On the approximation of a matrix
arXiv:2108.13195v1
Originality Synthesis-oriented
AI Analysis
This work addresses matrix approximation, a fundamental problem in numerical linear algebra, but appears incremental as it builds on existing methods by introducing randomization.
The paper tackles the problem of approximating a given matrix F by proving that a randomized algorithm can compute matrices H and T such that their product HT yields a better approximation than a non-randomized method's output F*.
Let $F^{*}$ be an approximation of a given $(a \times b)$ matrix $F$ derived by methods that are not randomized. We prove that for a given $F$ and $F^{*}$, $H$ and $T$ can be computed by randomized algorithm such that $(HT)$ is an approximation of $F$ better than $F^{*}$.