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I'm trying to btter understand the formalism under the following compact formulation of a single-layer perceptron. If we consider $V=\mathbb{R}^d$, then

$$\hat{f}(x_1, \dots, x_d) = \sum_{i=1}^Nc_i\sigma\left(\sum_{j=1}^dw_{ij}x_j + h_i\right)$$

So the initial $d$ is the dimension of the input vector, let's say $d=3$ and we have $N=4$ hidden neurons. This means that each single neuron $i$ in the hidden layer can be written as

$$\sum_{j=1}^3w_{ij}x_j = \sigma(w_{i1}x_1+w_{i2}x_2+w_{i3}x_3 + h_i) \, \, \, \, , \, \, i = \{1,2,3,4\}.$$

But then what do coefficients $c_i$ actually represent here? Maybe they are the weights connecting the $i$-th neuron of the hidden layer to a single output? So maybe this formalism is describing a single-layer perceptron with a single final output?

And if yes how would I modify the above formulation to describe a multi-output perceptron? Maybe I could intend it coordinate-wise.

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    $\begingroup$ Draw out a small network with one predictor, two hidden neurons, and one output neuron, and see if you can figure it out there. // I believe a parameter is missing from your first equation. $\endgroup$
    – Dave
    Nov 22 '21 at 16:01
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This is not single layer perceptron, which doesn't have a hidden layer. That said, $c_i$ represents the weight of the $i$-th hidden neuron's output. You could have a multi-output perceptron if you remove the last accumulation, but it should be something your problem requires.

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  • $\begingroup$ Apologize I meant single-hidden-layer $\endgroup$ Nov 22 '21 at 20:57
  • $\begingroup$ Ok, to have multiple outputs, you need to have multiple output neurons. Currently, you only have one. And, that should be something your problem can accommodate, i.e. what does the output of your model signify? $\endgroup$
    – gunes
    Nov 22 '21 at 21:03
  • $\begingroup$ How should I mathematically modify the formulation that I have provided in order to accomodate multiple neurons? Let's say that maybe the output of my model signifies a multi-target regression, or a multi-label classification. $\endgroup$ Nov 24 '21 at 10:27
  • $\begingroup$ You need multiple output neurons and it would be easier with matrix notation, e.g. $y=W^Tx$, where $y$ would represent a vector of outputs. $\endgroup$
    – gunes
    Nov 24 '21 at 10:59

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