Some related lower bounds include the work of Backurs et al. [2017] that solving kernel Support V ector Machines (SVM), ridge regression, or Principal Component Analysis (PCA) problems to high accuracy or approximating kernel density estimates up to a constant factor for kernels with

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We give a faster algorithm for computing an approximate John ellipsoid around $n$ points in $d$ dimensions. The best known prior algorithms are based on repeatedly computing the leverage scores of the points and reweighting them by these scores (Cohen et al., 2019). We show that this algorithm can be substantially sped up by delaying the computation of high accuracy leverage scores by using sampling, and then later computing multiple batches of high accuracy leverage scores via fast rectangular matrix multiplication. We also give low-space streaming algorithms for John ellipsoids using similar ideas.

Here are two important cases where this problem arises.

We study robust subspace estimation in the streaming and distributed settings.

A well-studied problem in machine learning and numerical linear algebra, with applications to recommendation systems, text mining, and computer vision, is that of computing a low-rank approximation of a matrix.

Linear regression is a fundamental problem in machine learning.

Such complexities are prohibitive for scaling to large datasets.

We consider a least squares regression problem where the data has been generated from a linear model, and we are interested to learn the unknown regression parameters. We consider "sketch-and-solve" methods that randomly project the