arXiv subjects

Zachary Grey

Publications and source records attributed to Zachary Grey.

5 recordsLinked to original sources

arXiv2026

Topology of Shape and Data in Material Microstructures

One of the challenges in microstructure analysis is the rigorous quantification of the shape, size, and spatial arrangement of the microstructure beyond comparison of average values. We expound on formal principles combining Topological Data Analysis (TDA) and non-Euclidean distances between curves to motivate novel perspectives on the form and nature of pattern and shape in images. Specifically, TDA descriptors extracting persistent topological structures are combined with product submanifold learning of separable shape tensors (SST) to offer unique insights about electron backscatter diffraction (EBSD) images of material microstructures through the lens of a dual-parameter filtration. Beyond standard approaches, our methodology highlights how different choices or permutations of shape distances can lead to distinct notions of topological persistence, thereby broadening the interpretive scope of TDA. The resulting visualizations of feature extraction are designed to be both principled and explanatory, offering novel tools for modern imaging science with applications to material metrology. More broadly, this framework has strong potential to impact domains where precise quantification of topology and shape is critical for uncovering fundamental image patterns and features, and enables additional data-driven tools for microstructure analysis.

cond-mat.mtrl-sci↗

arXiv2026

A Riemannian View on Active Subspaces

Active subspaces provide an explainable, eigenvalue-ordered principle for studying how scalar-valued quantities of interest change the most, on average, over a reduced basis of Euclidean domains. Composition with parallel transport generalizes this principle from Euclidean space to quantities of interest defined over Riemannian manifolds, and the resulting intrinsic formulation is contrasted with the extrinsic, embedding-based gradient average of manifold learning. Either strategy is studied in an intrinsically local sense, restricted to mean-centered geodesic-balls, and within that scope the two are not identical: on the central tangent space, eigenvalues agree to second order in the geodesic radius of the sampled domain, while dominant eigenspaces agree at the same order relative to the spectral gap. Extending activity beyond that central space then calls for either recomputed decompositions over changing tangent spaces or, intrinsically, parallel transport of a single central frame. Hyperspheres are emphasized throughout as a particular manifold of interest, motivated by applications over preshape spaces for statistical shape analysis. Numerical examples over the 2-sphere illustrate the formalism, including the derived ridge recovery at a curvature-limited quadratic rate.

math.NA↗

arXiv2024

Explainable Binary Classification of Separable Shape Ensembles

Scientists, engineers, biologists, and technology specialists universally leverage image segmentation to extract shape ensembles containing many thousands of curves representing patterns in observations and measurements. These large curve ensembles facilitate inferences about important changes when comparing and contrasting images. We introduce novel pattern recognition formalisms combined with inference methods over large ensembles of segmented curves. Our formalism involves accurately approximating eigenspaces of composite integral operators to motivate discrete, dual representations of curves collocated at quadrature nodes. Approximations are projected onto underlying matrix manifolds and the resulting separable shape tensors constitute rigid-invariant decompositions of curves into generalized (linear) scale variations and complementary (nonlinear) undulations. With thousands of curves segmented from pairs of images, we demonstrate how data-driven features of separable shape tensors inform explainable binary classification utilizing a product maximum mean discrepancy; absent labeled data, building interpretable feature spaces in seconds without high performance computation, and detecting discrepancies below cursory visual inspections.

cs.CV↗

arXiv2023

Separable Shape Tensors for Aerodynamic Design

Airfoil shape design is a classical problem in engineering and manufacturing. In this work, we combine principled physics-based considerations for the shape design problem with modern computational techniques using a data-driven approach. Modern and traditional analyses of 2D and 3D aerodynamic shapes reveal a flow-based sensitivity to specific deformations that can be represented generally by affine transformations (rotation, scaling, shearing, translation). We present a novel representation of shapes that decouples affine-style deformations over a submanifold and a product submanifold principally of the Grassmannian. As an analytic generative model, the separable representation, informed by a database of physically relevant airfoils, offers (i) a rich set of novel 2D airfoil deformations not previously captured in the data, (ii) an improved low-dimensional parameter domain for inferential statistics informing design/manufacturing, and (iii) consistent 3D blade representation and perturbation over a sequence of nominal 2D shapes.

cs.GR↗

arXiv2021

Multi-Criteria Radio Spectrum Sharing With Subspace-Based Pareto Tracing

Radio spectrum is a high-demand finite resource. To meet growing demands of data throughput for forthcoming and deployed wireless networks, new wireless technologies must operate in shared spectrum over unlicensed bands (coexist). As an example application, we consider a model of Long-Term Evolution (LTE) License-Assisted Access (LAA) with incumbent IEEE 802.11 wireless-fidelity (Wi-Fi) systems in a coexistence scenario. This scenario considers multiple LAA and Wi-Fi links sharing an unlicensed band; however, a multitude of other scenarios could be applicable to our general approach. We aim to improve coexistence by maximizing the key performance indicators (KPIs) of two networks simultaneously via dimension reduction and multi-criteria optimization. These KPIs are network throughputs as a function of medium access control (MAC) protocols and physical layer parameters. We perform an exploratory analysis of coexistence behavior by approximating active subspaces to identify low-dimensional structure in the optimization criteria, i.e., few linear combinations of parameters for simultaneously maximizing LAA and Wi-Fi throughputs. We take advantage of an aggregate low-dimensional subspace parametrized by approximate active subspaces of both throughputs to regularize a multi-criteria optimization. Additionally, a choice of two-dimensional subspace enables visualizations augmenting interpretability and explainability of the results. The visualizations and approximations suggest a predominantly convex set of KPIs over active coordinates leading to a regularized manifold of approximately Pareto optimal solutions. Subsequent geodesics lead to continuous traces through parameter space constituting non-dominated solutions in contrast to random grid search.

eess.SP↗