You are provided with the abstract of a scientific paper and the abstracts of the papers it references. Your task is to write a Related Work section for the main paper. This section should concisely summarize and integrate the key points from the cited abstracts, reflecting how they relate to the main paper's contributions and context. In your summary, use numerical citation markers like [1], [2], etc., to refer to the corresponding papers.

Main Abstract:
We describe a new algorithm called Frequent Directions for deterministic matrix sketching in the row-updates model. The algorithm is presented an arbitrary input matrix @math one row at a time. It performed @math operations per row and maintains a sketch matrix @math such that for any @math @math and @math . Here, @math stands for the minimizer of @math over all rank @math matrices (similarly @math ) and @math is the rank @math matrix resulting from projecting @math on the row span of @math . We show both of these bounds are the best possible for the space allowed. The summary is mergeable, and hence trivially parallelizable. Moreover, Frequent Directions outperforms exemplar implementations of existing streaming algorithms in the space-error tradeoff.

Cited Abstract(s):
[1]: We give near-optimal space bounds in the streaming model for linear algebra problems that include estimation of matrix products, linear regression, low-rank approximation, and approximation of matrix rank. In the streaming model, sketches of input matrices are maintained under updates of matrix entries; we prove results for turnstile updates, given in an arbitrary order. We give the first lower bounds known for the space needed by the sketches, for a given estimation error e. We sharpen prior upper bounds, with respect to combinations of space, failure probability, and number of passes. The sketch we use for matrix A is simply STA, where S is a sign matrix. Our results include the following upper and lower bounds on the bits of space needed for 1-pass algorithms. Here A is an n x d matrix, B is an n x d' matrix, and c := d+d'. These results are given for fixed failure probability; for failure probability δ>0, the upper bounds require a factor of log(1 δ) more space. We assume the inputs have integer entries specified by O(log(nc)) bits, or O(log(nd)) bits. (Matrix Product) Output matrix C with F(ATB-C) ≤ e F(A) F(B). We show that Θ(ce-2log(nc)) space is needed. (Linear Regression) For d'=1, so that B is a vector b, find x so that Ax-b ≤ (1+e) minx' ∈ Reald Ax'-b. We show that Θ(d2e-1 log(nd)) space is needed. (Rank-k Approximation) Find matrix tAk of rank no more than k, so that F(A-tAk) ≤ (1+e) F A-Ak , where Ak is the best rank-k approximation to A. Our lower bound is Ω(ke-1(n+d)log(nd)) space, and we give a one-pass algorithm matching this when A is given row-wise or column-wise. For general updates, we give a one-pass algorithm needing [O(ke-2(n + d e2)log(nd))] space. We also give upper and lower bounds for algorithms using multiple passes, and a sketching analog of the CUR decomposition.
It is worth noting that under the construction model, and allowing streaming turnstile updates to each element of the matrix, the hashing algorithm has been shown space optimal by Clarkson and Woodruff [1] (assuming each matrix entry requires @math bits, and otherwise off by only a @math factor). It is randomized and it constructs a decomposition of a rank @math matrix @math that satisfies @math , with probability at least @math . This provides a relative error construction bound of size @math bits. They also show an @math bits lower bound. Our paper shows that the row-update model is strictly easier with a lower upper bound.