Universal spaces for finite topological spaces simplify shape descriptions.
problem Describing shape properties of compact metric spaces.
method Inverse limits of finite spaces and Alexandroff extensions.
result Universal spaces simplify shape descriptions of compact metric spaces.
Study shows some complex shapes don't fit a certain property.
problem Some aspherical manifolds lack the Bounded Index Property (BIP).
method Analyzes specific aspherical manifolds to prove they don't have BIP.
result Proves existence of aspherical manifolds without BIP.
Paper characterizes star-shaped risk measures and their properties.
problem Characterizing risk measures in the presence of liquidity risk and competitive delegation.
method Characterization of star-shaped risk measures, study of their properties.
result Star-shaped risk measures include all practically used risk measures.
Direct proof of Alexander polynomial scaling for L-shaped representations.
problem Proving scaling property of Alexander polynomials for specific representations.
method Direct use of Reshetikhin-Turaev formalism to compute R-matrices.
result Normalized Alexander polynomial for one-hook representations scales with q∣R∣. Optimizes shapes on non-standard manifolds.
problem Optimization on non-standard infinite-dimensional manifolds.
method Develops gradient descent on weak Riemannian manifolds.
result Establishes foundational properties for optimization on various weak Riemannian manifolds.
In shape analysis, the concept of shape spaces has always been vague, requiring a case-by-case approach for every new type of shape. In this paper, we give a general definition for an abstract space of shapes in a manifold. This notion encompasses every shape space studied so far in the literature, and offers a rigorou…
Kernel-based tests for shape constraints in finance.
problem Enforcing shape relations on latent functions in financial econometrics.
method Kernel-based nonparametric framework for mean-variance optimization.
result Established statistical properties and a joint Wald-type statistic for testing shape constraints.
Paper controls shape stability in infinite Riemannian manifolds.
problem Characterizing optimal shapes in infinite-dimensional Riemannian manifolds.
method Uses Riemannian manifold framework and mean curvature analysis.
result Control on shape stability depends only on mean curvature.
A registration-free framework monitors shape and color in 4D point clouds.
problem Monitoring shape and color changes in complex parts without registration.
method Laplace-Beltrami operator spectral properties for geometric and color feature capture; combined monitoring scheme for shape and color anomalies.
result Effective detection of shape deformations and color anomalies without registration or mesh reconstruction.
The paper explores metrics and models for analyzing biological shapes.
problem Analyzing biological shapes using mathematical metrics.
method Review of Riemannian metrics and evolution equations, focusing on diffeomorphic shape analysis.
result Introduction of a new class of metrics involving optimization of a growth tensor.
New method for high-fidelity shape representations from raw data.
problem Creating accurate shape representations from raw data.
method A simple loss function encouraging neural network to vanish on input point cloud and have unit norm gradient.
result Our method produces high-fidelity, smooth, and natural zero level set surfaces.
METASET selects diverse unit cells for efficient data-driven metamaterial design.
problem Imbalanced datasets in unit cells can bias data-driven metamaterial design.
method METASET uses similarity metrics and DPPs to select diverse subsets of unit cells.
result Smaller, diverse subsets improve search process and structural performance.
This paper explores how environmental properties can simplify reinforcement learning in non-episodic settings.
problem Challenges in reinforcement learning with continuous interaction and sparse delayed rewards.
method Analysis of environment shaping and dynamism properties to simplify learning.
result Properties like environment shaping and dynamism can significantly ease learning in non-episodic, sparse reward settings.
We introduce a guide to help deep learning practitioners understand and manipulate convolutional neural network architectures. The guide clarifies the relationship between various properties (input shape, kernel shape, zero padding, strides and output shape) of convolutional, pooling and transposed convolutional layers…
Let M be a most singular orbit of the isotropy representation of a simple symmetric space. Let (νi,Φi) be an irreducible factor of the normal holonomy representation (νpM,Φ(p)). We prove that there exists a basis of a section Σi⊂νi of Φi such that the corresponding shape operators have rational…
We present and study a family of metrics on the space of compact subsets of RN (that we call ``shapes''). These metrics are ``geometric'', that is, they are independent of rotation and translation; and these metrics enjoy many interesting properties, as, for example, the existence of minimal geodesics. We view our s…
In this paper, we define a new metric structure on the shape space of a high genus surface. We introduce a rigorous definition of a shape of a surface and construct a metric based on two energies measuring the area distortion and the angle distortion of a quasiconformal homeomorphism. We show that the energy minimizer …
Modeling functional data, this study uncovers the size-and-shape of functions under noisy observations.
