Nonlinear RKHS Regression as a Dissipative Flow

A sketch of a function-space dynamical systems viewpoint on nonlinear regression.

Setup

Let \(u\) be an unknown function in a Hilbert space \(H\). A nonlinear regression problem can be written abstractly as

\[ \min_{u\in H} \frac12\|G(u)-y\|^2 + \frac{\alpha}{2}\|u\|_H^2. \]

The associated gradient flow is

\[ \dot{u} = - DG(u)^\ast(G(u)-y) - \alpha u. \]

This turns nonlinear regression into an infinite-dimensional dissipative dynamical system.

Analysis program

A natural paper structure is:

1. well-posedness of the flow;
2. existence of absorbing sets;
3. existence and regularity of a global attractor;
4. convergence to equilibria under a Łojasiewicz–Simon inequality;
5. exponential-in-time discretization;
6. spatial discretization by FEM or domain decomposition;
7. numerical experiments.