Topic
Penrose graphical notation
About: Penrose graphical notation is a research topic. Over the lifetime, 50 publications have been published within this topic receiving 1170 citations.
Papers
TL;DR: A tensor SVD is derived which is shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied.
Abstract: Higher order tensor inversion is possible for even order. This is due to the fact that a tensor group endowed with the contracted product is isomorphic to the general linear group of degree $n$. With these isomorphic group structures, we derive a tensor SVD which we have shown to be equivalent to well-known canonical polyadic decomposition and multilinear SVD provided that some constraints are satisfied. Moreover, within this group structure framework, multilinear systems are derived and solved for problems of high-dimensional PDEs and large discrete quantum models. We also address multilinear systems which do not fit the framework in the least-squares sense. These are cases when there is an odd number of modes or when each mode has distinct dimension. Numerically we solve multilinear systems using iterative techniques, namely, biconjugate gradient and Jacobi methods.
216 citations
112 citations
TL;DR: The tensor algebra and vector space geometry provide a unifying framework for multilinear data analysis, which simplifies notation and leads to economy of thought as mentioned in this paper, avoiding too much abstraction too soon in defining tensor products makes these concepts accessible.
58 citations
19 May 2012
TL;DR: In this article, the power of linear algebra in the context of multilinear computation has been examined and a super-polynomial separation between the two models has been established.
Abstract: This work deals with the power of linear algebra in the context of multilinear computation. By linear algebra we mean algebraic branching programs (ABPs) which are known to be computationally equivalent to two basic tools in linear algebra: iterated matrix multiplication and the determinant. We compare the computational power of multilinear ABPs to that of multilinear arithmetic formulas, and prove a tight super-polynomial separation between the two models. Specifically, we describe an explicit n-variate polynomial F that is computed by a linear-size multilinear ABP but every multilinear formula computing F must be of size nΩ(log n).
45 citations
1 Jan 2007
TL;DR: This thesis will study the problem of finding a best low rank approximation to a tensor, a symmetric Tensor, and a nonnegative tensor; define a notion of eigenvalues and eigenvectors for symmetric tensors and a concept of singular values and singular vectors for general tensors.
Abstract: The subject of this thesis is best described as the study of a few new problems in multilinear algebra that are analogous to important classical problems in numerical linear algebra. Among other things, we will study the problem of finding a best low rank approximation to a tensor, a symmetric tensor, and a nonnegative tensor; define a notion of eigenvalues and eigenvectors for symmetric tensors and a notion of singular values and singular vectors for general tensors; we apply our multilinear spectral theory to hypergraphs in a way that parallels spectral graph theory, and to nonnegative tensors in a way that parallels the results of Perron-Robenius; we will also study an extension of nonnegative matrix factorization to nonnegative tensors.
25 citations
Related Topics (5)
Performance Metrics
| Year | Papers |
|---|---|
| 2017 | 2 |
| 2014 | 3 |
| 2013 | 6 |
| 2012 | 3 |
| 2010 | 2 |
| 2009 | 1 |