Vapnik–Chervonenkis Dimension

Problem Statement

We define a parametric space of stochastic processes (analogous to a feature‑encoding space in supervised learning) so that a stochastic process realization (sample path) is produced by the composition of three stages:

• Time vector map that converts raw inputs (e.g., a sequence of heatmap matrices) into a per‑time vector representation.
  
• 1‑D temporal wavelet transform applied to each channel of the time vector, producing multiscale, time–frequency localized coefficients.
  
• Sequencewise dimensionality reduction (possibly nonlinear, e.g., nonlinear ICA / CEBRA) that maps the multiscale coefficients to a lower‑dimensional sequence representation.

For more formal exposition refer to the Compute heatmap modelling.

In our project, the parametric space of stochastic processes plays the role of a statistical classification model. VC dimension of the model is important as it measures its capacity, which helps to estimate the model’s generalization error. The goal is to find a balance where the model is complex enough to capture the underlying patterns but not so complex that it overfits the data.

Suggestions

This task is challenging due to the complexity of the model. It requires a rigorous mathematical analysis, which involves proving whether a set of points can be shattered by the hypothesis space. This is non-trivial, and moreover, can be computationally very intensive!