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Assume we are given a training dataset $D = \{ (x_i, y_i)\}_{i=1}^{N}$.

My question is: which is better?

  1. A multivariate regression with basis expansion with independent matrix $X$ and dependent matrix $Y$, such that $X \in K; K \subset \mathbb R^n$ and $Y \in \mathbb R^m$ with training data $D$.

Or

  1. A neural network which takes $n$ input variables and returns $m$ output with training data $D$

Without a doubt, the multivariate regression option is better with its basis polynomials because it can adapt any curve required in any dimension and doesn't need a large number of datasets than neural networks. Then, why neural networks are used more than multivariate regression?

Note: Prefer explaining the mechanism of neural network used as regression in your answers. To help us know the degree of flexibility of both.

Edit: You may prefer choosing your own loss function in case you need.

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  • $\begingroup$ This question is impossible to answer. There is no objective 'better' solution unless you know more about the shape of the data. It's like asking "what's better, a fork or a spoon?" -- depends on whether you want to eat soup or steak. $\endgroup$ – Oliver Mason Mar 24 '20 at 11:10
  • $\begingroup$ @OliverMason You may explain in which case which is better. $\endgroup$ – RewCie Mar 24 '20 at 11:41

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