Measuring Similarity between 2 Angular Distributions

Question

For a project I'm working on, I have two sets of samples, where each set has N x length 7 vectors. For context, each vector represents the joint parameter setting for a robot (the angle each joint of the robot is set to).

I'm interested in comparing the similarity of the distributions from which the vectors were drawn from. I understand typically you can use KL-Divergence/MMD/etc., but these methods aren't appropriate (I don't think at least) because of the circular topology of the data - I.e. the value 0 and 2pi are the same, but would be considered far apart by a standard similarity measure for probability distributions.

How can I calculate a numeric similarity between the two distributions from which the samples were drawn from given this circular topology?

Thanks!

brief extra: An idea I had was to convert each angle into a (x, y) pair, $\theta \rightarrow (cos(\theta), sin(\theta))$. Then the problem becomes comparing the distributions of vectors of (x,y) pairs, not sure that thats a great path to head down though

Answer 1

Your data lie on the 7-torus $T^7.$ It has many geometries, but two natural ones in this application would be based on the function

$$\delta(a) = \left(|a| + \pi \mod 2\pi\right) - \pi$$

with values in the interval $[-\pi, \pi).$ This is the unsigned value of the oriented angle $a.$

Where all variables are considered approximately of equal weight in the analysis, for the distance between two vectors $\mathrm{x}=(x_1, \ldots, x_7)$ and $\mathrm{y}=(y_1,\ldots, y_7)$ use a function of the vector

$$\delta(\mathrm{x}, \mathrm{y}) = (\delta(x_1-y_1), \ldots, \delta(x_7-y_7)),$$

such as its $L_p$ norm for $p \ge 1.$ You can weight this norm by multiplying the components by positive weights $\omega_1, \ldots, \omega_7$ before computing the norm.

---

These are all valid metrics: they are symmetric and satisfy the triangle inequality. One way to prove this follows your suggestion: these metrics can be expressed in terms of metrics on $\mathbb{R}^{14}$ induced by the embedding $\phi:T^7\to \mathbb{R}^{14}$ given by

$$\phi(\mathrm{x}) = (\cos(x_1), \sin(x_1), \cos(x_2), \ldots, \sin(x_7)).$$

However, you don't need actually to compute this embedding in order to compute $\delta,$ as you can see from its initial formula.