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1. Solving Sparse Linear Systems Faster than Matrix Multiplication

   https://doi.org/10.1137/1.9781611976465.31  

   Peng, Richard ; Vempala, Santosh S. ( January 2021 , Proceedings of the 2021 ACM-SIAM Symposium on Discrete Algorithms, SODA 2021, Virtual Conference, January 10 - 13, 2021)

   null (Ed.)

   Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear system $Ax=b$ is $\tilde{O}(n^\omega)$, where $\omega < 2.372864$ is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly$(n)$ condition number. In this paper, we present an algorithm that solves linear systems in sparse matrices asymptotically faster than matrix multiplication for any $\omega > 2$. This speedup holds for any input matrix $A$ with $o(n^{\omega -1}/\log(\kappa(A)))$ non-zeros, where $\kappa(A)$ is the condition number of $A$. For poly$(n)$-conditioned matrices with $\tilde{O}(n)$ nonzeros, and the current value of $\omega$, the bit complexity of our algorithm to solve to within any $1/\text{poly}(n)$ error is $O(n^{2.331645})$. Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorizations. It is inspired by the algorithm of [Eberly et al. ISSAC `06 `07] for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semi-random matrices. 

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2. Fast Regression for Structured Inputs

   Meyer, Raphael ; Musco, Cameron ; Musco, Christopher ; Woodruff, David P. ; Zhou, Samson ( April 2022 , International Conference on Learning Representations (ICLR))

   We study the $\ell_p$ regression problem, which requires finding $\mathbf{x}\in\mathbb R^{d}$ that minimizes $\|\mathbf{A}\mathbf{x}-\mathbf{b}\|_p$ for a matrix $\mathbf{A}\in\mathbb R^{n \times d}$ and response vector $\mathbf{b}\in\mathbb R^{n}$. There has been recent interest in developing subsampling methods for this problem that can outperform standard techniques when $n$ is very large. However, all known subsampling approaches have run time that depends exponentially on $p$, typically, $d^{\mathcal{O}(p)}$, which can be prohibitively expensive. We improve on this work by showing that for a large class of common \emph{structured matrices}, such as combinations of low-rank matrices, sparse matrices, and Vandermonde matrices, there are subsampling based methods for $\ell_p$ regression that depend polynomially on $p$. For example, we give an algorithm for $\ell_p$ regression on Vandermonde matrices that runs in time $\mathcal{O}(n\log^3 n+(dp^2)^{0.5+\omega}\cdot\text{polylog}\,n)$, where $\omega$ is the exponent of matrix multiplication. The polynomial dependence on $p$ crucially allows our algorithms to extend naturally to efficient algorithms for $\ell_\infty$ regression, via approximation of $\ell_\infty$ by $\ell_{\mathcal{O}(\log n)}$. Of practical interest, we also develop a new subsampling algorithm for $\ell_p$ regression for arbitrary matrices, which is simpler than previous approaches for $p \ge 4$. 

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3. Approximating Orthogonal Matrices with Effective Givens Factorization

   Frerix, Thomas ; Bruna, Joan ( June 2019 , International Conference on Machine Learning)

   We develop effective approximation methods for 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 an effective representation becomes advantageous. Although efficient Givens factorizations are 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 that our methods improve the approximate representation of the graph Fourier transform, the matrix obtained when diagonalizing a graph Laplacian. 

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4. Approximating Orthogonal Matrices with Effective Givens Factorization

   Frerix, Thomas ; Bruna, Joan ( June 2019 , international conference on machine learning)

   We develop effective approximation methods for 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 an effective representation becomes advantageous. Although efficient Givens factorizations are 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 that our methods improve the approximate representation of the graph Fourier transform, the matrix obtained when diagonalizing a graph Laplacian. 

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5. Kaleidoscope: An Efficient, Learnable Representation For All Structured Linear Maps

   Dao, Tri ; Sohoni, Nimit ; Gu, Albert ; Eichhorn, Matthew ; Blonder, Amit ; Leszczynski, Megan ; Rudra, Atri ; Ré, Christopher ( April 2020 , Eighth International Conference on Learning Representations)

   Modern neural network architectures use structured linear transformations, such as low-rank matrices, sparse matrices, permutations, and the Fourier transform, to improve inference speed and reduce memory usage compared to general linear maps. However, choosing which of the myriad structured transformations to use (and its associated parameterization) is a laborious task that requires trading off speed, space, and accuracy. We consider a different approach: we introduce a family of matrices called kaleidoscope matrices (K-matrices) that provably capture any structured matrix with near-optimal space (parameter) and time (arithmetic operation) complexity. We empirically validate that K-matrices can be automatically learned within end-to-end pipelines to replace hand-crafted procedures, in order to improve model quality. For example, replacing channel shuffles in ShuffleNet improves classification accuracy on ImageNet by up to 5%. K-matrices can also simplify hand-engineered pipelines---we replace filter bank feature computation in speech data preprocessing with a learnable kaleidoscope layer, resulting in only 0.4% loss in accuracy on the TIMIT speech recognition task. In addition, K-matrices can capture latent structure in models: for a challenging permuted image classification task, adding a K-matrix to a standard convolutional architecture can enable learning the latent permutation and improve accuracy by over 8 points. We provide a practically efficient implementation of our approach, and use K-matrices in a Transformer network to attain 36% faster end-to-end inference speed on a language translation task. 

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