Can we think of neurons as maps between matrices?
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Usually when we think about neurons, we imagine that they enact some kind of map between real numbers. For example, a neuron might take in real numbers $x_{i}$ and weight them with parameters $W_{ij}$, so that the neuron $j$ in a subsequent layer receives an input $sum_{i}W_{ij}x_{i}$ and outputs $f(sum_{i}W_{ij}x_{i}+b_{j})$, where $b_{j}$ is a bias and $f$ is the activation function. However, one can instead imagine a neuron that maps matrices to matrices by simply promoting the outputs $x_{i}$ and biases $b_{j}$ to matrices, so that they carry an additional index structure, e.g., $x_{i}rightarrow x_{i}^{alphabeta}$. In this convention, lower indices would then indicate the locations of objects in the network and upper indices would indicate matrix elements, so that $x_{i}^{alphabeta}$ denotes a matrix input from neuron $i$ with matrix elements indexed by $alpha$ and $beta$.
This kind of structure then tells us that an input to an arbitrary neuron, i.e. $sum_{i}W_{ij}x_{i}^{alphabeta}$, is just a linear combination of the matrices output by the previous layer of the network (note that the weights are still real numbers). Further, the activation function $f$ would now be a matrix function, by which I mean the output of a neuron would be defined by the Taylor series of that function evaluated on the input matrix.
Could such a scheme work, by which I mean could such a network be practically trainable? It seems to me that the answer should be yes if the activation functions in the network are well behaved. A simple example would be something like the matrix exponential function: suppose I want to compute, for a matrix $A$,
$$e^{A} = sum_{n=0}^{infty}frac{1}{n!}A^{n}$$
Further, $A$ can be written as some linear combination of other matrices $A_{i}$: $A = sum_{i}a_{i}A_{i}$, with the $a_{i}$ real numbers. If I fix $e^{A}$, shouldn't it be possible to train a network by adjusting the $a_{i}$ until a best fit is found?
This seems like a plausible idea but I'm not aware of neurons being used in this way. Has this been discussed before in the literature?
machine-learning linear-algebra
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mflynnmflynn
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Usually when we think about neurons, we imagine that they enact some kind of map between real numbers. For example, a neuron might take in real numbers $x_{i}$ and weight them with parameters $W_{ij}$, so that the neuron $j$ in a subsequent layer receives an input $sum_{i}W_{ij}x_{i}$ and outputs $f(sum_{i}W_{ij}x_{i}+b_{j})$, where $b_{j}$ is a bias and $f$ is the activation function. However, one can instead imagine a neuron that maps matrices to matrices by simply promoting the outputs $x_{i}$ and biases $b_{j}$ to matrices, so that they carry an additional index structure, e.g., $x_{i}rightarrow x_{i}^{alphabeta}$. In this convention, lower indices would then indicate the locations of objects in the network and upper indices would indicate matrix elements, so that $x_{i}^{alphabeta}$ denotes a matrix input from neuron $i$ with matrix elements indexed by $alpha$ and $beta$.
This kind of structure then tells us that an input to an arbitrary neuron, i.e. $sum_{i}W_{ij}x_{i}^{alphabeta}$, is just a linear combination of the matrices output by the previous layer of the network (note that the weights are still real numbers). Further, the activation function $f$ would now be a matrix function, by which I mean the output of a neuron would be defined by the Taylor series of that function evaluated on the input matrix.
Could such a scheme work, by which I mean could such a network be practically trainable? It seems to me that the answer should be yes if the activation functions in the network are well behaved. A simple example would be something like the matrix exponential function: suppose I want to compute, for a matrix $A$,
$$e^{A} = sum_{n=0}^{infty}frac{1}{n!}A^{n}$$
Further, $A$ can be written as some linear combination of other matrices $A_{i}$: $A = sum_{i}a_{i}A_{i}$, with the $a_{i}$ real numbers. If I fix $e^{A}$, shouldn't it be possible to train a network by adjusting the $a_{i}$ until a best fit is found?
