Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty.

Deep neural networks (DNNs) achieve remarkable performance on a wide range of tasks, yet their mathematical analysis remains fragmented: stability and generalization are typically studied in disparate frameworks and on a case-by-case basis. Architecturally, DNNs rely on the recursive application of parametrized functions, a mechanism that can be unstable and difficult to train, making stability a primary concern. Even when training succeeds, there are few rigorous results on how well such models generalize beyond the observed data, especially in the generative setting. In this work, we leverage the theory of stochastic Iterated Function Systems (IFS) and show that two important deep architectures can be viewed as, or canonically associated with, place-dependent IFS. This connection allows us to import results from random dynamical systems to (i) establish the existence and uniqueness of invariant measures under suitable contractivity assumptions, and (ii) derive a Wasserstein generalization bound for generative modeling. The bound naturally leads to a new training objective that directly controls the collage-type approximation error between the data distribution and its image under the learned transfer operator. We illustrate the theory on a controlled 2D example and empirically evaluate the proposed objective on standard image datasets (MNIST, CelebA, CIFAR-10).

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty.

Belief systems are rarely globally consistent, yet effective reasoning often persists locally. We propose a novel graph-theoretic framework that cleanly separates credibility--external, a priori trust in sources--from confidence--an internal, emergent valuation induced by network structure. Beliefs are nodes in a directed, signed, weighted graph whose edges encode support and contradiction. Confidence is obtained by a contractive propagation process that mixes a stated prior with structure-aware influence and guarantees a unique, stable solution. Within this dynamics, we define reasoning zones: high-confidence, structurally balanced subgraphs on which classical inference is safe despite global contradictions. We provide a near-linear procedure that seeds zones by confidence, tests balance using a parity-based coloring, and applies a greedy, locality-preserving repair with Jaccard de-duplication to build a compact atlas. To model belief change, we introduce shock updates that locally downscale support and elevate targeted contradictions while preserving contractivity via a simple backtracking rule. Re-propagation yields localized reconfiguration-zones may shrink, split, or collapse--without destabilizing the entire graph. We outline an empirical protocol on synthetic signed graphs with planted zones, reporting zone recovery, stability under shocks, and runtime. The result is a principled foundation for contradiction-tolerant reasoning that activates classical logic precisely where structure supports it.

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty.

-- Neural networks can be fragile to input noise and adversarial attacks. In this work, we consider Convolutional Neural Ordinary Differential Equations (NODEs) - a family of continuous-depth neural networks represented by dynamical systems - and propose to use contraction theory to improve their robustness. Contractive Convolutional NODEs can enjoy increased robustness as slight perturbations of the features do not cause a significant change in the output. Contractivity can be induced during training by using a regularization term involving the Jacobian of the system dynamics. T o reduce the computational burden, we show that it can also be promoted using carefully selected weight regularization terms for a class of NODEs with slope-restricted activation functions. The performance of the proposed regularizers is illustrated through benchmark image classification tasks on MNIST and Fashion-MNIST datasets, where images are corrupted by different kinds of noise and attacks.

Global stability and robustness guarantees in learned dynamical systems are essential to ensure well-behavedness of the systems in the face of uncertainty. The key feature of ELCD is a parametrization of the extended linearization of the nonlinear vector field. In its most basic form, ELCD is guaranteed to be (i) globally exponentially stable, (ii) equilibrium contracting, and (iii) globally contracting with respect to some metric. To allow for contraction with respect to more general metrics in the data space, we train diffeomorphisms between the data space and a latent space and enforce contractivity in the latent space, which ensures global contractivity in the data space. We demonstrate the performance of ELCD on the high dimensional LASA, multi-link pendulum, and Rosenbrock datasets.

Graph Neural Networks (GNNs) have gained significant popularity for learning representations of graph-structured data due to their expressive power and scalability. However, despite their success in domains such as social network analysis, recommendation systems, and bioinformatics, GNNs often face challenges related to stability, generalization, and robustness to noise and adversarial attacks. Regularization techniques have shown promise in addressing these challenges by controlling model complexity and improving robustness. Building on recent advancements in contractive GNN architectures, this paper presents a novel method for inducing contractive behavior in any GNN through SVD regularization. By deriving a sufficient condition for contractiveness in the update step and applying constraints on network parameters, we demonstrate the impact of SVD regularization on the Lipschitz constant of GNNs. Our findings highlight the role of SVD regularization in enhancing the stability and generalization of GNNs, contributing to the development of more robust graph-based learning algorithms dynamics.

We study the convex-concave bilinear saddle-point problem $\min_x \max_y f(x) + y^\top Ax - g(y)$, where both, only one, or none of the functions $f$ and $g$ are strongly convex, and suitable rank conditions on the matrix $A$ hold. The solution of this problem is at the core of many machine learning tasks. By employing tools from operator theory, we systematically prove the contractivity (in turn, the linear convergence) of several first-order primal-dual algorithms, including the Chambolle-Pock method. Our approach results in concise and elegant proofs, and it yields new convergence guarantees and tighter bounds compared to known results.

Explicit, momentum-based dynamics for optimizing functions defined on Lie groups was recently constructed, based on techniques such as variational optimization and left trivialization. We appropriately add tractable noise to the optimization dynamics to turn it into a sampling dynamics, leveraging the advantageous feature that the trivialized momentum variable is Euclidean despite that the potential function lives on a manifold. We then propose a Lie-group MCMC sampler, by delicately discretizing the resulting kinetic-Langevin-type sampling dynamics. The Lie group structure is exactly preserved by this discretization. Exponential convergence with explicit convergence rate for both the continuous dynamics and the discrete sampler are then proved under $W_2$ distance. Only compactness of the Lie group and geodesically $L$-smoothness of the potential function are needed. To the best of our knowledge, this is the first convergence result for kinetic Langevin on curved spaces, and also the first quantitative result that requires no convexity or, at least not explicitly, any common relaxation such as isoperimetry.