arXiv subjects

Gary P. T. Choi

Publications and source records attributed to Gary P. T. Choi.

At least 19 recordsLinked to original sources

arXiv2026

The Normalized Maximum Likelihood for Regular Non-Smooth Models: Measure-Theoretic Foundations and Geometric Sampling

The Normalized Maximum Likelihood (NML) codelength, or stochastic complexity, represents a principled criterion for universal coding. While recent coarea-based formulations provided a calculation method for smooth models, this framework collapses for the non-smooth estimators ubiquitous in modern machine learning (e.g., Lasso, Sparse SVMs). In this work, we provide a rigorous framework for computing the NML for regular path-differentiable Lipschitz (PDL) estimators. By applying classical geometric measure theory and bridging the coarea formula with conservative Jacobians, we prove that the stochastic complexity for non-smooth models is well-posed and theoretically consistent with the outputs of modern Automatic Differentiation. To compute this quantity exactly, we introduce the Propose-and-Project Metropolis-Hastings (PDL-PPMH) sampler, a geometric MCMC algorithm capable of traversing the non-differentiable level sets of the maximum likelihood estimator. We theoretically justify its components, including a stochastic tangent space proposal and a provably convergent non-smooth projection solver. We demonstrate the method's robustness by sampling from a high-dimensional Lasso posterior ($P=2000$), while simultaneously quantifying the computational scaling that governs the trade-off between exactness and mixing time. Crucially, we empirically demonstrate that our exact NML criterion provides a highly data-efficient alternative to cross-validation, achieving statistically indistinguishable predictive optima without requiring data splitting. Altogether, our work paves the way for the theoretical analysis of the NML codelength for regular non-smooth models.

cs.LG↗

arXiv2026

Explosive connectivity and mechanical rigidity in cubic lattice structures

We study explosive connectivity and mechanical rigidity in three-dimensional cubic lattice structures under Achlioptas-type product-rule dynamics. Our work combines extensive numerical simulation with a theoretical framework based on rigorous finite-size scaling. Using massive-scale simulations up to $L=192$ ($N \approx 7 \times 10^6$) with 20,000 independent realizations, we demonstrate that for $k \ge 8$, the peak susceptibility scales with an exponent of $γ= 1.000$, and the maximum single-step jump stabilizes at a macroscopic fraction. This confirms that while the transition is continuous in the infinite thermodynamic limit, it exhibits the exact finite-size scaling signatures of a first-order discontinuity in finite physical systems. For rigidity, we discover numerically that for richly-connected hosts, increasing the number of choices $k$ optimally enhances the efficiency of rigidification. To explain this phenomenon, we propose a theoretical model centered on a conditional progress function that links an edge's local product-rule score to its global mechanical utility. We show that while local rigidification efficiency monotonically increases, the global rigidity gap exhibits a ``Goldilocks'' minimum at intermediate $k$ due to the emergence of maximally floppy, tree-like components at large $k$. Altogether, our work provides new insights into the relationship between local dynamics and global connectivity and rigidity in cubic lattice structures via both theory and computation.

cond-mat.stat-mech↗

arXiv2026

Monotile kirigami

Kirigami, the art of paper cutting, has been widely used in the modern design of mechanical metamaterials. In recent years, many kirigami-based metamaterials have been designed based on different planar tiling patterns and applied to different science and engineering problems. However, it is natural to ask whether one can create deployable kirigami structures based on the simplest forms of tilings, namely the monotile patterns. In this work, we answer this question by proving the existence of periodic and aperiodic monotile kirigami structures via explicit constructions. In particular, we present a comprehensive collection of periodic monotile kirigami structures covering all 17 wallpaper groups and aperiodic monotile kirigami structures covering various quasicrystal patterns as well as polykite tilings. We further perform theoretical and computational analyses of monotile kirigami patterns in terms of their shape and size changes under deployment. Altogether, our work paves a new way for the design and analysis of a wider range of shape-morphing metamaterials.

cond-mat.soft↗

arXiv2026

An integrated geometric quantification and shape analysis framework for axillary lymph node metastasis in breast cancer patients