problem Uncertainty in recovering a fixed effect function from noisy observations.
method Bayesian functional mixed model with priors on unitary transformations.
result It is possible to recover the size-and-shape of a square-integrable function μ. This article provides an overview of various notions of shape spaces, including the space of parametrized and unparametrized curves, the space of immersions, the diffeomorphism group and the space of Riemannian metrics. We discuss the Riemannian metrics that can be defined thereon, and what is known about the propertie…
The paper proposes a deep learning approach to efficiently approximate diffeomorphisms for shape alignment.
problem Finding optimal reparameterizations of shapes for computing geodesic distances.
method The authors develop a neural network-based algorithm to construct approximations of diffeomorphisms using PyTorch.
result The proposed method achieves universal approximation properties and bounds on Lipschitz constants for the constructed diffeomorphisms.
In this paper, we obtain some properties of biconservative Lorentz hypersurface M1n in E1n+1 having shape operator with complex eigen values. We prove that every biconservative Lorentz hypersurface M1n in E1n+1 whose shape operator has complex eigen values with at most five distinct prin…
Curvature estimate for stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
problem Estimating curvature of stable free boundary minimal hypersurfaces in wedge-shaped manifolds.
method Compactness theorem and Schoen-Simon-Yau estimates.
result Curvature estimate for free boundary minimal hypersurfaces in wedge-shaped manifolds.
The classification of shapes is of great interest in diverse areas ranging from medical imaging to computer vision and beyond. While many statistical frameworks have been developed for the classification problem, most are strongly tied to early formulations of the problem - with an object to be classified described as …
We consider the problem of nonparametric regression under shape constraints. The main examples include isotonic regression (with respect to any partial order), unimodal/convex regression, additive shape-restricted regression, and constrained single index model. We review some of the theoretical properties of the least …
This work establishes properties on diffeological structures for set-valued maps and measures.
problem Establish rigorous properties on diffeological structures for set-valued maps and measures.
method Using diffeologies, the authors link various structures including set-valued maps, relations, gradients, measures, and shape analysis.
result Established rigorous properties on sample diffeologies.
A framework for generating 3D shapes by sequentially assembling primitives.
problem Combinatorial complexity in generating 3D shapes.
method Bayesian optimization for efficient exploration and exploitation of feasible combinations.
result Successfully generates realistic combinatorial 3D shapes.
This paper provides further investigation of the concept of shape msimpl-fibrators (previously introduced by the author). The main results identify shape msimpl-fibrators among direct products of Hopfian manifolds. First it is established that every closed orientable manifold homotopically determined …
In the elastic shape analysis approach to shape matching and object classification, plane curves are represented as points in an infinite-dimensional Riemannian manifold, wherein shape dissimilarity is measured by geodesic distance. A remarkable result of Younes, Michor, Shah and Mumford says that the space of closed p…
We study completeness properties of Sobolev metrics on the space of immersed curves and on the shape space of unparametrized curves. We show that Sobolev metrics of order n≥2 are metrically complete on the space In(S1,Rd) of Sobolev immersions of the same regularity and that any two curves i…
By introducing a shape manifold as a solution set to solve inverse obstacle scattering problems we allow the reconstruction of general, not necessarily star-shaped curves. The bending energy is used as a stabilizing term in Tikhonov regularization to gain independence of the parametrization. Moreover, we discuss how se…
This paper introduces a new reward shaping method for average-reward reinforcement learning.
problem Speeding up convergence to an optimal policy in average-reward reinforcement learning tasks.
method Developed a temporal logic-based approach to automatically generate reward shaping functions.
result The optimal policy can be recovered using the proposed reward shaping framework.
This paper connects monetary and star-shaped risk measures by showing their equivalence under certain conditions.
problem Understanding the relationship between monetary and star-shaped risk measures.
method Analyzing the acceptability of 0 and the normalization property.
result Monetary risk measures are only a translation away from star-shapedness under mild conditions.
