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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.06768 |
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| _version_ | 1866909606552272896 |
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| author | Chiantini, Luca D'Inverno, Giuseppe Alessio Marziali, Sara |
| author_facet | Chiantini, Luca D'Inverno, Giuseppe Alessio Marziali, Sara |
| contents | Any kind of network can be naturally represented by a Directed Acyclic Graph (DAG); additionally, activation functions can determine the reaction of each node of the network with respect to the signal(s) incoming. We study the characterization of the signal distribution in a network under the lens of tensor algebra. More specifically, we describe every activation function as tensor distributions with respect to the nodes, called \textit{activation tensors}. The distribution of the signal is encoded in the \textit{total tensor} of the network. We formally prove that the total tensor can be obtained by computing the \textit{Batthacharya-Mesner Product} (BMP), an $n$-ary operation for tensors of order $n$, on the set of the activation tensors properly ordered and processed via two basic operations, that we call \textit{blow} and \textit{forget}. Our theoretical framework can be validated through the related code developed in Python. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_06768 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Product of Tensors and Description of Networks Chiantini, Luca D'Inverno, Giuseppe Alessio Marziali, Sara Algebraic Geometry Any kind of network can be naturally represented by a Directed Acyclic Graph (DAG); additionally, activation functions can determine the reaction of each node of the network with respect to the signal(s) incoming. We study the characterization of the signal distribution in a network under the lens of tensor algebra. More specifically, we describe every activation function as tensor distributions with respect to the nodes, called \textit{activation tensors}. The distribution of the signal is encoded in the \textit{total tensor} of the network. We formally prove that the total tensor can be obtained by computing the \textit{Batthacharya-Mesner Product} (BMP), an $n$-ary operation for tensors of order $n$, on the set of the activation tensors properly ordered and processed via two basic operations, that we call \textit{blow} and \textit{forget}. Our theoretical framework can be validated through the related code developed in Python. |
| title | Product of Tensors and Description of Networks |
| topic | Algebraic Geometry |
| url | https://arxiv.org/abs/2402.06768 |