Nicholas Fisher

Victor Rielly, Kamel Lahouel, Ethan Lew et al.

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.

Victor Rielly, Kamel Lahouel, Chau Nguyen et al.

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.

Zachary Grey, Nicholas Fisher, Andrew Glaws et al.

Materials scientists utilize image segmentation of micrographs to create large curve ensembles representing grain boundaries of material microstructures. Observations of these collections of shapes can facilitate inferences about material properties and manufacturing processes. We seek to bolster this application, and related engineering/scientific tasks, using novel pattern recognition formalisms and inference over large ensembles of segmented curves -- i.e., facilitate principled assessments for quantifying differences in distributions of shapes. To this end, we apply a composite integral operator to motivate accurate and efficient numerical representations of discrete planar curves over matrix manifolds. The main result is a rigid-invariant orthonormal decomposition of curve component functions into separable forms of scale variations and complementary features of undulation. We demonstrate how these separable shape tensors -- given thousands of curves in an ensemble -- can inform explainable binary classification of segmented images by utilizing a product maximum mean discrepancy to distinguish the shape distributions; absent labelled data, building interpretable feature spaces in seconds without high performance computation, and detecting discrepancies below cursory visual inspections.