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1

A Simple and Efficient Tensor Calculus for Machine Learning

Laue Sören, Mitterreiter Matthias, Giesen Joachim

Fundamenta Informaticae

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2020

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Vol. 177, nr 2

157-179

EN

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 from deep learning 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. The methods that are described in this paper for computing derivatives of matrix and tensor expressions have been implemented in the online tool www.MatrixCalculus.org.

2

Application of Matrix Notation for the Analysis of Power Changes in a 3‑phase Circuit

Ładniak L.

Acta Energetica

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2016

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nr 2

149--152

EN

Using the matrix notation for voltages and currents of a 3-phase system, a description of changing the instantaneous power on terminals of electric circuit has been offered as a third rank matrix. As a result of the power matrix’ decomposition into a symmetric matrix and an antisymmetric matrix and upon defining norms for these matrices the state of the 3-phase circuit can be clearly determined.

PL

Korzystając z zapisu macierzowego napięć i prądów występujących w węźle obwodu 3-fazowego, zaproponowano opis zmian wartości chwilowych mocy na zaciskach rozpatrywanego układu w postaci macierzy trzeciego stopnia. W wyniku dekompozycji macierzy mocy na macierz symetryczną i antysymetryczną oraz po zdefiniowaniu norm dla tych macierzy można w jednoznaczny sposób określić stan układu 3-fazowego.

3

Ludwik Silberstein and the operator calculus

Flin P., Bajan K., Godłowski W.

Czasopismo Techniczne. Nauki Podstawowe

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2015

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Y. 112, iss. 2-NP

165--171

EN

In the article we outline the life of Ludwik Silberstein (1872‒1948). We present his approach to the matrix calculus and its application to the operator form of relativity. We also give the list of books on the subject written by him as well as translated by him.

PL

W artykule przedstawiamy zarys życiorysu Ludwika Silbersteina (1872‒1948). Omawiamy podejście do rachunku macierzowego i jego zastosowanie do podania operatorowej postaci teorii względności. Prezentujemy również listę książek i tłumaczeń z różnych języków dotyczących tej tematyki.

4

Metody optymalizacji kolejności analizy węzłów HAZOP

Sauk R., Moskal F.

Inżynieria i Aparatura Chemiczna

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2014

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Nr 1

33--35

PL

W artykule zaprezentowano metodykę ustalania optymalnej kolejności analizy węzłów HAZOP. Zaimplementowano w niej teorię grafów i rachunek macierzowy stosowane w inżynierii chemicznej i procesowej do modelowania matematycznego instalacji procesowych. Opisywane metody zostały z powodzeniem zastosowane w ramach przeprowadzonych analiz HAZOP instalacji kotła parowego z cyrkulującym złożem fluidalnym (CFB) oraz podziemnego magazynu gazu.

EN

The paper presents issues dealing with a methodology for the determination of optimal HAZOP nodes order by implementation of graph theory and matrix calculus, applied in chemical and process engineering for mathematical modelling of process systems. The specified methods were successfully applied in HAZOP analyses of a steam boiler with the circulating fluidized bed (CFB) and underground gas storage plant.

5

N-Body Potential Interaction as a Cost Function in the Elastic Model for SANET Cloud Computing

Chaczko Z., Resconi G., Chiu C., Aslanzadeh S.

International Journal of Electronics and Telecommunications

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2012

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Vol, 58, No. 1

63-70

EN

Given a connection graph of entities that send and receive a flow of data controlled by effort and given the parameters, the metric tensor is computed that is in the elastic relational flow to effort. The metric tensor can be represented by the Hessian of the interaction potential. Now the interaction potential or cost function can be among two entities: 3 entities or 'N' entities and can be separated into two main parts. The first part is the repulsion potential the entities move further from the others to obtain minimum cost, the second part is the attraction potential for which the entities move near to others to obtain the minimum cost. For Pauli's model [1], the attraction potential is a functional set of parameters given from the environment (all the elements that have an influence in the module can be the attraction of one entity to another). Now the cost function can be created in a space of macro-variables or macro-states that is less of all possible variables. Any macro-variable collect a set of micro-variables or microstates. Now from the hessian of the macro-variables, the Hessian is computed of the micro-variables in the singular points as stable or unstable only by matrix calculus without any analytical computation - possible when the macro-states are distant among entities. Trivially, the same method can be obtained by a general definition of the macro-variable or macro-states and micro-states or variables. As cloud computing for Sensor-Actor Networks (SANETS) is based on the bonding concept for complex interrelated systems; the bond valence or couple corresponds to the minimum of the interaction potential V and in the SANET cloud as the minimum cost.

6

A new method of aircraft evalution using matrix calculus

Błaszczyk J.

Research Bulletin / Warsaw University of Technology, Institute of Aeronautics and Applied Mechanics

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2001

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nr 11

17-20

EN

In the report an estimation method of technical object evaluation as well as comparison is presented and its utilisation for the military multirole aircraft On the base of air missions fulfilled by combats some areas of aircraft's features are determined e.g. survivability, manoeuvrability, mission flexibility as the components of combat effectiveness. These areas are described by me representative factors, measurements, parameters etc. Matrices define aircraft, where the rows represent he assessment areas while the columns contain proper factors. These matrices are normalised in some way. By the process of aggregation of fraction scores, the evaluated areas are specified consequently, to the order, specifying the group of the assessed aircraft.