arXiv research

Normalization methods play an important role in enhancing the performance of deep learning while their theoretical understandings have been limited. To theoretically elucidate the effectiveness of normalization, we quantify the geometry of the parameter space determined by the Fisher information matrix (FIM), which als…

Study 3D shapes in 5D space with sharp points.

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev $10$:$368-370$, $1968$) on a star-shaped bounded domain in $\mathbb{R}^2$. Let $Ω$ be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenv…

Proposes DILATE and STRIPE++ for precise time series forecasting.

Sharp reverse affine isoperimetric inequalities for asymmetric Wulff shapes and their polars are established, along with the characterization of all extremals. These new inequalities have as special cases previously obtained simplex inequalities by Ball, Barthe and Lutwak, Yang, and Zhang. In particular, they provide t…

Fine shape of local compacta represented by ordinary maps.

Sharp upper bounds derived for capacities in hyperbolic and Euclidean spaces.

Study on shape optimization for specific eigenvalue problems on domains.

Study reveals pathological eigenvalue spectra in FIM and its variants of DNNs.

Proves a Minkowski inequality for star-shaped hypersurfaces in warped cylinders.

The study finds minimal hypersurfaces in wedge-shaped manifolds with boundary.

In this paper, we use the inverse curvature flow to prove a sharp geometric inequality on star-shaped and two-convex hypersurface in hyperbolic space.

Local minimizers are convex and 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.

Sharp Veronese rigidity theorem for submanifolds of unit ball.

Novel method for shape optimization of non-smooth PDEs.

Fine shape theory extends strong shape to noncompact metrizable spaces.

Estimates true Sharpe ratio of selected assets with various methods.

Sharp stability of Alexandrov's theorem for $C^1$ domains in the small-excess regime

Truncated SGD with heavy-tailed noise eliminates sharp local minima.

PointGMM learns hGMMs from point clouds for 3D shape representation.

Study eigenvalues and shapes, proving sharp inequalities for Steklov eigenvalues.

Proves local maximizers for higher Ekeland-Hofer capacities in 4D star-shaped domains.

Let $Ω$ be a star-shaped bounded domain in $(\mathbb{S}^{n}, ds^{2})$ with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in $Ω.$ This result is the generalization of a result given by Kuttler and Sigillito for a star-shaped bounded doma…

The paper proves a Whitehead theorem for fine shape spaces.

The paper characterizes dynamic return and star-shaped risk measures via BSDEs.

The paper establishes sharp geometric inequalities for hypersurfaces in warped product manifolds.

The paper shows how to use fine shape to understand infinite-dimensional spaces.

Develops local curvature estimates for mean curvature flow.

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…

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the resu…

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…

We find a new monotone increasing quantity along smooth solutions to the inverse mean curvature flow in $\mathbb R^n$. As an application, we derive a sharp geometric inequality for mean convex, star-shaped hypersurfaces which relates the volume enclosed by a hypersurface to a weighted total mean curvature of the hypers…

The ACS criterion is verified for specific hypersurfaces in unit spheres.

A new model explains U- and Swoosh-shaped stock price recovery during the COVID-19.

The paper encodes local shapes of polynomial curves using permutations.

Flat minimal hypersurfaces found in wedge-shaped domains.

Quantitative metric spaces study function shapes and sphere diameters.

A new method shapes reinforcement learning environments by abstracting large state spaces.

Sharp conditions found for solving heat equation on Riemannian manifolds.

New curvature measures characterize non-convex Wulff shapes in normed spaces.

A new measure $k$-variance captures local distributional shape.

Sharp estimate for nodal domains intersecting a ball on a Riemannian manifold.

Sharp bounds for anisotropic p-capacity of Euclidean compact sets derived using flow methods.

The paper classifies minimizers for a specific inequality in Euclidean balls.

Improves CNN robustness by reducing texture bias.

The paper simplifies quickshift hyperparameter tuning for larger images.