Quantitative characterization of lymph node morphology is important for assessing axillary lymph node metastasis in breast cancer. However, surfaces reconstructed from computed tomography (CT) segmentation may contain geometric and topological defects that compromise subsequent analysis, while conventional shape descriptors predominantly characterize global morphology. To address these issues, we developed an integrated framework combining topology-aware surface processing with multi-resolution spherical harmonic (SH) analysis of CT-derived axillary lymph nodes. The processing pipeline produced topology-valid genus-0 surfaces with improved mesh quality, which were then represented at multiple SH degrees and characterized using 20 predefined geometric feature families. Geometric fidelity increased with SH degree, whereas predictive performance peaked at intermediate resolutions. Preferred SH degree also differed across feature families. A family-specific mixed-resolution model achieved an AUC of 0.918, compared with 0.884 for the conventional PyRadiomics Shape14 baseline, corresponding to an improvement of 0.0344. Controlled perturbation experiments showed that higher SH degrees transmitted more fine-scale geometric variation and yielded lower stability of curvature-based predictions. Representative geometric descriptors provided interpretable characterization of metastasis-associated surface morphology. Independent validation further supported the framework's transportability: label-free replication in a multicenter lymph node cohort reproduced the family-specific resolution effects, while a labeled LIDC-IDRI lung-nodule experiment reproduced the resolution-dependent relationship between SH degree and predictive performance. Altogether, the framework provides a topology-valid basis for quantitative characterization of lymph node morphology and metastasis-associated imaging phenotypes.

q-bio.QM↗

arXiv2026

Low-distortion planar embedding of rod-based structures

Rod-based structures are commonly used in practical applications in science and engineering. However, in many design, analysis, and manufacturing tasks, handling the rod-based structures in three dimensions directly is generally challenging. To simplify the tasks, it is usually more desirable to achieve a two-dimensional representation of the rod-based structures via some suitable geometric mappings. In this work, we develop a novel method for computing a low-distortion planar embedding of rod-based structures. Specifically, we identify geometrical constraints that aim to preserve key length and angle quantities of the 3D rod-based structures and prevent the occurrence of overlapping rods in the planar embedding. Experimental results with a variety of rod-based structures are presented to demonstrate the effectiveness of our approach. Moreover, our method can be naturally extended to the design and mapping of hybrid structures consisting of both rods and surface elements. Altogether, our approach paves a new way for the efficient design and fabrication of novel three-dimensional geometric structures for practical applications.

cs.CG↗

arXiv2026

A unified geometric design framework for kirigami structures

In recent years, kirigami metamaterials have been widely studied and applied in science and engineering. While various two- and three-dimensional kirigami design methods have been developed, most of them are only applicable to a limited class of kirigami structures. In this work, we develop a unified framework for kirigami design that encompasses a wide range of 2D-to-2D, 2D-to-3D, and 3D-to-3D shape-morphing effects, as well as additional geometric and physical properties such as compact reconfigurability and rigid deployability. In particular, by reformulating the design task as a length-based constrained optimization problem and solving it simultaneously for multiple target states of the kirigami structure, our unified design framework enables greater design flexibility and stronger theoretical support. Experimental results with a wide range of shape-morphing effects are presented to demonstrate the effectiveness of our framework. We further present a rigorous theoretical analysis of several key aspects of kirigami design, covering inertia transposition, aspect-ratio law, and angle defects, thereby elucidating important design rules and limitations. Altogether, our work paves a new way for the design of shape-morphing mechanical metamaterials.

cond-mat.soft↗

arXiv2026

Adaptive Volumetric Parameterization of Simply Connected 3-Manifolds with Applications

Volumetric parameterization, the process of mapping a 3-manifold onto a simplified volumetric domain, is important for many tasks in computer graphics and imaging science. However, most prior volumetric parameterization approaches have only utilized standardized domains such as a solid ball regardless of the overall shape of the given 3-manifolds, which introduces significant geometric distortion and affects the subsequent shape processing and analysis tasks. To overcome this issue, in this work we propose a novel volumetric parameterization framework for simply connected 3-manifolds. Specifically, the proposed framework jointly controls local shape and mass distortions, while adapting the target domain during the optimization process. It enables three progressively more flexible target-domain settings for the parameterization: a prescribed solid ellipsoid, a volume-normalized adaptive ellipsoid with variable radii, and a sea-embedded free-boundary domain. For each setting, the parameterization algorithm consists of a 3D quasi-conformality shape update, a diffusion-based density-equalizing update, and a geometric correction procedure for removing element foldings, thereby allowing for volumetric parameterizations with different desired effects. Experimental results are presented to demonstrate the effectiveness of our proposed framework. Moreover, our framework can be easily applied to multiresolution and localized adaptive volumetric remeshing, volumetric registration, and volumetric morphing. Altogether, our work provides a new way for the representation, processing, and analysis of 3-manifolds.

