Single-layer perceptron mathematical formulation

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.

• 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.
– Dave
Nov 22 '21 at 16:01

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.
• 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. Nov 24 '21 at 10:59