scholarly journals FAST ALGORITHMS AND FAST ARITHMETIC IN LINEAR ALGEBRA (COMPUTATIONAL COMPLEXITY).

1972 ◽  
Author(s):  
E. Bareiss ◽  
D. Mazukelli
Keyword(s):  
Computational Complexity ◽  
Linear Algebra ◽  
Fast Algorithms ◽  
Fast Arithmetic

DMRG Approach to Fast Linear Algebra in the TT-Format

2011 ◽  
Vol 11 (3) ◽  
pp. 382-393 ◽  
Author(s):  
Ivan Oseledets
Keyword(s):  
Linear Systems ◽  
Linear Algebra ◽  
Direct Method ◽  
Fast Algorithms ◽  
Approximate Solutions ◽  
Vector Product ◽  
The Matrix ◽  
Minimization Scheme

AbstractIn this paper, the concept of the DMRG minimization scheme is extended to several important operations in the TT-format, like the matrix-by-vector product and the conversion from the canonical format to the TT-format. Fast algorithms are implemented and a stabilization scheme based on randomization is proposed. The comparison with the direct method is performed on a sequence of matrices and vectors coming as approximate solutions of linear systems in the TT-format. A generated example is provided to show that randomization is really needed in some cases. The matrices and vectors used are available from the author or at http://spring.inm.ras.ru/osel

scholarly journals What Can You Do with a Rock? Affordance Extraction via Word Embeddings

2017 ◽  
Author(s):  
Nancy Fulda ◽  
Daniel Ricks ◽  
Ben Murdoch ◽  
David Wingate
Keyword(s):  
Reinforcement Learning ◽  
Computational Complexity ◽  
Linear Algebra ◽  
Autonomous Agents ◽  
Common Knowledge ◽  
Search Space ◽  
Word Embeddings ◽  
Knowledge Database ◽  
Learning Agent ◽  
Action Spaces

Autonomous agents must often detect affordances: the set of behaviors enabled by a situation. Affordance extraction is particularly helpful in domains with large action spaces, allowing the agent to prune its search space by avoiding futile behaviors. This paper presents a method for affordance extraction via word embeddings trained on a tagged Wikipedia corpus. The resulting word vectors are treated as a common knowledge database which can be queried using linear algebra. We apply this method to a reinforcement learning agent in a text-only environment and show that affordance-based action selection improves performance in most cases. Our method increases the computational complexity of each learning step but significantly reduces the total number of steps needed. In addition, the agent's action selections begin to resemble those a human would choose.

The computational complexity of some problems of linear algebra

1997 ◽  
pp. 451-462 ◽  
Author(s):  
Jonathan F. Buss ◽  
Gudmund S. Frandsen ◽  
Jeffrey O. Shallit
Keyword(s):  
Computational Complexity ◽  
Linear Algebra

Randomized algorithms in numerical linear algebra

Acta Numerica ◽  
2017 ◽  
Vol 26 ◽  
pp. 95-135 ◽  
Author(s):  
Ravindran Kannan ◽  
Santosh Vempala
Keyword(s):  
Linear Algebra ◽  
Randomized Algorithms ◽  
Matrix Multiplication ◽  
Fast Algorithms ◽  
Numerical Linear Algebra ◽  
Low Rank ◽  
Low Rank Approximation ◽  
Rank Approximation

This survey provides an introduction to the use of randomization in the design of fast algorithms for numerical linear algebra. These algorithms typically examine only a subset of the input to solve basic problems approximately, including matrix multiplication, regression and low-rank approximation. The survey describes the key ideas and gives complete proofs of the main results in the field. A central unifying idea is sampling the columns (or rows) of a matrix according to their squared lengths.

scholarly journals The Computational Complexity of Some Problems of Linear Algebra

1996 ◽  
Vol 3 (33) ◽  
Author(s):  
Jonathan F. Buss ◽  
Gudmund Skovbjerg Frandsen ◽  
Jeffery O. Shallit
Keyword(s):  
Computational Complexity ◽  
Commutative Ring ◽  
Linear Algebra ◽  
Polynomial Time ◽  
Matrix Rank ◽  
Maximum Rank ◽  
The Matrix ◽  
Np Complete

We consider the computational complexity of some problems dealing with matrix rank.<br /> Let E, S be subsets of a commutative ring R.<br />Let x1, x2, ..., xt be variables. Given a matrix M = M(x1, x2, ..., xt)<br />with entries chosen from E union {x1, x2, ..., xt}, we want to determine<br />maxrankS(M) = max rank M(a1, a2, ... , at)<br />and<br />minrankS(M) = min rank M(a1, a2, ..., at). <br />There are also variants of these problems that specify more about the<br />structure of M, or instead of asking for the minimum or maximum rank, <br />ask if there is some substitution of the variables that makes the matrix<br /> invertible or noninvertible.<br />Depending on E, S, and on which variant is studied, the complexity<br />of these problems can range from polynomial-time solvable to random<br />polynomial-time solvable to NP-complete to PSPACE-solvable to<br />unsolvable.

scholarly journals The Computational Complexity of Some Problems of Linear Algebra

1999 ◽  
Vol 58 (3) ◽  
pp. 572-596 ◽  
Author(s):  
Jonathan F Buss ◽  
Gudmund S Frandsen ◽  
Jeffrey O Shallit
Keyword(s):  
Computational Complexity ◽  
Linear Algebra
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