cs.GR↗

arXiv2026

Conformal tubular parameterization and toroidal bending of tube-like surfaces

Tube-like surfaces are widely encountered in geometry processing, engineering structures, and medical anatomy, yet their intrinsic longitudinal and circumferential topology is not well preserved by conventional planar annular or rectangular parameterization domains. In this work, we propose a new conformal parameterization framework for open tube-like surfaces with two boundary components. The proposed method first constructs a fixed-boundary tubular parameterization by cutting the input mesh, computing a disk-to-parallelogram conformal map, and lifting the result to a three-dimensional tubular domain. To reduce residual distortion introduced near the cut seam, we further introduce a localized quasi-conformal correction scheme formulated on an annular domain, which improves conformality while leaving regions away from the seam unchanged. To handle noisy or irregular input boundaries, we also develop a free-boundary variant based on boundary extension and cycle-Laplacian smoothing, allowing the prescribed boundary constraints to be imposed on artificial outer rings rather than directly on the original surface. Finally, we derive two conformal toroidal bending maps that transform the tubular parameterization into toroidal geometries while preserving the underlying tube topology. Experiments on synthetic tube meshes and real vascular surfaces demonstrate that the proposed framework produces low-distortion parameterizations, effectively mitigates seam-induced artifacts, improves robustness for boundary-noisy inputs, and provides flexible tubular and toroidal target domains for downstream surface processing tasks.

cs.GR↗

arXiv2026

Geometric quantification for nonlinear deformation in knitted fabrics

Knitted fabrics exemplify a broad class of architected materials capable of large deformations, enabling shape morphing, mechanical biocompatibility, and embedded multifunctionality without material damage. Although geometric nonlinearity has been intuitively utilized in their design, a quantitative description of stitch-resolved deformation and its temporal evolution remains lacking. Here, we introduce a geometric quantification framework that reconstructs smooth yarn centerlines and fabric surfaces from sparse yarn-level representations and extracts interpretable descriptors across dimensions. Applied to representative knitted structures, this framework resolves how global deformation is distributed among stitch reorientation, loop bending, surface bending, and dilation. Moreover, it reveals how regions of large geometric variation emerge, persist, and redistribute over time. Rather than directly measuring stress, these geometric descriptors define a unified geometric state space for comparing knitted structures and identifying candidate regions of mechanical localization. The framework provides a quantitative language for nonlinear deformation in knits and establishes a geometry-based representation that can be coupled to constitutive models, experimental measurements, and graph-based inverse-design workflows.

cond-mat.soft↗

arXiv2026

PyKirigami: An interactive Python simulator for kirigami structures

In recent years, the concept of kirigami has been used in creating deployable structures for various scientific and technological applications. While high-fidelity Finite Element Analysis (FEA) is the standard for analyzing stress distributions and material deformation, it is computationally intensive and often ill-suited for the rapid exploration of vast kinematic configuration spaces. In this work, we develop PyKirigami, a lightweight, open-source Python framework for the efficient deployment simulation of kirigami structures. Unlike continuum mechanics solvers, PyKirigami models tessellations as articulated rigid-body networks, allowing for the real-time simulation of global deployment trajectories and volumetric transformations. The tool incorporates collision detection and interactive actuation, enabling users to validate folding paths and identify geometric locking states in both 2D and 3D topologies. This framework serves as a fast kinematic prototyping tool for kirigami structures, allowing researchers to verify deployment mechanics and self-contacts prior to performing detailed mechanical analysis or physical fabrication.

cond-mat.soft↗

arXiv2026

Exact Schur-Sylvester Dimensionality Reductions for Non-Smooth Stochastic Complexity and Manifold Sampling

The exact computation of the Normalized Maximum Likelihood (NML) codelength for regular non-smooth estimators (e.g., Lasso) has been historically limited by the cubic scaling walls of manifold-constrained projection and volume integration. At each step of the geometric Propose-and-Project Metropolis--Hastings (PPMH) sampler, evaluating the projection operator requires inverting an $(N+k) \times (N+k)$ generalized KKT matrix, while calculating the volume factor requires the determinant of an $(N-k) \times (N-k)$ Gram matrix. This paper presents an exact, mathematically equivalent formulation that bypasses both bottlenecks by utilizing the block Schur complement and Sylvester's determinant identity. We prove that the computational complexity of both operations collapses from $\mathcal{O}(N^3)$ to $\mathcal{O}(k^3 + N^2 k)$ per step. We generalize this reduction to Sparse Support Vector Machines (SVMs), Elastic Net, and Group Lasso. Finally, we provide a rigorous numerical stability analysis and evaluate the sampler's efficiency using the Effective Sample Size (ESS) per second. Our empirical benchmarks on high-dimensional datasets confirm a constant speedup exceeding $14{,}100\times$ while maintaining double-precision numerical equivalence, rendering exact non-smooth NML estimation highly tractable for large-scale statistical inference.

