Related papers: Kleinberg navigation on anisotropic lattices
We initiate the study of the inherent tradeoffs between the size of a neural network and its robustness, as measured by its Lipschitz constant. We make a precise conjecture that, for any Lipschitz activation function and for most datasets,…
Small-world networks are the focus of recent interest because they appear to circumvent many of the limitations of either random networks or regular lattices as frameworks for the study of interaction networks of complex systems. Here, we…
We present theoretical and numerical results concerning the problem to find the path that minimizes the time to navigate between two given points in a complex fluid under realistic navigation constraints. We contrast deterministic Optimal…
Bottleneck Steiner networks model energy consumption in wireless ad-hoc networks. The task is to design a network spanning a given set of terminals and at most $k$ Steiner points such that the length of the longest edge is minimised. The…
Due to the computational complexity of finding almost shortest simple paths, we propose that identifying a larger collection of (nonbacktracking) paths is more efficient than finding almost shortest simple paths on positively weighted…
We consider random lattice triangulations of $n\times k$ rectangular regions with weight $\lambda^{|\sigma|}$ where $\lambda>0$ is a parameter and $|\sigma|$ denotes the total edge length of the triangulation. When $\lambda\in(0,1)$ and $k$…
Motile microorganisms develop effective swimming gaits to adapt to complex biological environments. Translating this adaptability to smart microrobots presents significant challenges in motion planning and stroke design. In this work, we…
We show continuity of the martingale optimal transport optimisation problem as a functional of its marginals. This is achieved via an estimate on the projection in the nested/causal Wasserstein distance of an arbitrary coupling on to the…
We apply statistical physics to study the task of resource allocation in random sparse networks with limited bandwidths for the transportation of resources along the links. Useful algorithms are obtained from recursive relations.…
Finding optimal trajectories for multiple traffic demands in a congested network is a challenging task. Optimal transport theory is a principled approach that has been used successfully to study various transportation problems. Its usage is…
We study the small-regularisation limit of the entropic optimal transport problem on the line with distance cost. While convergence of entropic minimizers is well understood in the discrete setting and in the case where the cost is…
Lattice reduction is a combinatorial optimization problem aimed at finding the most orthogonal basis in a given lattice. The Lenstra-Lenstra-Lov\'asz (LLL) algorithm is the best algorithm in the literature for solving this problem. In light…
The second smallest eigenvalue of the Laplacian matrix is determinative in characterizing many network properties and is known as algebraic connectivity. In this paper, we investigate the problem of maximizing algebraic connectivity in…
Entropic optimal transport -- the optimal transport problem regularized by KL diver\-gence -- is highly successful in statistical applications. Thanks to the smoothness of the entropic coupling, its sample complexity avoids the curse of…
In this work we determine a process-level Large Deviation Principle (LDP) for a model of interacting particles indexed by a lattice $\mathbb{Z}^d$. The connections are random, sparse and unscaled, so that the system converges in the large…
Entropic regularization provides a generalization of the original optimal transport problem. It introduces a penalty term defined by the Kullback-Leibler divergence, making the problem more tractable via the celebrated Sinkhorn algorithm.…
We consider the optimal paths in a $d$-dimensional lattice, where the bonds have isotropically correlated random weights. These paths can be interpreted as the ground state configuration of a simplified polymer model in a random potential.…
In this paper, we show that although the minimizers of cross-entropy and related classification losses are off at infinity, network weights learned by gradient flow converge in direction, with an immediate corollary that network…
Transport of spherical Brownian particles of finite size possessing radii through narrow channels with varying cross-section area is considered. Applying the so-called Fick-Jacobs approximation, i.e. assuming fast equilibration in…
We provide a visual and intuitive introduction to effectively calculating in 2-groups along with explicit examples coming from non-abelian 1- and 2-form gauge theory. In particular, we utilize string diagrams, tools similar to tensor…