On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix

Weiss, Stephan; Pestana, Jennifer; Proudler, Ian K.

doi:10.1109/tsp.2018.2812747

Cited by 54 publications

(119 citation statements)

References 32 publications

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“…A PEVD produces eigenvalues λ m (e jΩ ) and eigenvectors q m (e jΩ ), where λ m (e jΩ ) is the mth diagonal element of D(z)| z=e jΩ and q m (e jΩ ) is the mth column of Q(z)| z=e jΩ . For the eigenvectors q m (e jΩ ) to be compact in the time domain, they must be maximally smooth on the unit circle [10]. In this section, we propose a novel metric to measure the smoothness of eigenvectors.…”

Section: Smoothness Metricmentioning

confidence: 99%

“…Noting that the decompositions in (1) and (14) are only unique up to permutations and phase shifts [10], an advantage of the frequency-based approach proposed here is the option to rearrange the eigenvalues and eigenvectors at each frequency bin if desired. If the eigenvalues in D[k] are arranged in descending order, approximate spectral majorisation of the resulting polynomial eigenvalues occurs.…”

Section: Proposed Algorithm a Overviewmentioning

confidence: 99%

“…II, eigenvectors containing such phase discontinuities are not smooth and return high values of χ (P ) . For a short paraunitary matrix Q(z), these discontinuities must be smoothed [10] and χ (P ) decreased. This is achieved through the use of a phase alignment function, described in Sec.…”

Section: Proposed Algorithm a Overviewmentioning

confidence: 99%

“…(1) Equation (1) has only approximate equality, as the PEVD of a finite order polynomial matrix is generally not of finite order. Note that the decomposition in (1) is unique up to permutations and arbitrary allpass filters applied to the eigenvectors, provided that Q(z) and D(z) are selected analytic [10].…”

Section: Introductionmentioning

confidence: 99%

“…Here, we present a novel frequency-based PEVD algorithm which can compute an accurate PEVD without requiring an estimate of the paraunitary filter length. It has been demonstrated in [10] that the lowest order approximation to (1) is possible if we attempt to approximate analytic eigenvectors whichon the unit circle -are characterised by being infinitely differentiable. We utilise a cost function that measures the power in derivatives based on the Fourier coefficients of their discrete samples on the unit circle.…”

Section: Introductionmentioning

confidence: 99%

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Enforcing Eigenvector Smoothness for a Compact DFT-Based Polynomial Eigenvalue Decomposition

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A variety of algorithms have been developed to compute an approximate polynomial matrix eigenvalue decomposition (PEVD). As an extension of the ordinary EVD to polynomial matrices, the PEVD will generate paraunitary matrices that diagonalise a parahermitian matrix. While iterative PEVD algorithms that compute a decomposition in the time domain have received a great deal of focus and algorithmic improvements in recent years, there has been less research in the field of frequency-based PEVD algorithms. Such algorithms have shown promise for the decomposition of problems of finite order, but the state-of-the-art requires a priori knowledge of the length of the polynomial matrices required in the decomposition. This paper presents a novel frequency-based PEVD algorithm which can compute an accurate decomposition without requiring this information. Also presented is a new metric for measuring a function's smoothness on the unit circle, which is utilised within the algorithm to maximise eigenvector smoothness for a compact decomposition, such that the polynomial eigenvectors have low order. We demonstrate through the use of simulations that the algorithm can achieve superior levels of decomposition accuracy to a state-of-the-art frequency-based method.

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Section: Smoothness Metricmentioning

confidence: 99%

Section: Proposed Algorithm a Overviewmentioning

confidence: 99%

Section: Proposed Algorithm a Overviewmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Application of the sequential matrix diagonalization algorithm to high-dimensional functional MRI data

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This paper introduces an adaptation of the sequential matrix diagonalization (SMD) method to high-dimensional functional magnetic resonance imaging (fMRI) data. SMD is currently the most efficient statistical method to perform polynomial eigenvalue decomposition. Unfortunately, with current implementations based on dense polynomial matrices, the algorithmic complexity of SMD is intractable and it cannot be applied as such to high-dimensional problems. However, for certain applications, these polynomial matrices are mostly filled with null values and we have consequently developed an efficient implementation of SMD based on a GPU-parallel representation of sparse polynomial matrices. We then apply our "sparse SMD" to fMRI data, i.e. the temporal evolution of a large grid of voxels (as many as 29,328 voxels). Because of the energy compaction property of SMD, practically all the information is concentrated by SMD on the first polynomial principal component. Brain regions that are activated over time are thus reconstructed with great fidelity through analysis-synthesis based on the first principal component only, itself in the form of a sequence of polynomial parameters.

show abstract

On the Existence and Uniqueness of the Eigenvalue Decomposition of a Parahermitian Matrix

Cited by 54 publications

References 32 publications

Enforcing Eigenvector Smoothness for a Compact DFT-Based Polynomial Eigenvalue Decomposition

Enforcing Eigenvector Smoothness for a Compact DFT-Based Polynomial Eigenvalue Decomposition

High-Performance System-on-Chip-Based Accelerator System for Polynomial Matrix Multiplications

Application of the sequential matrix diagonalization algorithm to high-dimensional functional MRI data

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