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Suppose a matrix $A \in \mathbb{R}^{m \times n}$ of rank $r$ with singularvalue decomposition $A = U_{A}\Sigma_{A} V_{A}^{T}$, where $U_{A} \in\mathbb{R}^{m \times r}$, $V_{A} \in \mathbb{R}^{n \times r}$ are orthonormaland $\Sigma_{A} \in \mathbb{R}^{r \times r}$ is a diagonal matrix. Thestatistical leverage scores of a matrix $A$ are the squared row-norms definedby $\ell_{i} = \|(U_{A})_{i,:}\|_2^2$, where $i \in [m]$, and the matrixcoherence is the largest statistical leverage score. These quantities play animportant role in machine learning algorithms such as matrix completion andNystr\"{o}m-based low rank matrix approximation as well as large-scalestatistical data analysis applications, whose usual algorithm complexity ispolynomial in the dimension of the matrix $A$. As an alternative to theconventional approach, and inspired by recent development on dequantizationtechniques, we propose a quantum-inspired algorithm for approximating thestatistical leverage scores. We then analyze the accuracy of the algorithm andperform numerical experiments to illustrate the feasibility of our algorithm.Theoretical analysis shows that our novel algorithm takes time polynomial in aninteger $k$, condition number $\kappa$ and logarithm of the matrix size.

A quantum-inspired algorithm for approximating statistical leverage scores