A note on the inversion of complex matrices

Smith, W. B.; Erdman, S.

doi:10.1109/tac.1974.1100466

Cited by 20 publications

(11 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…But it sometimes fails due to the poor estimate of the factor that results in a singular matrix. In this paper, a more stable matrix inverse formula [36] is implemented, which converts the problem of complex matrix inverse Φ −1 = (A+ iB) −1 ∈ C m×m into the inverse of a 2m × 2m real matrix:…”

Section: Attacking the Numerical Instability Issuementioning

confidence: 99%

End-to-End Dereverberation, Beamforming, and Speech Recognition with Improved Numerical Stability and Advanced Frontend

Zhang

Boeddeker²,

Watanabe

et al. 2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

View full text Add to dashboard Cite

Recently, the end-to-end approach has been successfully applied to multi-speaker speech separation and recognition in both singlechannel and multichannel conditions. However, severe performance degradation is still observed in the reverberant and noisy scenarios, and there is still a large performance gap between anechoic and reverberant conditions. In this work, we focus on the multichannel multi-speaker reverberant condition, and propose to extend our previous framework for end-to-end dereverberation, beamforming, and speech recognition with improved numerical stability and advanced frontend subnetworks including voice activity detection like masks. The techniques significantly stabilize the end-to-end training process. The experiments on the spatialized wsj1-2mix corpus show that the proposed system achieves about 35% WER relative reduction compared to our conventional multi-channel E2E ASR system, and also obtains decent speech dereverberation and separation performance (SDR = 12.5 dB) in the reverberant multi-speaker condition while trained only with the ASR criterion.

show abstract

Section: Attacking the Numerical Instability Issuementioning

confidence: 99%

End-to-End Dereverberation, Beamforming, and Speech Recognition with Improved Numerical Stability and Advanced Frontend

Zhang

Boeddeker²,

Watanabe

et al. 2021

ICASSP 2021 - 2021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP)

View full text Add to dashboard Cite

show abstract

“…Furthermore, using (33), there is again a coupled iteration for the square root and inverse square root as well as their derivatives that is based on (26). Now recall the iterations involving the conjugate transpose ( 19), ( 20) and (27) for the computation of P(A) for complex matrices A. As mentioned before, the mapping X → X H is not Fréchet differentiable, thus making the update function g not Fréchet differentiable for any of the three iterative schemes.…”

Section: Coupled Iterationsmentioning

confidence: 99%

On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

Werner¹

2021

Preprint

View full text Add to dashboard Cite

We show, that the Complex Step approximation Im(f (A + ihE))/h to the Fréchet derivative of matrix functions f : R m,n → R m,n is applicable to the matrix sign, square root and polar mapping using iterative schemes. While this property was already discovered for the matrix sign using Newtons method, we extend the research to the family of Padé iterations, that allows us to introduce iterative schemes for finding function and derivative values while approximately preserving automorphism group structure.

show abstract

“…The proof of this theorem is obtained by substituting (9) and (10) into (7) and noting that the n colnmnsofthematrixlU/vl are linearly independent. It can be seen that the first term on the right-hand side of (10) gives the inhomogeneous solution of (7), while the second term gives n linearly independent homogeneous solutions.…”

Section: Solution Proceduresmentioning

confidence: 99%

Solution of complex matrix equations with changing phase

Csendes

1976

J Optim Theory Appl

View full text Add to dashboard Cite

A method is presented for solving a succession of complex matrix equations in which the phase of the real and imaginary components changes. The method is more efficient than the technique obtained by using complex Gaussian elimination on each of the matrix equations separately. In addition, some interesting theoretical relationships are presented for the solution of complex matrix equations in general, using only real-valued arithmetic operations.Key Words. Linear systems, unconstrained minimization, mathematical programming, finite-difference-finite-element methods, pseudoinverse solutions.t T h i s w o r k w a s p e r f o r m e d while the a u t h o r w a s with the Electrical E n g i n e e r i n g D e p a r t m e n t ,McGitl University, M o n t r e a l , C a n a d a . F i n a n c i a l s u p p o r t w a s p r o v i d e d b y the N a t i o n a l

show abstract

A note on the inversion of complex matrices

Cited by 20 publications

References 0 publications

End-to-End Dereverberation, Beamforming, and Speech Recognition with Improved Numerical Stability and Advanced Frontend

End-to-End Dereverberation, Beamforming, and Speech Recognition with Improved Numerical Stability and Advanced Frontend

On using the complex step method for the approximation of Fréchet derivatives of matrix functions in automorphism groups

Solution of complex matrix equations with changing phase

Contact Info

Product

Resources

About