arXiv research

Mathematical framework for minimum enclosing ball problem.

Paper offers a method for finding the smallest sphere enclosing a set in d-dimensional space.

In this paper we study the curvature flow of a curve in a plane endowed with a minkowskian norm whose unit ball is smooth. We show that many of the properties known in the euclidean case can be extended (with due adaptations) to this new situation. In particular, we show that simple, closed, strictly convex, smooth cur…

We present a streaming model for large-scale classification (in the context of $\ell_2$-SVM) by leveraging connections between learning and computational geometry. The streaming model imposes the constraint that only a single pass over the data is allowed. The $\ell_2$-SVM is known to have an equivalent formulation in …

Smooth surface encloses less volume than a ball.

Let X be a data matrix of rank ρ, whose rows represent n points in d-dimensional space. The linear support vector machine constructs a hyperplane separator that maximizes the 1-norm soft margin. We develop a new oblivious dimension reduction technique which is precomputed and can be applied to any input matrix X. We pr…

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather in…

We study the stability of capillary hypersurfaces in a unit Euclidean ball. It is proved that if the mass center of the generalized body enclosed by the immersed capillary hypersurface and the wetted part of the sphere is located at the origin, then the hypersurface is unstable. An immediate result is that all known ex…

In this paper, we study a mean curvature type flow with capillary boundary in the unit ball. Our flow preserves the volume of the bounded domain enclosed by the hypersurface, and monotonically decreases an energy functional $E$. We show that it has the longtime existence and subconverges to spherical caps. As an applic…

We give two provably accurate feature-selection techniques for the linear SVM. The algorithms run in deterministic and randomized time respectively. Our algorithms can be used in an unsupervised or supervised setting. The supervised approach is based on sampling features from support vectors. We prove that the margin i…

New shapes enclose less volume than the sphere, surprising in 3D.

For any closed Riemannian manifold $X$ we prove that large isoperimetric regions in $X\times{\mathbb R}^n$ are of the form $X\times$(Euclidean ball). We prove that if $X$ has non-negative Ricci curvature then the only soap bubbles enclosing a large volume are the products $X\times$(Euclidean sphere). We give an example…

Proves spheres with bounded curvatures must contain a unit ball.

Let $M^n$ be a closed convex hypersurface lying in a convex ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close and almost isometric to a…

New proof shows origin-centred balls are unique solutions to curvature problems.

Training model to generate data has increasingly attracted research attention and become important in modern world applications. We propose in this paper a new geometry-based optimization approach to address this problem. Orthogonal to current state-of-the-art density-based approaches, most notably VAE and GAN, we pres…

Large PL surfaces in homology balls can have arbitrarily high genus.

We propose a new topic modeling procedure that takes advantage of the fact that the Latent Dirichlet Allocation (LDA) log likelihood function is asymptotically equivalent to the logarithm of the volume of the topic simplex. This allows topic modeling to be reformulated as finding the probability simplex that minimizes …

Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.

Let $M^n$ be a closed immersed hypersurface lying in a contractible ball $B(p,R)$ of the ambient $(n+1)$-manifold $N^{n+1}$. We prove that, by pinching Heintze-Reilly's inequality via sectional curvature upper bound of $B(p,R)$, 1st eigenvalue and mean curvature of $M$, not only $M$ is Hausdorff close to a geodesic sph…

We introduce the non-pure versions of simplicial balls and spheres with minimum number of vertices. These are a special type of non-homogeneous balls and spheres (NH-balls and NH-spheres) satisfying a minimality condition on the number of maximal simplices. The main result is that minimal NH-balls and NH-spheres are pr…

Upper bound found for first nonzero Steklov eigenvalue.

Paper calculates ball number of links using Lorentz geometry and circle packing.

The paper explores why a specific type of predictor works well in noisy data.

