Related papers: Constrained Rough Paths
The theory of classical types of curves in normed planes is not strongly developed. In particular, the knowledge on existing concepts of curvatures of planar curves is widespread and not systematized in the literature. Giving a…
In this paper, rough approximations of Cayley graphs are studied and rough edge Cayley graphs are introduced. Furthermore, a new algebraic definition called pseudo-Cayley graphs containing Cayley graphs is proposed. Rough approximation is…
The main objects under consideration in this thesis are called maps, a certain class of graphs embedded on surfaces. Our problems have a powerful relatively recent tool in common, the so-called topological recursion (TR) introduced by…
A notion of dual curve for pseudoholomorphic curves in 4--manifolds turns out to be possible only if the notion of almost complex structure structure is slightly generalized. The resulting structure is as easy (perhaps easier) to work with,…
Existence and uniqueness for rough flows, transport and continuity equations driven by general geometric rough paths are established.
In this paper, practically computable low-order approximations of potentially high-dimensional differential equations driven by geometric rough paths are proposed and investigated. In particular, equations are studied that cover the linear…
Deep Neural Networks are widely used for solving complex problems in several scientific areas, such as speech recognition, machine translation, image analysis. The strategies employed to investigate their theoretical properties mainly rely…
We develop a rough-path framework for two-parameter rough differential equations on rectangular and simplicial domains, motivated by the signature kernel and Schwinger--Dyson kernel equations. The theory is formulated in spaces of jointly…
Manifolds discovered by machine learning models provide a compact representation of the underlying data. Geodesics on these manifolds define locally length-minimising curves and provide a notion of distance, which are key for reduced-order…
The simplest (3+1)D Regge calculus model (with three-dimensional discrete space and continuous time) is considered which describes evolution of the simplest closed two-tetrahedron piecewise flat manifold in the continuous time. The measure…
Effective path planning is a pivotal challenge across various domains, from robotics to logistics and beyond. This research is centred on the development and evaluation of the Dynamic Curvature-Constrained Path Planning Algorithm (DCCPPA)…
In this manuscript we consider Intrinsic Stochastic Differential Equations on manifolds and constrain it to a level set of a smooth function. Such type of constraints are known as explicit algebraic constraints. The system of differential…
In recent years, substantial progress was made towards understanding convergence of fast-slow deterministic systems to stochastic differential equations. In contrast to more classical approaches, the assumptions on the fast flow are very…
Motion planning with constraints is an important part of many real-world robotic systems. In this work, we study manifold learning methods to learn such constraints from data. We explore two methods for learning implicit constraint…
This paper focuses on the mathematical framework for reducing the complexity of models using path signatures. The structure of these signatures, which can be interpreted as collections of iterated integrals along paths, is discussed and…
We introduce a training-efficient framework for time-series learning that combines random features with controlled differential equations (CDEs). In this approach, large randomly parameterized CDEs act as continuous-time reservoirs, mapping…
We develop a differential geometric framework for parallel transport over path spaces and a corresponding discrete theory, an integrated version of the continuum theory, using a category-theoretic framework.
We define and study exact, efficient representations of realization spaces Euclidean Distance Constraint Systems (EDCS), which includes Linkages and Frameworks. Each representation corresponds to a choice of Cayley parameters and yields a…
Neural controlled differential equations (NCDEs), which are continuous analogues to recurrent neural networks (RNNs), are a specialized model in (irregular) time-series processing. In comparison with similar models, e.g., neural ordinary…
Graph Representation Learning (GRL) has experienced significant progress as a means to extract structural information in a meaningful way for subsequent learning tasks. Current approaches including shallow embeddings and Graph Neural…