Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Matrix Multiplication, Trilinear Decompositions, APA Algorithms, and
  Summation   Matrix multiplication (hereafter we use the acronym MM) is among the most
fundamental operations of modern computations. The efficiency of its
performance depends on various factors, in particular vectorization, data
movement and arithmetic complexity of the computations, but here we focus just
on the study of the arithmetic cost and the impact of this study on other areas
of modern computing. In the early 1970s it was expected that the
straightforward cubic time algorithm for MM will soon be accelerated to enable
MM in nearly quadratic arithmetic time, with some far fetched implications.
While pursuing this goal the mainstream research had its focus on the decrease
of the classical exponent 3 of the complexity of MM towards its lower bound 2,
disregarding the growth of the input size required to support this decrease.
Eventually, surprising combinations of novel ideas and sophisticated techniques
enabled the decrease of the exponent to its benchmark value of about 2.38, but
the supporting MM algorithms improved the straightforward one only for the
inputs of immense sizes. Meanwhile, the communication complexity, rather than
the arithmetic complexity, has become the bottleneck of computations in linear
algebra. This development may seem to undermine the value of the past and
future research aimed at the decrease of the arithmetic cost of MM, but we feel
that the study should be reassessed rather than closed and forgotten. We review
the old and new work in this area in the present day context, recall some major
techniques introduced in the study of MM, discuss their impact on the modern
theory and practice of computations for MM and beyond MM, and link one of these
techniques to some simple algorithms for inner product and summation.

20
The categories of this paper are: acc-phys
31
The categories of this paper are: ao-sci
21
The categories of this paper are: adap-org
11
The categories of this paper are: hep-lat
5
The categories of this paper are: gr-qc
33
The categories of this paper are: plasm-ph
29
The categories of this paper are: funct-an
15
The categories of this paper are: nucl-ex
8
The categories of this paper are: hep-ex
28
The categories of this paper are: alg-geom
34
The categories of this paper are: chem-ph
9
The categories of this paper are: nlin
37
The categories of this paper are: supr-con
13
The categories of this paper are: quant-ph
30
The categories of this paper are: q-alg
18
The categories of this paper are: econ