Neural network as a nonlinear system?
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I defined a very simple neural network of $2$ inputs, $1$ hidden layer with $2$ nodes, and one output node.
For each input pattern $x⃗ ∈ ℝ×ℝ$
and associated output $o∈ℝ$, the resulting nonlinear equation is:
$wo_{0} σ(x_0 Wi{00} + x_1 Wi{10}) + wo{1} σ(x_0 Wi{01} + x_1 Wi{11}) = o$
where $Wi$ is the weight matrix of order $2×2$, where each element $Wi_{jk} in ℝ$, of input connections, $σ(x)=frac{1}{1+exp(−x)}$, and $vec{wo}$, with $wo_{i} in ℝ$, is the weight vector of the two output connections before the output node.
Given a dataset of $n$ (pattern, output) examples, there will be $n$ nonlinear equations.
I'm asking how to find the solutions of those nonlinear systems, as an alternative method to solve the learning problem, without backpropagation. I've implemented an optimizer for the stated probem. If someone is interested I can provide the relative C sources (email: fportera2@gmail.com).
nonlinear-system
asked 14 hours ago
Filippo Portera
112
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Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
add a comment |
up vote
1
down vote
favorite
I defined a very simple neural network of $2$ inputs, $1$ hidden layer with $2$ nodes, and one output node.
For each input pattern $x⃗ ∈ ℝ×ℝ$
and associated output $o∈ℝ$, the resulting nonlinear equation is:
$wo_{0} σ(x_0 Wi{00} + x_1 Wi{10}) + wo{1} σ(x_0 Wi{01} + x_1 Wi{11}) = o$
where $Wi$ is the weight matrix of order $2×2$, where each element $Wi_{jk} in ℝ$, of input connections, $σ(x)=frac{1}{1+exp(−x)}$, and $vec{wo}$, with $wo_{i} in ℝ$, is the weight vector of the two output connections before the output node.
Given a dataset of $n$ (pattern, output) examples, there will be $n$ nonlinear equations.
I'm asking how to find the solutions of those nonlinear systems, as an alternative method to solve the learning problem, without backpropagation. I've implemented an optimizer for the stated probem. If someone is interested I can provide the relative C sources (email: fportera2@gmail.com).
nonlinear-system
asked 14 hours ago
Filippo Portera
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Genetic algorithms?
– N74
14 hours ago
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
I defined a very simple neural network of $2$ inputs, $1$ hidden layer with $2$ nodes, and one output node.
For each input pattern $x⃗ ∈ ℝ×ℝ$
and associated output $o∈ℝ$, the resulting nonlinear equation is:
$wo_{0} σ(x_0 Wi{00} + x_1 Wi{10}) + wo{1} σ(x_0 Wi{01} + x_1 Wi{11}) = o$
where $Wi$ is the weight matrix of order $2×2$, where each element $Wi_{jk} in ℝ$, of input connections, $σ(x)=frac{1}{1+exp(−x)}$, and $vec{wo}$, with $wo_{i} in ℝ$, is the weight vector of the two output connections before the output node.
Given a dataset of $n$ (pattern, output) examples, there will be $n$ nonlinear equations.
I'm asking how to find the solutions of those nonlinear systems, as an alternative method to solve the learning problem, without backpropagation. I've implemented an optimizer for the stated probem. If someone is interested I can provide the relative C sources (email: fportera2@gmail.com).
nonlinear-system
asked 14 hours ago
Filippo Portera
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
I defined a very simple neural network of $2$ inputs, $1$ hidden layer with $2$ nodes, and one output node.
For each input pattern $x⃗ ∈ ℝ×ℝ$
and associated output $o∈ℝ$, the resulting nonlinear equation is:
$wo_{0} σ(x_0 Wi{00} + x_1 Wi{10}) + wo{1} σ(x_0 Wi{01} + x_1 Wi{11}) = o$
where $Wi$ is the weight matrix of order $2×2$, where each element $Wi_{jk} in ℝ$, of input connections, $σ(x)=frac{1}{1+exp(−x)}$, and $vec{wo}$, with $wo_{i} in ℝ$, is the weight vector of the two output connections before the output node.
Given a dataset of $n$ (pattern, output) examples, there will be $n$ nonlinear equations.
I'm asking how to find the solutions of those nonlinear systems, as an alternative method to solve the learning problem, without backpropagation. I've implemented an optimizer for the stated probem. If someone is interested I can provide the relative C sources (email: fportera2@gmail.com).
nonlinear-system
nonlinear-system
asked 14 hours ago
Filippo Portera
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
Filippo Portera
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
edited 4 hours ago
asked 14 hours ago
Filippo Portera
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
asked 14 hours ago
Filippo Portera
112
asked 14 hours ago
Filippo Portera
112
112
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
New contributor
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Filippo Portera is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.
Genetic algorithms?
– N74
14 hours ago
add a comment |
Genetic algorithms?
– N74
14 hours ago
Genetic algorithms?
– N74
14 hours ago
Genetic algorithms?
– N74
14 hours ago
add a comment |
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Filippo Portera is a new contributor. Be nice, and check out our Code of Conduct.
Filippo Portera is a new contributor. Be nice, and check out our Code of Conduct.
Filippo Portera is a new contributor. Be nice, and check out our Code of Conduct.
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Genetic algorithms?
– N74
14 hours ago