

A Simple and Efficient Tensor Calculus
Part of a collection
  

Author(s)
Sören Laue
, Matthias Mitterreiter
, Joachim Giesen

Published in The ThirtyFourth AAAI Conference on Artificial Intelligence, AAAI 2020, New York, NY, USA, February 712, 2020

Year 2020 
Publisher AAAI Press 
Abstract Computing derivatives of tensor expressions, also known as tensor calculus, is a fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions and their derivatives that hinges on the representation of these expressions. Recently, an algorithm for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been introduced that is a few orders of magnitude faster than previous stateoftheart approaches. Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into automatic differentiation frameworks like TensorFlow, PyTorch, autograd, or JAX that use the simpler Einstein notation. This leaves two options, to either change the underlying tensor representation in these frameworks or to develop a new, provably correct algorithm based on Einstein notation. Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that using Ricci notation is not necessary for an efficient tensor calculus and develop an equally efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation enables further improvements that lead to even better efficiency. 
AD Theory and Techniques Hierarchical Approach 
BibTeX
@INPROCEEDINGS{
Laue2020ASa,
author = "S{\"{o}}ren Laue and Matthias Mitterreiter and Joachim Giesen",
title = "A Simple and Efficient Tensor Calculus",
pages = "45274534",
publisher = "{AAAI} Press",
year = "2020",
url = "https://aaai.org/ojs/index.php/AAAI/article/view/5881",
doi = "10.1609/aaai.v34i04.5881",
ad_theotech = "Hierarchical Approach",
abstract = "Computing derivatives of tensor expressions, also known as tensor calculus, is a
fundamental task in machine learning. A key concern is the efficiency of evaluating the expressions
and their derivatives that hinges on the representation of these expressions. Recently, an algorithm
for computing higher order derivatives of tensor expressions like Jacobians or Hessians has been
introduced that is a few orders of magnitude faster than previous stateoftheart approaches.
Unfortunately, the approach is based on Ricci notation and hence cannot be incorporated into
automatic differentiation frameworks like TensorFlow, PyTorch, autograd, or JAX that use the simpler
Einstein notation. This leaves two options, to either change the underlying tensor representation in
these frameworks or to develop a new, provably correct algorithm based on Einstein notation.
Obviously, the first option is impractical. Hence, we pursue the second option. Here, we show that
using Ricci notation is not necessary for an efficient tensor calculus and develop an equally
efficient method for the simpler Einstein notation. It turns out that turning to Einstein notation
enables further improvements that lead to even better efficiency.",
booktitle = "The ThirtyFourth {AAAI} Conference on Artificial Intelligence, {AAAI} 2020, New
York, NY, USA, February 712, 2020"
}
 
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