) restrictions over the spatial domain (for an arbitrary antenna array geometry), then a further sample-support reduction could be expected.…”

mentioning

confidence: 99%

“…(2.1) Fairly recent results of Woerdeman et al [18] , which then has to undergo further modifications to meet those structural requirements. While possible in principle, but not in the maximum-likelihood sense (e.g., see [18]), it is uncertain whether this is the only model that can achieve the dramatic sample-support reduction ( 8), and at the same time, produce an -variate covariance matrix estimate appropriate for STAP applications.…”

mentioning

confidence: 99%

See 1 more Smart Citation

Two-Dimensional Multivariate Parametric Models for Radar Applications—Part I: Maximum-Entropy Extensions for Toeplitz-Block Matrices

Abramovich

Johnson

Spencer³

2008

IEEE Trans. Signal Process.

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In a series of two papers, a new class of parametric models for two-dimensional multivariate (matrix-valued, space-time) adaptive processing is introduced. This class is based on the maximum-entropy extension and/or completion of partially specified matrix-valued Hermitian covariance matrices in both the space and time dimensions. This first paper considers the more restricted class of Toeplitz Hermitian covariance matrices that model stationary clutter. If the clutter is stationary only in time then we deal with a Toeplitz-block matrix, whereas clutter that is stationary in time and space is described by a Toeplitz-block-Toeplitz matrix. We first derive exact expressions for this new class of 2-D models that act as approximations for the unknown true covariance matrix. Second, we propose suboptimal (but computationally simpler) relaxed 2-D time-varying autoregressive models ("relaxations") that directly use the non-Toeplitz Hermitian sample covariance matrix. The high efficiency of these parametric models is illustrated by simulation results using true ground-clutter covariance matrices provided by the DARPA KASSPER Dataset 1, which is a trusted phenomenological airborne radar model, and a complementary AFRL dataset.

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A numerical algorithm for stable 2D autoregressive filter design

Cited by 7 publications

References 13 publications

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Multivariable quantum signal processing (M-QSP): prophecies of the two-headed oracle

Two-Dimensional Multivariate Parametric Models for Radar Applications—Part I: Maximum-Entropy Extensions for Toeplitz-Block Matrices

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