# Approximating Orthogonal Matrices with Effective Givens Factorization

arXiv: https://arxiv.org/abs/1905.05796 · HTML: https://mlsanity.com/paper/1905.05796/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 May 2019.

## Built on by (most cited first, 7 of 7)

- [projUNN: efficient method for training deep networks with unitary matrices](https://mlsanity.com/paper/2203.05483/) (2022)
- [Parameter Efficient Quasi-Orthogonal Fine-Tuning via Givens Rotation](https://mlsanity.com/paper/2404.04316/) (2024)
- [Sparser, Better, Deeper, Stronger: Improving Sparse Training with Exact Orthogonal Initialization](https://mlsanity.com/paper/2406.01755/) (2024)
- [A Block Coordinate Descent Method for Nonsmooth Composite Optimization under Orthogonality Constraints](https://mlsanity.com/paper/2304.03641/) (2023)
- [On sampling determinantal and Pfaffian point processes on a quantum computer](https://mlsanity.com/paper/2305.15851/) (2023)
- [Online Orthogonal Dictionary Learning Based on Frank-Wolfe Method](https://mlsanity.com/paper/2103.01484/) (2021)
- [L2G-Net: Local to Global Spectral Graph Neural Networks via Cauchy Factorizations](https://mlsanity.com/paper/2602.18837/) (2026)

## Abstract (arXiv)

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.