This seems like a plausible idea but I'm not aware of neurons being used in this way. Has this been discussed before in the literature?
machine-learning linear-algebra
asked 5 mins ago
mflynnmflynn
61
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mflynn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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Usually when we think about neurons, we imagine that they enact some kind of map between real numbers. For example, a neuron might take in real numbers $x_{i}$ and weight them with parameters $W_{ij}$, so that the neuron $j$ in a subsequent layer receives an input $sum_{i}W_{ij}x_{i}$ and outputs $f(sum_{i}W_{ij}x_{i}+b_{j})$, where $b_{j}$ is a bias and $f$ is the activation function. However, one can instead imagine a neuron that maps matrices to matrices by simply promoting the outputs $x_{i}$ and biases $b_{j}$ to matrices, so that they carry an additional index structure, e.g., $x_{i}rightarrow x_{i}^{alphabeta}$. In this convention, lower indices would then indicate the locations of objects in the network and upper indices would indicate matrix elements, so that $x_{i}^{alphabeta}$ denotes a matrix input from neuron $i$ with matrix elements indexed by $alpha$ and $beta$.
This kind of structure then tells us that an input to an arbitrary neuron, i.e. $sum_{i}W_{ij}x_{i}^{alphabeta}$, is just a linear combination of the matrices output by the previous layer of the network (note that the weights are still real numbers). Further, the activation function $f$ would now be a matrix function, by which I mean the output of a neuron would be defined by the Taylor series of that function evaluated on the input matrix.
Could such a scheme work, by which I mean could such a network be practically trainable? It seems to me that the answer should be yes if the activation functions in the network are well behaved. A simple example would be something like the matrix exponential function: suppose I want to compute, for a matrix $A$,
$$e^{A} = sum_{n=0}^{infty}frac{1}{n!}A^{n}$$
Further, $A$ can be written as some linear combination of other matrices $A_{i}$: $A = sum_{i}a_{i}A_{i}$, with the $a_{i}$ real numbers. If I fix $e^{A}$, shouldn't it be possible to train a network by adjusting the $a_{i}$ until a best fit is found?
This seems like a plausible idea but I'm not aware of neurons being used in this way. Has this been discussed before in the literature?
machine-learning linear-algebra
asked 5 mins ago
mflynnmflynn
61
New contributor
mflynn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
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Usually when we think about neurons, we imagine that they enact some kind of map between real numbers. For example, a neuron might take in real numbers $x_{i}$ and weight them with parameters $W_{ij}$, so that the neuron $j$ in a subsequent layer receives an input $sum_{i}W_{ij}x_{i}$ and outputs $f(sum_{i}W_{ij}x_{i}+b_{j})$, where $b_{j}$ is a bias and $f$ is the activation function. However, one can instead imagine a neuron that maps matrices to matrices by simply promoting the outputs $x_{i}$ and biases $b_{j}$ to matrices, so that they carry an additional index structure, e.g., $x_{i}rightarrow x_{i}^{alphabeta}$. In this convention, lower indices would then indicate the locations of objects in the network and upper indices would indicate matrix elements, so that $x_{i}^{alphabeta}$ denotes a matrix input from neuron $i$ with matrix elements indexed by $alpha$ and $beta$.
This kind of structure then tells us that an input to an arbitrary neuron, i.e. $sum_{i}W_{ij}x_{i}^{alphabeta}$, is just a linear combination of the matrices output by the previous layer of the network (note that the weights are still real numbers). Further, the activation function $f$ would now be a matrix function, by which I mean the output of a neuron would be defined by the Taylor series of that function evaluated on the input matrix.
Could such a scheme work, by which I mean could such a network be practically trainable? It seems to me that the answer should be yes if the activation functions in the network are well behaved. A simple example would be something like the matrix exponential function: suppose I want to compute, for a matrix $A$,
$$e^{A} = sum_{n=0}^{infty}frac{1}{n!}A^{n}$$
Further, $A$ can be written as some linear combination of other matrices $A_{i}$: $A = sum_{i}a_{i}A_{i}$, with the $a_{i}$ real numbers. If I fix $e^{A}$, shouldn't it be possible to train a network by adjusting the $a_{i}$ until a best fit is found?
This seems like a plausible idea but I'm not aware of neurons being used in this way. Has this been discussed before in the literature?
machine-learning linear-algebra
machine-learning linear-algebra
asked 5 mins ago
mflynnmflynn
61
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mflynn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 5 mins ago
mflynnmflynn
61
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mflynn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 5 mins ago
mflynnmflynn
61
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mflynn is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
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asked 5 mins ago
mflynnmflynn
61
asked 5 mins ago
mflynnmflynn
61
61
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