Fine shape of local compacta represented by ordinary maps.
problem Representing fine shape of local compacta.
method Constructing a space ∣X∣ for each local compactum X such that fine shape classes correspond to homotopy classes of maps to ∣X∣. result Fine shape classes from any locally compact metrizable space Y to X bijectively correspond to homotopy classes of maps from Y to ∣X∣. The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.
problem Finding minimal hypersurfaces in wedge-shaped manifolds with boundary.
method Developed a min-max theory for locally wedge-shaped manifolds with boundary.
result Proved existence of smooth free boundary minimal hypersurfaces in wedge-shaped manifolds.
Local minimizers are convex and close to Wulff shapes.
problem Finding local minimizers in anisotropic isoperimetric problems.
method Showed local minimizers are geodesically convex and small smooth perturbations of tangent Wulff shapes.
result Local minimizers are quantitatively close to Wulff shapes.
Representing 3D shape deformations by linear models in high-dimensional space has many applications in computer vision and medical imaging, such as shape-based interpolation or segmentation. Commonly, using Principal Components Analysis a low-dimensional (affine) subspace of the high-dimensional shape space is determin…
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.
Novel method for shape optimization of non-smooth PDEs.
problem Optimizing shapes governed by non-smooth PDEs.
method Functional variational approach and sensitivity analysis.
result Necessary conditions for locally optimal shapes.
Fine shape theory extends strong shape to noncompact metrizable spaces.
problem Computational complexity in extending strong shape to noncompact spaces.
method Introducing FDR-embeddings and mapping cylinders to extend SSDR-maps to noncompact spaces.
result Fine shape category can be represented as a left fraction localization.
PointGMM learns hGMMs from point clouds for 3D shape representation.
problem Lack of shape priors and non-local information in point cloud representations.
method Neural network that learns hierarchical Gaussian mixture models (hGMMs) for 3D shapes.
result Generative model learns meaningful latent space for interpolations and novel shape synthesis.
Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.
problem Finding local maximizers for higher Ekeland-Hofer capacities in specific domains.
method Analogous to 4D local Viterbo conjecture, proving maximizers for rational ellipsoids.
result Local maximizers of the k-th Ekeland-Hofer capacities are symplectomorphic to rational ellipsoids.
The paper proves a Whitehead theorem for fine shape spaces.
problem Proving a Whitehead theorem for fine shape spaces.
method Using Steenrod-Sitnikov homotopy groups and ind-groups.
result Fine shape morphisms are equivalences if they induce isomorphisms on π_i.
The paper characterizes dynamic return and star-shaped risk measures via BSDEs.
problem Characterizing dynamic return and star-shaped risk measures.
method Characterization of star-shaped functionals and BSDEs.
result Existence of convex BSDEs with non-empty set of supersolutions.
The paper shows how to use fine shape to understand infinite-dimensional spaces.
problem Understanding infinite-dimensional metrizable spaces and their homology theories.
method Obtained results indicating fine shape is tractable and can be used for Polish spaces.
result Every Polish space is fine shape equivalent to the limit of an inverse sequence of simplicial maps.
We introduce a method called multi-scale local shape analysis, or MLSA, for extracting features that describe the local structure of points within a dataset. The method uses both geometric and topological features at multiple levels of granularity to capture diverse types of local information for subsequent machine lea…
Difficult image segmentation problems, for instance left atrium MRI, can be addressed by incorporating shape priors to find solutions that are consistent with known objects. Nonetheless, a single multivariate Gaussian is not an adequate model in cases with significant nonlinear shape variation or where the prior distri…
The paper encodes local shapes of polynomial curves using permutations.
problem Measuring non-convexity of real algebraic plane curves.
method Generic projections avoiding specific tangencies.
result Local shapes of curves can be encoded in alternating permutations.
Flat minimal hypersurfaces found in wedge-shaped domains.
problem Finding minimal surfaces in wedge-shaped domains.
method Proving stability and flatness of C1,1-to-edge minimal hypersurfaces. result Stable minimal hypersurfaces are flat in wedge-shaped domains.
