Advances in Matrix Algorithms and Numerical Methods

The field of matrix algorithms and numerical methods is experiencing significant developments, with a focus on improving the efficiency and accuracy of various algorithms. Researchers are exploring new approaches to solving complex problems, such as the Sinkhorn-Knopp algorithm and the Lanczos method, and are making progress in understanding the theoretical foundations of these algorithms. Additionally, there is a growing interest in developing new numerical methods for solving large-scale problems, including low-rank matrix estimation and linear systems with low-rank structure. These advances have important implications for a range of applications, from computer vision to machine learning. Noteworthy papers in this area include:

  • A paper on the phase transition of the Sinkhorn-Knopp algorithm, which establishes a sharp phase transition at a density threshold of 1/2.
  • A paper on approaching optimality for solving dense linear systems with low-rank structure, which presents new high-accuracy randomized algorithms with improved running times.

Sources

Phase transition of the Sinkhorn-Knopp algorithm

A structural bound for cluster robustness of randomized small-block Lanczos

Numerically Computing Galois Groups of Minimal Problems

On the Complexity of the Skolem Problem at Low Orders

Approaching Optimality for Solving Dense Linear Systems with Low-Rank Structure

A quasi-interpolation operator yielding fully computable error bounds

The Role of Rank in Mismatched Low-Rank Symmetric Matrix Estimation

Probabilistic algorithm for computing all local minimizers of Morse functions on a compact domain

Generalized Scattering Matrix Framework for Modeling Implantable Antennas in Multilayered Spherical Media

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