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So, I was wondering about the following question: Is there any neural network architecture that it is good for predicting a dense vector of values between 0 and 1 (multiple linear regression)? And if so, is there any tutorial or paper that describes it?

I've seen neural networks been used in classification problems where the output of the network is usually 1 for one neuron and 0 for the others. I have also seen some examples where the network tries to predict (regression) one output value (e.g given features about a house, estimate how mush it is worth it). However, if I want to, given an input, predict a vector with values between 0 and 1...is that possible? Or networks are just bad at it?

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Sure, people use neural networks for regression all the time. A recent Google search for "multivariate neural network regression" returns more than $4,760,000$ results. Predicting more than 1 output is simply a matter of having more than 1 output neuron, as in What's the point in neural networks for multivariate regression?

Since you want to bound your output values in [0,1], a nice trick is using a sigmoidal output layer. This does all the normalization for you. Typically people use squared error as a regression loss, but there are many other options.

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