Quantitative metric spaces study function shapes and sphere diameters.
problem Understanding function shapes and sphere diameters in metric spaces.
method Quantitative analysis of transport-rays decompositions using localization method.
result Bounding the deficit between manifold and sphere diameters.
A new method shapes reinforcement learning environments by abstracting large state spaces.
problem Learning in large, noisy environments with sparse feedback.
method Environment shaping using state abstraction.
result Agent's policy in shaped environment preserves near-optimal behavior in original environment.
A new measure k-variance captures local distributional shape.
problem Summarizing distributional shape with local information.
method Random bipartite matchings and stochastic approximation.
result Easily approximated k-variance measures capture local distributional properties. Improves CNN robustness by reducing texture bias.
problem CNNs' reliance on local texture over global shape.
method Inspired by human vision, InfoDrop decorrelates model output from local texture.
result Enhanced robustness across various scenarios.
Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
Estimates mean curvature, scalar curvature, shape operator in warped products.
problem Estimating geometric properties in warped product spaces.
method Local and global upper estimates for curvature and shape operator.
result Results on pseudo-hyperbolic spaces and space forms.
New method allows sheets to morph into multiple shapes via spatially varying stimuli.
problem Limitation of current shape-programmed sheets to achieve only one target geometry.
method Patterning the stimulus itself for spatiotemporal control over local deformation magnitudes.
result A single physical sample can be induced to traverse a continuous family of target geometries.
We develop a new route through which to explore kerΨX, the kernel of the π1-shape group homomorphism determined by a general space X, and establish, for each locally path connected, paracompact Hausdorff space X, kerΨX is precisely the Spanier group of X.
A new method for 3D surface registration using dynamic programming.
problem Elastic shape registration of 3D surfaces.
method Optimization over a subset of reparametrizations using dynamic programming.
result Proposes an algorithm that produces a solution closer to optimal than gradient-based methods.
The paper characterizes law-invariant star-shaped risk measures.
problem Understanding and characterizing law-invariant star-shaped risk measures.
method Developed characterizations for positively homogeneous and star-shaped functionals, derived Kusuoka-type representations, and offered representations of general law-invariant star-shaped functionals.
result Characterizations of law-invariant star-shaped functionals, including their connections to Value-at-Risk and Expected Shortfall.
funLOCI identifies clusters in functional data.
problem Identifying similar behavior in functional data.
method Divisive hierarchical clustering with additive model.
result funLOCI reduces the number of local clusters.
Let M be a manifold or (more generally) a locally compact, metrizable ANR. If K is an attractor for a flow in M, with basin of attraction A(K), it is well known that the inclusion i:K⊆A(K) is always a shape equivalence. In this paper we investigate to what extent this generaliz…
Study of star-shaped hypersurfaces with capillary boundary using constrained mean curvature flow.
problem Understanding the evolution of hypersurfaces with capillary boundaries.
method Locally constrained mean curvature flow for star-shaped hypersurfaces in the half-space.
result Established new Alexandrov-Fenchel inequalities for convex hypersurfaces with capillary boundary.
We introduce and develop fine shape, which has a very simple definition and aims to supersede all previously known shape theories for metrizable spaces. The problem with known shape theories of metrizable spaces is illustrated by the following bizarre situation. Čech cohomology is an invariant of shape, and a fortiori …
In two former papers, the authors independently proved that the space of hyperbolic cone-3-manifolds with cone angles less than 2π and fixed singular locus is locally parametrized by the cone angles. In this sequel, we investigate the local shape of the deformation space when the singular locus is no longer fixed, i.e.…
The space of shapes of a polyhedron with given total angles less than 2πat each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take place a finite distance from a nonsingular point. The metric completion is a comple…
Study on real hypersurfaces in products of complex space forms, proving rigidity and nonexistence results.
problem Existence and properties of totally umbilical real hypersurfaces in complex space forms.
method Analyzing shape operators and local product structures in products of complex space forms.
result Nonexistence and rigidity results for totally umbilical real hypersurfaces in products of complex space forms.
