Tootfinder

Dimensionality reduction and width of deep neural networks based on topological degree theory
Xiao-Song Yang
https://arxiv.org/abs/2511.06821 https://arxiv.org/pdf/2511.06821 https://arxiv.org/html/2511.06821
arXiv:2511.06821v1 Announce Type: new
Abstract: In this paper we present a mathematical framework on linking of embeddings of compact topological spaces into Euclidean spaces and separability of linked embeddings under a specific class of dimension reduction maps. As applications of the established theory, we provide some fascinating insights into classification and approximation problems in deep learning theory in the setting of deep neural networks.
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Riccati-ZORO: An efficient algorithm for heuristic online optimization of internal feedback laws in robust and stochastic model predictive control
Florian Messerer, Yunfan Gao, Jonathan Frey, Moritz Diehl
https://arxiv.org/abs/2511.10473 https://arxiv.org/pdf/2511.10473 https://arxiv.org/html/2511.10473
arXiv:2511.10473v1 Announce Type: new
Abstract: We present Riccati-ZORO, an algorithm for tube-based optimal control problems (OCP). Tube OCPs predict a tube of trajectories in order to capture predictive uncertainty. The tube induces a constraint tightening via additional backoff terms. This backoff can significantly affect the performance, and thus implicitly defines a cost of uncertainty. Optimizing the feedback law used to predict the tube can significantly reduce the backoffs, but its online computation is challenging.
Riccati-ZORO jointly optimizes the nominal trajectory and uncertainty tube based on a heuristic uncertainty cost design. The algorithm alternates between two subproblems: (i) a nominal OCP with fixed backoffs, (ii) an unconstrained tube OCP, which optimizes the feedback gains for a fixed nominal trajectory. For the tube optimization, we propose a cost function informed by the proximity of the nominal trajectory to constraints, prioritizing reduction of the corresponding backoffs. These ideas are developed in detail for ellipsoidal tubes under linear state feedback. In this case, the decomposition into the two subproblems yields a substantial reduction of the computational complexity with respect to the state dimension from $\mathcal{O}(n_x^6)$ to $\mathcal{O}(n_x^3)$, i.e., the complexity of a nominal OCP.
We investigate the algorithm in numerical experiments, and provide two open-source implementations: a prototyping version in CasADi and a high-performance implementation integrated into the acados OCP solver.
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