Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results. We provide provable guarantees for deterministic column sampling using ridge leverage scores. The matrix sketch returned by our algorithm is a column subset of the original matrix, yielding additional interpretability. Like the randomized counterparts, the deterministic algorithm provides $(1+\epsilon)$ error column subset selection, $(1+\epsilon)$ error projection-cost preservation, and an additive-multiplicative spectral bound.

While ridge regression provides shrinkage for the regression coefficients, manyofthecoefficients remain smallbutnon-zero. Performing ridgeregression with the matrix sketch returned by our algorithm and a particular regularization parameter forces coefficients to zero and has a provable(1+) bound on the statisticalrisk.

Typically this mapping is implicit. A kernel function is used to compute inner products in the high-dimensional kernel space, without ever actually mapping original data points to the space.

Ridge leverage scores provide a balance between low-rank approximation and regularization, and are ubiquitous in randomized linear algebra and machine learning. Deterministic algorithms are also of interest in the moderately big data regime, because deterministic algorithms provide interpretability to the practitioner by having no failure probability and always returning the same results.

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.

The authors provide an algorithm which learns a provably accurate low-rank approximation to a PSD matrix in sublinear time. Specifically, it learns an approximation that has low additive error with high probability by sampling columns from the matrix according to a certain importance measure, then forming a Nystrom approximation using these kernels. The importance measure used is an estimate of the ridge leverage scores, which are expensive to compute exactly ( O(n 3)). Their algorithm recursively estimates these leverage score by starting from a set of columns, using those to estimate the leverage scores, sampling a set of columns according to those probabilities, and repeating ... the authors show that when this process is done carefully, the leverage score estimates are accurate enough that they can be used to get almost as good an approximation as using the true ridge leverage scores. The cost of producing the final approximation is O(ns 2) computation time and O(ns) computations of entries in the PSD matrix.

The paper proposes an efficient second-order online kernel learning mainly by combining KONS and Nystrom method. NOVELTY The novelty is limited on both the methodological and theoretical contributions. The achieved results do not have profound implication for the advancement of theory and practice. WRITING QUALITY The English writing and organization of this paper are relatively good. The reviewer strongly suggests the authors arrange Table 2 in the main paper rather than in Appendix because the experimental results in Table 2 are the core material.

We give the first algorithm for kernel Nyström approximation that runs in linear time in the number of training points and is provably accurate for all kernel matrices, without dependence on regularity or incoherence conditions.

The $\ell_p$ subspace approximation problem is an NP-hard low rank approximation problem that generalizes the median hyperplane problem ($p = 1$), principal component analysis ($p = 2$), and the center hyperplane problem ($p = \infty$). A popular approach to cope with the NP-hardness of this problem is to compute a strong coreset, which is a small weighted subset of the input points which simultaneously approximates the cost of every $k$-dimensional subspace, typically to $(1+\varepsilon)$ relative error for a small constant $\varepsilon$. We obtain the first algorithm for constructing a strong coreset for $\ell_p$ subspace approximation with a nearly optimal dependence on the rank parameter $k$, obtaining a nearly linear bound of $\tilde O(k)\mathrm{poly}(\varepsilon^{-1})$ for $p<2$ and $\tilde O(k^{p/2})\mathrm{poly}(\varepsilon^{-1})$ for $p>2$. Prior constructions either achieved a similar size bound but produced a coreset with a modification of the original points [SW18, FKW21], or produced a coreset of the original points but lost $\mathrm{poly}(k)$ factors in the coreset size [HV20, WY23]. Our techniques also lead to the first nearly optimal online strong coresets for $\ell_p$ subspace approximation with similar bounds as the offline setting, resolving a problem of [WY23]. All prior approaches lose $\mathrm{poly}(k)$ factors in this setting, even when allowed to modify the original points.

We consider sketching algorithms which first compress data by multiplication with a random sketch matrix, and then apply the sketch to quickly solve an optimization problem, e.g., low-rank approximation and regression. In the learning-based sketching paradigm proposed by~\cite{indyk2019learning}, the sketch matrix is found by choosing a random sparse matrix, e.g., CountSketch, and then the values of its non-zero entries are updated by running gradient descent on a training data set. Despite the growing body of work on this paradigm, a noticeable omission is that the locations of the non-zero entries of previous algorithms were fixed, and only their values were learned. In this work, we propose the first learning-based algorithms that also optimize the locations of the non-zero entries. Our first proposed algorithm is based on a greedy algorithm. However, one drawback of the greedy algorithm is its slower training time. We fix this issue and propose approaches for learning a sketching matrix for both low-rank approximation and Hessian approximation for second order optimization. The latter is helpful for a range of constrained optimization problems, such as LASSO and matrix estimation with a nuclear norm constraint. Both approaches achieve good accuracy with a fast running time. Moreover, our experiments suggest that our algorithm can still reduce the error significantly even if we only have a very limited number of training matrices.