0
$\begingroup$

My understanding of the Lagrange interpolating polynomial is that given $\space n \space$ points, we can fit a polynomial of degree $\space n+1 \space$ as a means of approximating the values between the points.

This polynomial is defined as:

$$L_n(x) = {(x-x_1)(x-x_2)...(x-x_n) \over (x_0-x_1)(x_0-x_2)...(x_0-x_n)}y_0 +...+ {(x-x_1)(x-x_2)...(x-x_{n-1}) \over (x_n-x_1)(x_n-x_2)...(x_n-x_{n-1})}y_n$$

Could this be more reliable in a machine learning algorithm that deals with one numerical input and one numerical output, where the training data is a set of points $\space (x,y) \space$, with $\space x=input \space$ and $\space y=output \space$ . Rather than using neural networks?

$\endgroup$
  • $\begingroup$ First, the Lagrange form is really bad to actually evaluate for $n$ larger than about $5$. There are better representations of the same polynomial, such as the Newton form, out there. Second, the big problem with exact interpolating polynomials of high degree (rather than, for example, inexact interpolating polynomials of low degree, or spline polynomials with low degree pieces) is large oscillations in between nodes. As a result you tend to get very inaccurate results for predictions of outputs from inputs in between the nodes. $\endgroup$ – Ian Aug 25 '18 at 14:20
  • $\begingroup$ @Ian - Ahh I figured your second point would be an issue but wasn't too sure. Thank you for the clear and concise response :) $\endgroup$ – UndercoverCoder Aug 25 '18 at 14:27
0
$\begingroup$

The problem with interpolation polynomials is that "they are trying too hard to fit the data points" and end up missing the structure of the dataset. This problem is called overfitting and you should be aware of it. It happens when you use a larger than necerssary degree polynomial for regression.

$\endgroup$
  • $\begingroup$ Thank you so much for responding :). Been studying Neural Networks and having a look at Interpolating polynomials wondering why we don't just default to them. Just looked at overfitting, and the issue is clear. Thank you again. $\endgroup$ – UndercoverCoder Aug 25 '18 at 14:36
  • $\begingroup$ You are very welcome! $\endgroup$ – Μάρκος Καραμέρης Aug 25 '18 at 14:38

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.