Quantum-inspired sublinear classical algorithms for solving low-rank linear systems
This work addresses the challenge of efficient linear system solving for large-scale data in computational mathematics and machine learning, offering a classical alternative to quantum methods with potential applications in recommendation systems and numerical analysis.
The paper tackles the problem of solving low-rank linear systems classically with sublinear time complexity, inspired by quantum algorithms, and presents two algorithms with query and time complexity O(poly(k, κ, ‖A‖_F, 1/ε) polylog(m, n)), providing samples or estimates of entries in the solution vector.
We present classical sublinear-time algorithms for solving low-rank linear systems of equations. Our algorithms are inspired by the HHL quantum algorithm for solving linear systems and the recent breakthrough by Tang of dequantizing the quantum algorithm for recommendation systems. Let $A \in \mathbb{C}^{m \times n}$ be a rank-$k$ matrix, and $b \in \mathbb{C}^m$ be a vector. We present two algorithms: a "sampling" algorithm that provides a sample from $A^{-1}b$ and a "query" algorithm that outputs an estimate of an entry of $A^{-1}b$, where $A^{-1}$ denotes the Moore-Penrose pseudo-inverse. Both of our algorithms have query and time complexity $O(\mathrm{poly}(k, κ, \|A\|_F, 1/ε)\,\mathrm{polylog}(m, n))$, where $κ$ is the condition number of $A$ and $ε$ is the precision parameter. Note that the algorithms we consider are sublinear time, so they cannot write and read the whole matrix or vectors. In this paper, we assume that $A$ and $b$ come with well-known low-overhead data structures such that entries of $A$ and $b$ can be sampled according to some natural probability distributions. Alternatively, when $A$ is positive semidefinite, our algorithms can be adapted so that the sampling assumption on $b$ is not required.