Neural networks predict shapes of first passage percolation sets.
problem Predicting the shape of first passage percolation sets.
method Used a neural network to predict the shape of the set of discovered sites from the distribution of passage times.
result Neural networks can quickly predict the shape of the set of discovered sites from the distribution of passage times.
New method uses Riemannian geometry to describe molecular shapes.
problem Predicting drug-like molecules using shape similarity.
method Riemannian geometry applied to molecular surfaces.
result RGMolSA method captures molecular shape effectively.
We describe all families of star-shaped n-polygons in the Euclidean plane with prescribed perimeter and area ; they are leaves of a foliation F on the space of star-shaped n-polygons. By the way, we study some geometric properties of convex polygons, for instance their inscriptibility in a circle and their regularity i…
ML predicts alloy properties considering chemistry, processing, and data transformations.
problem Designing and predicting alloy properties in high-dimensional design space.
method Physics-informed machine learning with engineered features from chemistry and heat treatment.
result ML models accurately predict alloy properties, including hysteresis in shape memory alloys.
New method groups similar functional covariates for better modeling.
problem Analyzing functional covariates with similar shapes.
method Coefficient shape alignment regularization approach.
result True grouping structure can be accurately identified under certain conditions.
Statistical shape analysis can be done in a Riemannian framework by endowing the set of shapes with a Riemannian metric. Sobolev metrics of order two and higher on shape spaces of parametrized or unparametrized curves have several desirable properties not present in lower order metrics, but their discretization is stil…
Homotopy theory of differentiable sheaves connects manifold properties to underlying homotopy types.
problem Understanding the homotopy type of manifolds using differentiable sheaves.
method Developed model structures and homotopical calculi on the ∞-category Diff∞ to compute and compare shapes. result The shape of any manifold coincides with various other notions of underlying homotopy types.
VCML learns concepts and metaconcepts from images and questions.
problem Learning concepts and metaconcepts from visual data.
method Bidirectional connection between visual concepts and metaconcepts.
result VCML can generalize from limited data and noisy inputs.
The paper presents a method for analyzing shape graphs using specific features.
problem Analyzing geometric and topological variations in shape graphs.
method Curated set of topological, geometric, and directional features for shape graph analysis.
result The feature representation is effective for tasks like group comparison and classification.
New metrics for surface shapes incorporating curve properties.
problem Developing metrics for surface shape spaces.
method Incorporates geodesic and normal curvatures of curves on surfaces.
result Explicitly defined 6-parameter family of metrics.
A new method monitors unstructured 3D shapes without registration.
problem Error-prone registration and mesh reconstruction steps in PCD monitoring.
method Intrinsic geometric properties of shapes, using Laplacian and geodesic distances.
result Effective monitoring of defects without registration and mesh reconstruction.
Study properties of hyperbolic Yamabe solitons in submanifolds.
problem Characterize and understand hyperbolic Yamabe solitons in submanifolds.
method Define hyperbolic Yamabe flow, prove properties, and classify solitons.
result Prove that hyperbolic Yamabe solitons are pseudosymmetric or metallic shaped.
New metrics on curve spaces improve shape analysis.
problem Discretization of curve spaces and metric completeness.
method Sobolev metrics on discrete regular curves, completeness analysis.
result The finite-dimensional Riemannian manifolds are complete.
Proposes a new metric for comparing shapes in different spaces.
problem Comparing shapes in different metric spaces with unequal mass.
method Developed a Partial Gromov-Wasserstein (PGW) metric and algorithms to solve it.
result PGW is a well-defined metric between metric measure spaces.
The moving sofa problem, posed by L. Moser in 1966, asks for the planar shape of maximal area that can move around a right-angled corner in a hallway of unit width, and is conjectured to have as its solution a complicated shape derived by Gerver in 1992. We extend Gerver's techniques by deriving a family of six differe…
A novel 3D shape registration method using spectral graph embedding and probabilistic matching.
problem Challenges in 3D shape analysis and registration, especially with large variability.
method Combining spectral graph matching with Laplacian embedding for large graphs, using commute-time embedding and PCA.
result A method to register shapes with different samplings and isometric deformations.