# Non-linear dimensionality reduction for detecting coordinate systems [closed]

I am trying to find a way to automatically find the appropriate coordinate system for a physical problem.

For example, in the case of a simple pendulum, polar coordinates are the most appropriate ones. I have the data for the x,y cartesian coordinates of the pendulum at various times. I would like to be able to jump from this to the angular data at a given time.

I feel that this is a form of dimensionality reduction. Could we use auto-encoders for this or are methods such as ISOMAP or CDA more appropriate?

I feel that PCA would not work well for this problem as I need to perform a non-linear dimensionality reduction.

• Is this task a learning exercise or are you really pondering about using autoencoders to obtain phase information from pendulum simulations? – Firebug Jul 18 '19 at 0:19
• Because there's an analytical solution to that, no need for fancy machine learning. – Firebug Jul 18 '19 at 0:19
• How is this dimensionality reduction? You start with two parameters (angle and length) and you end up with two (x and y)> What is it that you want to automate? – Peter Flom Jul 18 '19 at 11:34
• Yes but the problem has one degree of freedom. Ok dimensionality reduction may not be the correct word. But I feel that autoencoders could work for this. – TriposG Jul 18 '19 at 13:32

## 1 Answer

For example, in the case of a simple pendulum, polar coordinates are the most appropriate ones. I have the data for the x,y cartesian coordinates of the pendulum at various times. I would like to be able to jump from this to the angular data at a given time. I feel that this is a form of dimensionality reduction. Could we use auto-encoders for this or are methods such as ISOMAP or CDA more appropriate?

And you're correct. An autoencoder could work here, since it's a nonlinear problem, but would not simply tell you that the latent dimensionality of the two-dimensional position distribution is $$\theta=\operatorname{tg}((x-\bar x)/(y-\bar y))$$. It would only allow you to estimate the intrinsic dimensionality of your problem, one, since the radius is invariant.

I feel that PCA would not work well for this problem as I need to perform a non-linear dimensionality reduction.

Your intuition is right, since PCA is a linear dimensionality reduction method.

• Ok, I understand that. So it isn't possible to get the value of the angular positions just from the x,y cartesian coordinates? Perhaps by varying the architecture of the autoencoder neural network. – TriposG Jul 17 '19 at 14:09
• @TriposG it's not uniquely defined. A 1-D space where $t=\theta$ is just as good at explaining your data as a 1-D space where $t = 1000 \theta$ – Firebug Jul 17 '19 at 23:02
• right but atleast the ratios would be the same? – TriposG Jul 17 '19 at 23:26
• @TriposG no, the autoencoder could learn a nonlinear transformation of the phase as well, such as $t=\theta^3$, which has equal explanatory power to all the other approximations – Firebug Jul 17 '19 at 23:28
• ok. Do you have any suggestion on how to go from x,y coordinates to theta using autoencoders? Or is this a very difficult task? – TriposG Jul 17 '19 at 23:46