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Abstract
In this work, we show that neural networks can be represented via the mathematical theory of quiver representations. More specifically, we prove that a neural network is a quiver representation with activation functions, a mathematical object that we represent using a network quiver. Furthermore, we show that network quivers gently adapt to common neural network concepts such as fully connected layers, convolution operations, residual connections, batch normalization, pooling operations and even randomly wired neural networks. We show that this mathematical representation is by no means an approximation of what neural networks are as it exactly matches reality. This interpretation is algebraic and can be studied with algebraic methods. We also provide a quiver representation model to understand how a neural network creates representations from the data. We show that a neural network saves the data as quiver representations, and maps it to a geometrical space called the moduli space, which is given in terms of the underlying oriented graph of the network, i.e., its quiver. This results as a consequence of our defined objects and of understanding how the neural network computes a prediction in a combinatorial and algebraic way. Overall, representing neural networks through the quiver representation theory leads to 9 consequences and 4 inquiries for future research that we believe are of great interest to better understand what neural networks are and how they work.
Highlights
TeodoroNeural networks have achieved unprecedented performances in almost every area where machine learning is applicable [1,2,3]
We presented the theoretical foundations for a different understanding of neural networks using their combinatorial and algebraic nature, while explaining current intuitions in deep learning by relying only on the mathematical consequences of the computations of the network during inference
We use quiver representation theory to represent neural networks and their data processing; This representation of neural networks scales to modern deep architectures like conv layers, pooling layers, residual layers, batch normalization and even randomly wired neural networks [32]; Theorem 1 shows that neural networks are algebraic objects, in the sense that the maps preserving the algebraic structure preserve the computations of the network
Summary
Neural networks have achieved unprecedented performances in almost every area where machine learning is applicable [1,2,3]. Based on quiver representation theory, we provide a new mathematical footing to represent neural networks as well as the data they process. We do not focus on how neural networks learn, but rather on the intrinsic properties of their architectures and their forward pass of data. We study the combinatorial and algebraic nature of neural networks by using ideas coming from the mathematical theory of quiver representations [9,10]. This paper is based on two observations that expose the algebraic nature of neural networks and how it is related to quiver representations: 2. We present the theoretical interpretation of data in terms of the architecture of the neural network and of quiver representations. We provide constructions and results supporting existing intuitions in deep learning while discarding others, and bring new concepts to the table
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