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The goal is to learn a function $f$ of the form:

$f:\mathbb{R} \rightarrow \mathbb{R}^n$, $n \ge 1$

Are there machine learning techniques that can do it?

Thank you.

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closed as too broad by Juho Kokkala, COOLSerdash, Nick Cox, gung, Sean Easter Dec 5 '15 at 16:15

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ Can you please tell us what you have tried? For example a ANN is generality regarded as a universal function approximator. Is this something like you are looking for? $\endgroup$ – usεr11852 Dec 5 '15 at 7:17
  • $\begingroup$ I have done some research and I think that the problem is of computing functions whose range is in a Hilbert space (paper). I have also found that multi-label classification might solve the problem. I am not sure which one is the best. $\endgroup$ – rmas Dec 5 '15 at 7:48
  • $\begingroup$ What is $D$? What kind of data have you observed? $\endgroup$ – Juho Kokkala Dec 5 '15 at 8:27
  • $\begingroup$ Let's say the observed data are of the form $(x, y_1,..., y_n)_i$, $i \in [1,m]$ $\endgroup$ – rmas Dec 5 '15 at 8:39
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Yes.

One simple, though not necessarily the most effective, way is to view it as $n$ regular univariate regression problems.

In general, this is sometimes called multitask regression. There are many ways to share knowledge among the tasks. For example, Solnon, Arlot, and Bach, JMLR 2012 (which I've only skimmed) use kernel ridge regression with an estimated covariance structure on the dimensions of the output task.

If you have sufficient data, it's particularly easy to phrase these problems as a neural network; just have multiple outputs at the end of your architecture, and use e.g. $L_2$ loss.

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