We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] for low-rank matrices [ Chakraborty et al., ICALP'19 ], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation. Namely, if the input model supports efficient \ell_2-norm importance sampling, and given A \in C^{m\times n} with minimum singular value \sigma and  b \in C^m as input , we can output a description of an x  such that \|x - A^+b\| \leq \varepsilon\|A^+b\| in \mathcal{O} %DIFDELCMD < \left(%%% \frac{\|A\|_{\mathrm{F}^6\|A\|^2}{\sigma^8\varepsilon^4}}%DIFDELCMD < \right) %%% \Big(}^6\|A\|^2}{\sigma^8\varepsilon^4}}\Big) time, improving on previous "quantum-inspired" algorithms in this line of research by a factor of \|A\|^{14}{\sigma^{14}\varepsilon^2} [Chia et al., STOC'20]. The algorithm is stochastic gradient descent, and the analysis bears similarities to results of [Gupta and Sidford, NeurIPS'18]. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired setting, for comparison against future quantum computers.
We give a classical algorithm for linear regression analogous to the quantum matrix inversion algorithm [Harrow, Hassidim, and Lloyd, Physical Review Letters'09] for low-rank matrices [ Wossnig et al., Physical Review Letters'18 ], when the input matrix A is stored in a data structure applicable for QRAM-based state preparation. Namely, given an A \in C^{m\times n} with minimum singular value \sigma and which supports certain efficient \ell_2-norm importance sampling queries, along with a b \in C^m , we can output a description of an x \in \mathbb{C such that \|x - A^+b\| \leq \varepsilon\|A^+b\| in \mathcal{O} %DIFDELCMD < \left(%%% }^6\|A\|^2}{\sigma^8\varepsilon^4}}%DIFDELCMD < \right) %%% \Big(\frac{\|A\|_{\mathrm{F}^6\|A\|^2}{\sigma^8\varepsilon^4}}\Big) time, improving on previous "quantum-inspired" algorithms in this line of research by a factor of \|A\|^{14}{\sigma^{14}\varepsilon^2} [Chia et al., STOC'20]. The algorithm is stochastic gradient descent, and the analysis bears similarities to those of optimization algorithms for regression in the usual setting [Gupta and Sidford, NeurIPS'18]. Unlike earlier works, this is a promising avenue that could lead to feasible implementations of classical regression in a quantum-inspired setting, for comparison against future quantum computers.
R
Chakraborty
Wossnig
R
ICALP'19
Physical Review Letters'18
R
if the input model supports efficient \ell_2-norm importance sampling, and given
given an
A
which supports certain efficient \ell_2-norm importance sampling queries, along with a
D
as input
A
\in \mathbb{C
D
\frac{\|A\|_{\mathrm{F
A
\frac{\|A\|_{\mathrm{F
R
results of
those of optimization algorithms for regression in the usual setting