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.
Crochet creates precise 2D shapes from 1D material.
problem Creating precise 2D shapes from 1D material.
method Using crochet to generate constant flat, spherical, or hyperbolic shapes.
result Crochet is the most flexible and precise method for building dynamical systems with high curvature precision.
Study shows how weak inverse anisotropic mean curvature flow behaves at infinity.
problem Understanding the asymptotic behavior of anisotropic mean curvature flow.
method Established local gradient estimates for anisotropic p-harmonic functions and weak solutions of IAMCF. result Weak IAMCF is asymptotic to the expanding Wulff shape solution at infinity.
The goal of this paper is to measure the non-convexity of compact and smooth connected components of real algebraic plane curves. We study these curves first in a general setting and then in an asymptotic one. In particular, we consider sufficiently small levels of a real bivariate polynomial in a small enough neighbou…
Study finds a non-locally contractible r-convex set.
problem Find an r-convex set which is not locally contractible. method Constructs a counterexample of a non-locally contractible r-convex set. result Proves that the class of supports with positive reach of absolutely continuous distributions includes strictly the class of r-convex supports. 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…
The paper introduces a new flow to converge to a Wulff shape from a smooth convex hypersurface.
problem Proving Alexandrov-Fenchel inequalities for anisotropic mixed volumes.
method Introducing a fully nonlinear locally constrained anisotropic curvature flow.
result The flow converges smoothly and exponentially to a scaled Wulff shape.
Unified treatment of elastic metrics for curves in any dimension.
problem Defining metrics on spaces of Euclidean curves for statistical analysis.
method Developing a unified approach to elastic metrics, extending results on existence of solutions and algorithms for computing distances and geodesics.
result Unified treatment of elastic metrics for all parameter choices, extending previous work.
Study on anisotropic curvature flow for noncompact convex hypersurfaces.
problem Anisotropic curvature flow of noncompact convex hypersurfaces.
method Flow of complete noncompact convex hypersurfaces with anisotropy determined by a Wulff shape.
result The flow exists for all positive time for initial conditions.
Purpose: Lung nodules have very diverse shapes and sizes, which makes classifying them as benign/malignant a challenging problem. In this paper, we propose a novel method to predict the malignancy of nodules that have the capability to analyze the shape and size of a nodule using a global feature extractor, as well as …
Training shapes the geometry of neural network feature maps, revealing local area magnification.
problem Understanding how training affects the geometric structure of neural network feature maps.
method Analyzing the Riemannian geometry induced by neural network feature maps at infinite width and after training.
result Training breaks the symmetry of the geometry induced by random neural network feature maps, magnifying local areas along decision boundaries.
We analyze a monetary system of random money transfer on the basis of double entry bookkeeping. Without boundary conditions, we do not reach a price equilibrium and violate text-book formulas of economists quantity theory (MV=PQ). To match the resulting quantity of money with the model assumption of a constant price, w…
We introduce the notion of multiscale covariance tensor fields (CTF) associated with Euclidean random variables as a gateway to the shape of their distributions. Multiscale CTFs quantify variation of the data about every point in the data landscape at all spatial scales, unlike the usual covariance tensor that only qua…
A new method matches similar regions in non-rigid shapes using spectra of differential operators.
problem Evaluating similarity of non-rigid shapes with partiality.
method Alignment of spectra of differential operators (SI-LBO and regular LBO) on a manifold with multiple metrics.
result Matching spectra outperforms competing methods on standard benchmarks.
Method flattens complex surfaces with consistent density and shape.
problem Shape deformations and local geometric distortions in density-equalizing maps for multiply-connected surfaces.
method Formulates density diffusion as a quasiconformal flow, solving an energy minimization problem involving the Beltrami coefficient to ensure bijectivity and control distortion.
result Achieves optimal parameterization of multiply-connected surfaces with bijective and controlled geometric distortions.