cs.LG↗

arXiv2026

PhyloSDF: Phylogenetically-Conditioned Neural Generation of 3D Skull Morphology via Residual Flow Matching

Generating novel, biologically plausible three-dimensional morphological structures is a fundamental challenge in computational evolutionary biology, hampered by extreme data scarcity and the requirement that generated shapes respect phylogenetic relationships among species. In this work, we present PhyloSDF, a phylogenetically-conditioned neural generative model for 3D biological morphology that integrates two innovations: (1) a DeepSDF auto-decoder regularized by a novel Phylogenetic Consistency Loss that structures the latent space to correlate with evolutionary distances (Pearson r=0.993); (2) a Residual Conditional Flow Matching (Residual CFM) architecture that factorizes generation into analytic species-centroid lookup and learned residual prediction, enabling generation from as few as ~4 specimens per species. We evaluate PhyloSDF on 100 micro-CT-scanned skulls of Darwin's Finches and their relatives across 24 species. The model generates novel meshes achieving 88-129% of real intra-species variation at the code level, with all 180 generated meshes verified as non-memorized. Residual CFM surpasses denoising diffusion (which fails entirely at this scale), standard flow matching (which mode-collapses to 3-6% variation), and a Gaussian mixture baseline in both fidelity (Chamfer Distance 0.00181 vs. 0.00190) and morphometric Fréchet distance (10,641 vs. 13,322). Leave-one-species-out experiments across 18 species demonstrate phylogenetic extrapolation capability, and smooth latent interpolations produce biologically plausible ancestral skull reconstructions.

q-bio.QM↗

arXiv2026

Learning the Geometry of Data: A Mathematical Review of Shape Space Analysis

A central objective of machine learning is to identify structure and patterns in data. Advances in data acquisition have increasingly produced datasets whose observations possess rich geometric form, giving rise to shape spaces that encode variability in object geometry. Such datasets arise across a wide range of disciplines, including biology, medicine, anthropology, and computer vision, where subtle geometric differences often carry important scientific information. Traditional machine learning methods, however, are frequently ill-equipped to account for the nonlinear geometric structure underlying these data. This survey synthesizes a rapidly growing body of work on shape space analysis, which provides a mathematical and computational framework for the study of geometric data. Drawing on ideas from differential geometry, statistics, and machine learning, we organize the literature around a common analytical pipeline: shape representation and parameterization, the rigorous construction of robust geodesic metrics, statistical analysis on shape spaces, and geometry-aware learning methods. We discuss how these tools enable the characterization of shape variability, the comparison of geometric objects, and the analysis of structural trajectories across populations and time. To illustrate the breadth of the field, we highlight applications spanning multiple scales of biological organization, including studies of subcellular morphology and primate tooth evolution. Across these and many other domains, researchers face common challenges arising from complex, nonlinear, and often unaligned geometric variation. The review concludes by identifying key theoretical and computational challenges, as well as emerging opportunities driven by increasingly large and diverse geometric datasets.

math.ST↗

arXiv2026

Robust Parametric Estimation of Avian Cranial Morphology

Understanding the growth and form of complex morphological structures is one of the most fundamental problems in biology. While many prior works have analyzed the beak morphology of Darwin's finches, other cranial features are relatively less explored. In this work, we develop geometric and statistical methods for analyzing the skull morphology of Darwin's finches and their relatives, focusing on the relationship between their skull dimensions, orbit curvature, and neurocranial geometries. Unlike traditional landmark-based approaches that scale linearly with human labor, our framework is fully unsupervised. Specifically, by utilizing tools in computational geometry, differential geometry, and numerical optimization, we develop efficient algorithms for quantifying various key geometric features of the skull. We then perform a statistical analysis and discover a strong correlation between skull size and orbit curvature. Based on our findings, we further establish a predictive model that can estimate the orbit curvature using easily obtainable linear skull measurements. Our results show that the predictive model is highly effective and capable of explaining 85.48\% of the variance in curvature with an average prediction error of only 6.35\%. Altogether, our work establishes a rigorous foundation for the digital estimation and high-throughput phenotyping of large-scale museum collections, overcoming the scalability bottlenecks of manual methods.

