# Which one is better: multivariate regression with basis expansion or neural networks?

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.

• 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. – Oliver Mason Mar 24 '20 at 11:10
• @OliverMason You may explain in which case which is better. – RewCie Mar 24 '20 at 11:41