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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. Quantum Time-Space Tradeoffs for Matrix Problems

   https://doi.org/10.1145/3618260.3649700  

   Beame, Paul; Kornerup, Niels; Whitmeyer, Michael (June 2024, ACM)

   We prove lower bounds on the time and space required for quantum computers to solve a wide variety of problems involving matrices, many of which have only been analyzed classically in prior work. Using a novel way of applying recording query methods we show that for many linear algebra problems—including matrix-vector product, matrix inversion, matrix multiplication and powering—existing classical time-space tradeoffs also apply to quantum algorithms with at most a constant factor loss. For example, for almost all fixed matrices A, including the discrete Fourier transform (DFT) matrix, we prove that quantum circuits with at most T input queries and S qubits of memory require T=Ω(n^2/S) to compute matrix-vector product Ax for x ∈ {0,1}^n. We similarly prove that matrix multiplication for nxn binary matrices requires T=Ω(n^3/√S). Because many of our lower bounds are matched by deterministic algorithms with the same time and space complexity, our results show that quantum computers cannot provide any asymptotic advantage for these problems at any space bound. We also improve the previous quantum time-space tradeoff lower bounds for n× n Boolean (i.e. AND-OR) matrix multiplication from T=Ω(n^2.5/S^0.5) to T=Ω(n^2.5/S^0.25) which has optimal exponents for the powerful query algorithms to which it applies. Our method also yields improved lower bounds for classical algorithms. 

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3. Superfast Direct Inversion of the Nonuniform Discrete Fourier Transform via Hierarchically Semiseparable Least Squares

   https://doi.org/10.1137/24M1656694  

   Wilber, Heather; Epperly, Ethan N; Barnett, Alex H (June 2025, SIAM Journal on Scientific Computing)

   A direct solver is introduced for solving overdetermined linear systems involving nonuniform discrete Fourier transform matrices. Such matrices can be transformed into a Cauchy-like form that has hierarchical low rank structure. The rank structure of this matrix is explained, and it is shown that the ranks of the relevant submatrices grow only logarithmically with the number of columns of the matrix. A fast rank-structured hierarchical approximation method based on this analysis is developed, along with a hierarchical least-squares solver for these and related systems. This result is a direct method for inverting nonuniform discrete transforms with a complexity that is usually nearly linear with respect to the degrees of freedom in the problem. This solver is benchmarked against various iterative and direct solvers in the setting of inverting the one-dimensional type-II (or forward) transform, for a range of condition numbers and problem sizes (up to 4 × 10 by 2 × 10 ). These experiments demonstrate that this method is especially useful for large problems with multiple right-hand sides. 

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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. 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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