In this paper, we study the problem minx Rd,Nx=v Pn i=1 f((Ax b)i)for a quasiself-concordant function f: R R, where A,N are n d and m d matrices, b,v are vectors of length n and m with n d. We show an algorithm based on a trust-region method with an oracle that can be implemented using eO(d1/3)linear system solves, improving the eO(n1/3) oracle by [Adil-Bullins-Sachdeva, NeurIPS 2021]. Our implementation of the oracle relies on solving the overdetermined ℓ regression problem minx Rd,Nx=v Ax b . We provide an algorithm that finds a (1+ε)-approximate solution to this problem using O((d1/3/ε+1/ε2)log(n/ε)) linear system solves. This algorithm leverages ℓ Lewis weight overestimates and achieves this iteration complexity via a simple lightweight IRLS approach, inspired by the work of [Ene-Vladu, ICML 2019]. Experimentally, we demonstrate that our algorithm significantly improves the runtime of the standard CVX solver.

We present a unified framework for quantum sensitivity sampling, extending the advantages of quantum computing to a broad class of classical approximation problems. Our unified framework provides a streamlined approach for constructing coresets and offers significant runtime improvements in applications such as clustering, regression, and low-rank approximation. Our contributions include: * $k$-median and $k$-means clustering: For $n$ points in $d$-dimensional Euclidean space, we give an algorithm that constructs an $ε$-coreset in time $\widetilde O(n^{0.5}dk^{2.5}~\mathrm{poly}(ε^{-1}))$ for $k$-median and $k$-means clustering. Our approach achieves a better dependence on $d$ and constructs smaller coresets that only consist of points in the dataset, compared to recent results of [Xue, Chen, Li and Jiang, ICML'23]. * $\ell_p$ regression: For $\ell_p$ regression problems, we construct an $ε$-coreset of size $\widetilde O_p(d^{\max\{1, p/2\}}ε^{-2})$ in time $\widetilde O_p(n^{0.5}d^{\max\{0.5, p/4\}+1}(ε^{-3}+d^{0.5}))$, improving upon the prior best quantum sampling approach of [Apers and Gribling, QIP'24] for all $p\in (0, 2)\cup (2, 22]$, including the widely studied least absolute deviation regression ($\ell_1$ regression). * Low-rank approximation with Frobenius norm error: We introduce the first quantum sublinear-time algorithm for low-rank approximation that does not rely on data-dependent parameters, and runs in $\widetilde O(nd^{0.5}k^{0.5}ε^{-1})$ time. Additionally, we present quantum sublinear algorithms for kernel low-rank approximation and tensor low-rank approximation, broadening the range of achievable sublinear time algorithms in randomized numerical linear algebra.