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1. 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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2. Randomized numerical linear algebra: Foundations and algorithms

   https://doi.org/10.1017/S0962492920000021  

   Martinsson, Per-Gunnar; Tropp, Joel A. (May 2020, Acta Numerica)

   null (Ed.)

   This survey describes probabilistic algorithms for linear algebraic computations, such as factorizing matrices and solving linear systems. It focuses on techniques that have a proven track record for real-world problems. The paper treats both the theoretical foundations of the subject and practical computational issues. Topics include norm estimation, matrix approximation by sampling, structured and unstructured random embeddings, linear regression problems, low-rank approximation, subspace iteration and Krylov methods, error estimation and adaptivity, interpolatory and CUR factorizations, Nyström approximation of positive semidefinite matrices, single-view (‘streaming’) algorithms, full rank-revealing factorizations, solvers for linear systems, and approximation of kernel matrices that arise in machine learning and in scientific computing. 

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3. Generalized Matrix Local Low Rank Representation by Random Projection and Submatrix Propagation

   https://doi.org/10.1145/3580305.3599361  

   Dang, Pengtao; Zhu, Haiqi; Guo, Tingbo; Wan, Changlin; Zhao, Tong; Salama, Paul; Wang, Yijie; Cao, Sha; Zhang, Chi (August 2023, ACM)

   Matrix low rank approximation is an effective method to reduce or eliminate the statistical redundancy of its components. Compared with the traditional global low rank methods such as singular value decomposition (SVD), local low rank approximation methods are more advantageous to uncover interpretable data structures when clear duality exists between the rows and columns of the matrix. Local low rank approximation is equivalent to low rank submatrix detection. Unfortunately,existing local low rank approximation methods can detect only submatrices of specific mean structure, which may miss a substantial amount of true and interesting patterns. In this work, we develop a novel matrix computational framework called RPSP (Random Probing based submatrix Propagation) that provides an effective solution for the general matrix local low rank representation problem. RPSP detects local low rank patterns that grow from small submatrices of low rank property, which are determined by a random projection approach. RPSP is supported by theories of random projection. Experiments on synthetic data demonstrate that RPSP outperforms all state-of-the-art methods, with the capacity to robustly and correctly identify the low rank matrices when the pattern has a similar mean as the background, background noise is heteroscedastic and multiple patterns present in the data. On real-world datasets, RPSP also demonstrates its effectiveness in identifying interpretable local low rank matrices. 

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4. Determinant-Preserving Sparsification of SDDM Matrices with Applications to Counting and Sampling Spanning Trees

   https://doi.org/10.1109/FOCS.2017.90  

   Durfee, David; Peebles, John; Peng, Richard; Rao, Anup B. (November 2017, 58th IEEE Annual Symposium on Foundations of Computer Science, FOCS 2017, Berkeley, CA, USA, October 15-17, 2017)

   We show variants of spectral sparsification routines can preserve the total spanning tree counts of graphs, which by Kirchhoff's matrix-tree theorem, is equivalent to determinant of a graph Laplacian minor, or equivalently, of any SDDM matrix. Our analyses utilizes this combinatorial connection to bridge between statistical leverage scores / effective resistances and the analysis of random graphs by [Janson, Combinatorics, Probability and Computing `94]. This leads to a routine that in quadratic time, sparsifies a graph down to about $$n^{1.5}$$ edges in ways that preserve both the determinant and the distribution of spanning trees (provided the sparsified graph is viewed as a random object). Extending this algorithm to work with Schur complements and approximate Choleksy factorizations leads to algorithms for counting and sampling spanning trees which are nearly optimal for dense graphs. We give an algorithm that computes a $$(1 \pm \delta)$$ approximation to the determinant of any SDDM matrix with constant probability in about $$n^2 \delta^{-2}$$ time. This is the first routine for graphs that outperforms general-purpose routines for computing determinants of arbitrary matrices. We also give an algorithm that generates in about $$n^2 \delta^{-2}$$ time a spanning tree of a weighted undirected graph from a distribution with total variation distance of $$\delta$$ from the $$w$$-uniform distribution . 

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5. Preparing Hamiltonians for quantum simulation: A computational framework for Cartan decomposition via Lax dynamics

   https://doi.org/10.1090/mcom/4056  

   Chu, Moody (February 2025, Mathematics of Computation)

   Quantum algorithms usually are described via quantum circuits representable as unitary operators. Synthesizing the unitary operators described mathematically in terms of the unitary operators recognizable as quantum circuits is essential. One such a challenge lies in the Hamiltonian simulation problem, where the matrix exponential of a large-scale skew-Hermitian matrix is to be computed. Most current techniques are prone to approximation errors, whereas the parametrization of the underlying Hamiltonian via the Cartan decomposition is more promising. To prepare for such a simulation, this work proposes to tackle the Cartan decomposition by means of the Lax dynamics. The advantages include not only that it is numerically feasible with no matrices involved, but also that this approach offers a genuine unitary synthesis within the integration errors. This work contributes to the theoretic and algorithmic foundations in three aspects: exploiting the quaternary representation of Hamiltonian subalgebras; describing a common mechanism for deriving the Lax dynamics; and providing a mathematical theory of convergence. 

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