An Efficient Method of Computing Higher-Order Bond Price Perturbation Approximations

2011 ◽  
Author(s):  
Martin M. Andreasen ◽  
Pawel Zabczyk
Keyword(s):  
Efficient Method ◽  
Higher Order ◽  
Bond Price

scholarly journals A Simple and Efficient Tensor Calculus

2020 ◽  
Vol 34 (04) ◽  
pp. 4527-4534
Author(s):  
Sören Laue ◽  
Matthias Mitterreiter ◽  
Joachim Giesen
Keyword(s):  
Machine Learning ◽  
Efficient Method ◽  
Automatic Differentiation ◽  
State Of The Art ◽  
Higher Order ◽  
Tensor Representation ◽  
Tensor Calculus ◽  
Correct Algorithm ◽  
Previous State ◽  
Derivatives Of

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 state-of-the-art 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.

Higher order asymptotic bond price valuation for interest rates with non-Gaussian dependent innovations

2010 ◽  
Vol 7 (1) ◽  
pp. 60-69 ◽  
Author(s):  
Tetsuhiro Honda ◽  
Kenichiro Tamaki ◽  
Takayuki Shiohama
Keyword(s):  
Interest Rates ◽  
Higher Order ◽  
Bond Price ◽  
Non Gaussian

An efficient method to compute the Moore–Penrose inverse

2018 ◽  
Vol 9 (2) ◽  
pp. 143-152
Author(s):  
Hamid Esmaeili ◽  
Raziyeh Erfanifar ◽  
Mahdis Rashidi
Keyword(s):  
Random Matrices ◽  
Fourth Order ◽  
Efficient Method ◽  
Higher Order ◽  
Type Method ◽  
Higher Order Methods ◽  
Numerical Comparisons ◽  
Cpu Time

AbstractA new Schulz-type method to compute the Moore–Penrose inverse of a matrix is proposed. Every iteration of the method involves four matrix multiplications. It is proved that this method always converge with fourth-order. A wide set of numerical comparisons of the proposed method with nine higher order methods shows that the average number of matrix multiplications and the average CPU time of our method are considerably less than those of other methods. For each of sizes{n\times n}and{n\times(n+10)},{n=200,400,600,800,1000,1200}, ten random matrices were chosen to make these comparisons.

Dual systems for all: Higher-order, role-based relational reasoning as a uniquely derived feature of human cognition

2019 ◽  
Vol 42 ◽  
Author(s):  
Daniel J. Povinelli ◽  
Gabrielle C. Glorioso ◽  
Shannon L. Kuznar ◽  
Mateja Pavlic
Keyword(s):  
Temporal Reasoning ◽  
Higher Order ◽  
Important Case ◽  
Human Cognition ◽  
Relational Reasoning ◽  
Dual Systems ◽  
First Order ◽  
Reasoning System ◽  
Role Based

Abstract Hoerl and McCormack demonstrate that although animals possess a sophisticated temporal updating system, there is no evidence that they also possess a temporal reasoning system. This important case study is directly related to the broader claim that although animals are manifestly capable of first-order (perceptually-based) relational reasoning, they lack the capacity for higher-order, role-based relational reasoning. We argue this distinction applies to all domains of cognition.

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