Incremental stability in $p=1$ and $p=\infty$: classification and synthesis
This work addresses the challenge of efficiently synthesizing and classifying contracting dynamical systems for applications in control and machine learning, though it is incremental as it builds on existing contraction theory with a focus on specific norms.
The paper tackled the problem of certifying Lipschitz dynamics with weak infinitesimal contraction (WIC) in non-Euclidean norms by proving a structure theorem that enables efficient parameterization using neural networks, resulting in a method that reduces certification costs to O(d²) operations, as demonstrated in numerical experiments on flow-fitting and opinion network tasks.
All Lipschitz dynamics with the weak infinitesimal contraction (WIC) property can be expressed as a Lipschitz nonlinear system in proportional negative feedback -- this statement, a ``structure theorem,'' is true in the $p=1$ and $p=\infty$ norms. Equivalently, a Lipschitz vector field is WIC if and only if it can be written as a scalar decay plus a Lipschitz-bounded residual. We put this theorem to use using neural networks to approximate Lipschitz functions. This results in a map from unconstrained parameters to the set of WIC vector fields, enabling standard gradient-based training with no projections or penalty terms. Because the induced $1$- and $\infty$-norms of a matrix reduce to row or column sums, Lipschitz certification costs only $O(d^2)$ operations -- the same order as a forward pass and appreciably cheaper than eigenvalue or semidefinite methods for the $2$-norm. Numerical experiments on a planar flow-fitting task and a four-node opinion network demonstrate that the parameterization (re-)constructs contracting dynamics from trajectory data. In a discussion of the expressiveness of non-Euclidean contraction, we prove that the set of $2\times 2$ systems that contract in a weighted $1$- or $\infty$-norm is characterized by an eigenvalue cone, a strict subset of the Hurwitz region that quantifies the cost of moving away from the Euclidean norm.