Approximating Orthogonal Matrices with Effective Givens Factorization

Thomas Frerix, Joan Bruna
Proceedings of the 36th International Conference on Machine Learning, PMLR 97:1993-2001, 2019.

Abstract

We analyze effective approximation of unitary matrices. In our formulation, a unitary matrix is represented as a product of rotations in two-dimensional subspaces, so-called Givens rotations. Instead of the quadratic dimension dependence when applying a dense matrix, applying such an approximation scales with the number factors, each of which can be implemented efficiently. Consequently, in settings where an approximation is once computed and then applied many times, such a representation becomes advantageous. Although effective Givens factorization is not possible for generic unitary operators, we show that minimizing a sparsity-inducing objective with a coordinate descent algorithm on the unitary group yields good factorizations for structured matrices. Canonical applications of such a setup are orthogonal basis transforms. We demonstrate numerical results of approximating the graph Fourier transform, which is the matrix obtained when diagonalizing a graph Laplacian.

Cite this Paper

BibTeX
@InProceedings{pmlr-v97-frerix19a, title = {Approximating Orthogonal Matrices with Effective Givens Factorization}, author = {Frerix, Thomas and Bruna, Joan}, booktitle = {Proceedings of the 36th International Conference on Machine Learning}, pages = {1993--2001}, year = {2019}, editor = {Chaudhuri, Kamalika and Salakhutdinov, Ruslan}, volume = {97}, series = {Proceedings of Machine Learning Research}, month = {09--15 Jun}, publisher = {PMLR}, pdf = {http://proceedings.mlr.press/v97/frerix19a/frerix19a.pdf}, url = {https://proceedings.mlr.press/v97/frerix19a.html}, abstract = {We analyze effective approximation of unitary matrices. In our formulation, a unitary matrix is represented as a product of rotations in two-dimensional subspaces, so-called Givens rotations. Instead of the quadratic dimension dependence when applying a dense matrix, applying such an approximation scales with the number factors, each of which can be implemented efficiently. Consequently, in settings where an approximation is once computed and then applied many times, such a representation becomes advantageous. Although effective Givens factorization is not possible for generic unitary operators, we show that minimizing a sparsity-inducing objective with a coordinate descent algorithm on the unitary group yields good factorizations for structured matrices. Canonical applications of such a setup are orthogonal basis transforms. We demonstrate numerical results of approximating the graph Fourier transform, which is the matrix obtained when diagonalizing a graph Laplacian.} }
Endnote
%0 Conference Paper %T Approximating Orthogonal Matrices with Effective Givens Factorization %A Thomas Frerix %A Joan Bruna %B Proceedings of the 36th International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2019 %E Kamalika Chaudhuri %E Ruslan Salakhutdinov %F pmlr-v97-frerix19a %I PMLR %P 1993--2001 %U https://proceedings.mlr.press/v97/frerix19a.html %V 97 %X We analyze effective approximation of unitary matrices. In our formulation, a unitary matrix is represented as a product of rotations in two-dimensional subspaces, so-called Givens rotations. Instead of the quadratic dimension dependence when applying a dense matrix, applying such an approximation scales with the number factors, each of which can be implemented efficiently. Consequently, in settings where an approximation is once computed and then applied many times, such a representation becomes advantageous. Although effective Givens factorization is not possible for generic unitary operators, we show that minimizing a sparsity-inducing objective with a coordinate descent algorithm on the unitary group yields good factorizations for structured matrices. Canonical applications of such a setup are orthogonal basis transforms. We demonstrate numerical results of approximating the graph Fourier transform, which is the matrix obtained when diagonalizing a graph Laplacian.
APA
Frerix, T. & Bruna, J.. (2019). Approximating Orthogonal Matrices with Effective Givens Factorization. Proceedings of the 36th International Conference on Machine Learning, in Proceedings of Machine Learning Research 97:1993-2001 Available from https://proceedings.mlr.press/v97/frerix19a.html.

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