q-bio.QM↗

arXiv2026

How geometry of subduction zones correlates with earthquake dynamics

Subduction zones on the surface of the Earth, where abrupt sliding leads to earthquakes, are generally curved and localized. How does the geometry of these zones influence the occurrence of megathrust earthquakes? Here we use a combination of simple scaling arguments and data analysis using the differential geometry of surfaces to examine the relationship between the earthquake productivity of subduction zones and their shape. A scaling argument suggests how interface curvature changes both the accumulation and release of stress relative to planar interfaces; conformable sliding along relatively flat subduction zones should lead to rare but large events, while curved subduction zones should lead to frequent smaller events. To test this, we leverage global geometry datasets and analyze the correlation between the surface curvatures of the subduction zones and the frequency and magnitude of earthquakes therein. Our analysis shows that weakly curved slab geometries are associated with rarer larger magnitude events, while slab geometries with a larger relative dispersion in curvature are associated with frequent but smaller magnitude events. Using different scale-dependent shape metrics of the subduction zones, we show that the earthquake productivity is influenced by the conformability of the overriding and downgoing plates. More broadly, our results suggest the need to incorporate the large-scale geometry of subduction zones in computational models and predictive frameworks for earthquake risk.

physics.geo-ph↗

arXiv2026

Resolution-free neural surrogates for geometric parameterization and mapping with spatially varying fields

Many imaging problems require computing spatial transformations induced by spatially varying intensity, feature, or density fields. Canonical examples include distortion correction, deformable image registration, atlas-based segmentation, and deformation-driven image analysis. These tasks can be formulated as geometric mapping problems in which the transformation is constrained to preserve local structure, control boundary behavior, or regulate angular distortion. Such formulations typically lead to variational models, diffusion processes, or elliptic partial differential equations. However, repeatedly solving high-resolution systems becomes computationally expensive when the underlying parameter fields vary across instances. In this work, we propose a resolution-free neural surrogate for geometric parameterization and mapping problems. Given a spatially varying parameter field $p:Ω\to\mathbb{R}^m$ and query locations $\{x_i\}_{i=1}^N\subsetΩ$, the model predicts mapped locations $\{u(x_i)\}_{i=1}^N$ on arbitrary structured or unstructured point sets. To avoid dependence on a fixed grid, we use a multi-resolution geometric encoding strategy that conditions the network on coordinate-augmented samples of the parameter field. The model is trained without labeled solution data by enforcing geometry-aware constraints derived from variational energies, diffusion-based density equalization, and quasi-conformal theory. Experimental results on quasi-conformal mapping and density-equalizing mapping problems are presented to demonstrate the effectiveness of our proposed method.

cs.CV↗

arXiv2026

Planar morphometry via functional shape data analysis and quasi-conformal mappings

The study of shapes is one of the most fundamental problems in life sciences. Although numerous methods have been developed for the morphometry of planar biological shapes over the past several decades, most of them focus solely on either the outer silhouettes or the interior features of the shapes without capturing the coupling between them. Moreover, many existing shape mapping techniques are limited to establishing correspondence between planar structures without further allowing for the quantitative analysis or modelling of shape changes. In this work, we introduce FDA-QC, a novel planar morphometry method that combines functional shape data analysis (FDA) techniques and quasi-conformal (QC) mappings, taking both the boundary and interior of the planar shapes into consideration. Specifically, closed planar curves are represented by their square-root velocity functions and registered by elastic matching in the function space. The induced boundary correspondence is then extended to the entire planar domains by a quasi-conformal map, optionally with landmark constraints. Moreover, the proposed FDA-QC method can naturally lead to a unified framework for shape morphing and shape variation quantification. We apply the FDA-QC method to various leaf and insect wing datasets, and the experimental results show that the proposed combined approach captures morphological variation more effectively than purely boundary-based or interior-based descriptions. Altogether, our work paves a new way for understanding the growth and form of planar biological shapes.

q-bio.QM↗

arXiv2026

Optimization of Constrained Quasiconformal Mapping for Origami Design

Origami structures, particularly Miura-ori patterns, offer unique capabilities for surface approximation and deployable designs. In this study, a constrained mapping optimization algorithm is designed for designing surface-aligned Miura-ori via a narrow band approximation of the input surface. The Miura-fold, embedded in the narrow band, is parameterized to a planar domain, and a mapping is computed on the parameter pattern by optimizing certain energy terms and constraints. Extensive experiments are conducted, showing the significance and flexibility of our methods.

cs.CG↗