Orthonormal and unitary matrices

Energy preserving operators

In which I think about parameterisations and implementations energy-presercing operators as matrices.

A particular nook in the the linear feedback process library.

Uses include maintaining stable gradients in recurrent neural networks (Arjovsky, Shah, and Bengio 2016; Jing et al. 2017; Mhammedi et al. 2017) and efficient normalising flows. (Berg et al. 2018; Hasenclever, Tomczak, and Welling 2017)

Also, parameterising stable Multi-Input-Multi-Output (MIMO) delay networks in signal processing.

(Berg et al. 2018) gives a short summary of this area.

Iterative normalising

Attributed to (Björck and Bowie 1971; Kovarik 1970)

\[ \mathbf{Q}^{(k+1)}=\mathbf{Q}^{(k)}\left(\mathbf{I}+\frac{1}{2}\left(\mathbf{I}-\mathbf{Q}^{(k) \top} \mathbf{Q}^{(k)}\right)\right) \] which reputedly converges if

\(\left\|\mathbf{Q}^{(0) \top} \mathbf{Q}^{(0)}-\mathbf{I}\right\|_{2}<1\)

Householder reflections

(Only for square matrices.) We can apply successive reflections about hyperplanes

\[ H(\mathbf{z})=\mathbf{z}-\frac{\mathbf{v} \mathbf{v}^{T}}{\|\mathbf{v}\|^{2}} \mathbf{z} \] 🏗

Givens rotation

One obvious parameterisation of unitary matrices is composing Givens rotations. 🏗

Arjovsky, Martin, Amar Shah, and Yoshua Bengio. 2016. “Unitary Evolution Recurrent Neural Networks.” In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48, 1120–8. ICML’16. New York, NY, USA: JMLR.org. http://arxiv.org/abs/1511.06464.

Berg, Rianne van den, Leonard Hasenclever, Jakub M. Tomczak, and Max Welling. 2018. “Sylvester Normalizing Flows for Variational Inference.” In UAI18. http://arxiv.org/abs/1803.05649.

Björck, Å., and C. Bowie. 1971. “An Iterative Algorithm for Computing the Best Estimate of an Orthogonal Matrix.” SIAM Journal on Numerical Analysis 8 (2): 358–64. https://doi.org/10.1137/0708036.

De Sena, Enzo, Huseyin Haciihabiboglu, Zoran Cvetkovic, and Julius O. Smith. 2015. “Efficient Synthesis of Room Acoustics via Scattering Delay Networks.” IEEE/ACM Transactions on Audio, Speech, and Language Processing 23 (9): 1478–92. https://doi.org/10.1109/TASLP.2015.2438547.

Hasenclever, Leonard, Jakub M Tomczak, and Max Welling. 2017. “Variational Inference with Orthogonal Normalizing Flows,” 4.

Hendeković, J. 1974. “On Parametrization of Orthogonal and Unitary Matrices with Respect to Their Use in the Description of Molecules.” Chemical Physics Letters 28 (2): 242–45. https://doi.org/10.1016/0009-2614(74)80063-1.

Jarlskog, C. 2005. “A Recursive Parametrization of Unitary Matrices.” Journal of Mathematical Physics 46 (10): 103508. https://doi.org/10.1063/1.2038607.

Jing, Li, Yichen Shen, Tena Dubcek, John Peurifoy, Scott Skirlo, Yann LeCun, Max Tegmark, and Marin Soljačić. 2017. “Tunable Efficient Unitary Neural Networks (EUNN) and Their Application to RNNs.” In PMLR, 1733–41. http://proceedings.mlr.press/v70/jing17a.html.

Kovarik, Zdislav. 1970. “Some Iterative Methods for Improving Orthonormality.” SIAM Journal on Numerical Analysis 7 (3): 386–89. https://doi.org/10.1137/0707031.

Menzer, Fritz, and Christof Faller. 2010. “Unitary Matrix Design for Diffuse Jot Reverberators.” https://www.researchgate.net/publication/230757792_Unitary_Matrix_Design_for_Diffuse_Jot_Reverberators.

Mhammedi, Zakaria, Andrew Hellicar, Ashfaqur Rahman, and James Bailey. 2017. “Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections.” In PMLR, 2401–9. http://proceedings.mlr.press/v70/mhammedi17a.html.

Regalia, P., and M. Sanjit. 1989. “Kronecker Products, Unitary Matrices and Signal Processing Applications.” SIAM Review 31 (4): 586–613. https://doi.org/10.1137/1031127.

Schroeder, Manfred R. 1961. “Improved Quasi-Stereophony and ‘Colorless’ Artificial Reverberation.” The Journal of the Acoustical Society of America 33 (8): 1061–4. https://doi.org/10.1121/1.1908892.

Schroeder, Manfred R., and B. Logan. 1961. “"Colorless" Artificial Reverberation.” Audio, IRE Transactions on AU-9 (6): 209–14. https://doi.org/10.1109/TAU.1961.1166351.

Tilma, Todd, and E C G Sudarshan. 2002. “Generalized Euler Angle Paramterization for SU(N).” Journal of Physics A: Mathematical and General 35 (48): 10467–10501. https://doi.org/10.1088/0305-4470/35/48/316.

Valimaki, v., and T. I. Laakso. 2012. “Fractional Delay Filters-Design and Applications.” In Nonuniform Sampling: Theory and Practice, edited by Farokh Marvasti. Springer Science & Business Media. http://books.google.com?id=n3fgBwAAQBAJ.