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.

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.
