April 21st – Marek Kaluba

Łukasz Garncarek

Non-injectivity via persistence topology

Consider cloud of points in \mathbb{R}^n. We would like to know if the points has been sampled from a (local) patch of a manifold. Such sampling usually occurs with some addition of Noise, so we may think of points as values of a (stochastic) “function”: Y = F(X,N), where N = \text{Noise} is assumed to be

  1. uni-modal, and
  2. (strongly) independent of X.

The basic problem approached can be summarized as follows. Given a set of points \{(X_i,Y_i)\}_i \subset \mathbb{R}^n = \mathbb{R}^k \times \mathbb{R}^{n-k}, determine if there is a function f\colon \mathbb{R}^k \to \mathbb{R}^{n-k} or g\colon \mathbb{R}^{n-k} \to \mathbb{R}^k of which a “noised sample” F(X,N) represents the data.

I will present easy 2-dimensional case where the persistence homology of appropriately thresholded Delaunay complex can supply computationally cheap hints to the true answer. Higher dimensional cases are also tractable with Rips complex. These considerations have some applications to the causality problem, where the task to infer the underlying casual structure of two random variables X, Y when only the set of points is given. Surprisingly though the presented approach works for non-functional relations (e.g. in the presence of con-founder).

We meet at 10.15 in room 403.