arXiv subjects

Victor Rielly

Publications and source records attributed to Victor Rielly.

5 recordsLinked to original sources

arXiv2026

Time Series Classification through Diffeomorphic Time Warping (DiffTW)

Time series classification involves learning a mapping from a continuous, temporally ordered sequence of real-valued observations to discrete response variables, like class labels. This task is fundamental in domains, including health monitoring, where temporal structure is critical for prediction. Dynamic Time Warping (DTW) is a standard technique for measuring similarity between sequences varying in time or speed. However, DTW is restricted to discrete point matching. Moving beyond pairwise alignment, we propose a theoretical framework learning mappings between real-valued functions. These mappings approximate the flow associated with the characteristic curves of a linear transport equation with a space-dependent velocity field, providing a diffeomorphic transformation between time series. Using the method of characteristics, we transform this partial differential equation into ordinary differential equations (ODEs) modeling system dynamics. The objective function to learn these ODEs derives from the fundamental theorem of calculus. To enable flexible, expressive representations of the velocity field, we utilize reproducing kernel Hilbert spaces and optimal control methods. Our method, Diffeomorphic Time Warping (DiffTW), provides a theoretically grounded dissimilarity measure. Using a 1-nearest neighbor classifier, DiffTW outperforms unconstrained DTW on 39 of 85 datasets with 3 ties; however, constrained DTW outperforms DiffTW on 48 of 85 datasets with 5 ties.

stat.ML↗

arXiv2025

MOCK: an Algorithm for Learning Nonparametric Differential Equations via Multivariate Occupation Kernel Functions

Learning a nonparametric system of ordinary differential equations from trajectories in a $d$-dimensional state space requires learning $d$ functions of $d$ variables. Explicit formulations often scale quadratically in $d$ unless additional knowledge about system properties, such as sparsity and symmetries, is available. In this work, we propose a linear approach, the multivariate occupation kernel method (MOCK), using the implicit formulation provided by vector-valued reproducing kernel Hilbert spaces. The solution for the vector field relies on multivariate occupation kernel functions associated with the trajectories and scales linearly with the dimension of the state space. We validate through experiments on a variety of simulated and real datasets ranging from 2 to 1024 dimensions. MOCK outperforms all other comparators on 3 of the 9 datasets on full trajectory prediction and 4 out of the 9 datasets on next-point prediction.

stat.ML↗

arXiv2025

ROCK: A variational formulation for occupation kernel methods in Reproducing Kernel Hilbert Spaces

We present a Representer Theorem result for a large class of weak formulation problems. We provide examples of applications of our formulation both in traditional machine learning and numerical methods as well as in new and emerging techniques. Finally we apply our formulation to generalize the multivariate occupation kernel (MOCK) method for learning dynamical systems from data proposing the more general Riesz Occupation Kernel (ROCK) method. Our generalized methods are both more computationally efficient and performant on most of the benchmarks we test against.

stat.ML↗

arXiv2023

Learning nonparametric ordinary differential equations from noisy data

Learning nonparametric systems of Ordinary Differential Equations (ODEs) dot x = f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator and provide experimental comparisons with the state-of-the-art.

stat.ML↗

arXiv2017

Classification of Minimal Separating Sets in Low Genus Surfaces

Consider a surface $S$ and let $M\subset S$. If $S\setminus M$ is not connected, then we say $M$ \emph{separates} $S$, and we refer to $M$ as a \emph{separating set} of $S$. If $M$ separates $S$, and no proper subset of $M$ separates $S$, then we say $M$ is a \emph{minimal separating set} of $S$. In this paper we use methods of computational combinatorial topology to classify the minimal separating sets of the orientable surfaces of genus $g=2$ and $g=3$. The classification for genus 0 and 1 was done in earlier work, using methods of algebraic topology.

math.CO↗