In this article we study the shape of a compact surface of constant mean curvature of Euclidean space whose boundary is contained in a round sphere. We consider the case that the boundary is prescribed or that the surface meets the sphere with a constant angle. We study under what geometric conditions the surface must …

A curve of minimum length to enclose a unit sphere in 3D is at least 4π.

A coreset (or core-set) of an input set is its small summation, such that solving a problem on the coreset as its input, provably yields the same result as solving the same problem on the original (full) set, for a given family of problems (models, classifiers, loss functions). Over the past decade, coreset constructio…

The concordance genus of a knot K is the minimum Seifert genus of all knots smoothly concordant to K. Concordance genus is bounded below by the 4-ball genus and above by the Seifert genus. We give a lower bound for the concordance genus of K coming from the knot Floer complex of K. As an application, we prove that ther…

Formula derived for enclosed volume of CMC surfaces in 3-sphere.

We define the concordance crosscap number of a knot as the minimum crosscap number among all the knots concordant to the knot. The four-dimensional crosscap number is the minimum first Betti number of non-orientable surfaces smoothly embedded in 4-dimensional ball, bounding the knot. Clearly the 4-dimensional crosscap …

The generalized soap bubble problem seeks the least perimeter way to enclose and separate n given volumes in R^m. We study the possible configurations for perimeter minimizing bubble complexes enclosing more than two regions. We prove that perimeter minimizing planar bubble complexes with equal pressure regions and wit…

In this article we study point configurations minimizing the discrete energy on a compact Riemannian manifold, where the energy kernel is taken to be the Green's function for the Laplacian. We show that every point in a minimizing configuration lies inside an open set called harmonic ball where no other point can enter…

New examples show flip distance and polyhedron triangulation numbers differ, with ratio close to 3/2.

The paper quantifies how much of a 4-ball must be removed to squeeze into a cylinder, proving a lower bound on the Minkowski dimension.

Optimizes minimum-volume prediction sets for multivariate regression.

The classical isoperimetric inequality in R^3 states that the surface of smallest area enclosing a given volume is a sphere. We show that the least area surface enclosing two equal volumes is a double bubble, a surface made of two pieces of round spheres separated by a flat disk, meeting along a single circle at an ang…

Uniform convergence of interpolators proven for Gaussian data.

Generic smooth boundaries for isoperimetric regions in 8D manifolds.

Study non-orientable 4-genus for 11-crossing non-alternating knots.

In blind hyperspectral unmixing (HU), the pure-pixel assumption is well-known to be powerful in enabling simple and effective blind HU solutions. However, the pure-pixel assumption is not always satisfied in an exact sense, especially for scenarios where pixels are heavily mixed. In the no pure-pixel case, a good blind…

We show that any star-shaped convex hypersurface with constant Weingarten curvature in the deSitter-Schwarzschild manifold is a sphere of symmetry. Moreover, we study an isoperimetric problem for bounded domains in the doubled Schwarzschild manifold. We prove the existence of an isoperimetric surface for any value of t…

Study on stability of network flow shrinkers with findings on instability of specific shapes.

Study $σ_2$-curvature and volume of compact manifolds, proving conditions for Einstein metrics and geodesic balls.

We investigate adversarial robustness of Gaussian Process Classification (GPC) models. Given a compact subset of the input space $T\subseteq \mathbb{R}^d$ enclosing a test point $x^*$ and a GPC trained on a dataset $\mathcal{D}$, we aim to compute the minimum and the maximum classification probability for the GPC over …

Optimization algorithms help overparameterized neural networks achieve high performance.

We analyze a gradient flow of closed planar curves minimizing the anisoperimetric ratio. For such a flow the normal velocity is a function of the anisotropic curvature and it also depends on the total interfacial energy and enclosed area of the curve. In contrast to the gradient flow for the isoperimetric ratio, we sho…

Develop an ABP approach to Sobolev and Michael-Simon inequalities beyond Euclidean volume growth.

We prove that the standard double bubble provides the least-area way to enclose and separate two regions of prescribed volume in \Bbb R^3.