arXiv research

The paper characterizes sets with infinite hyperbolic convex hull volume.

Study shows non-compact convex hulls in certain metric spaces.

Deep learning models generalize by extending decision boundaries outside the convex hull of training data.

The paper transforms a convex hull into a concave surface around a point cloud.

New proofs given for space curves with totally positive torsion.

Estimates convex hulls of smooth function images with error bounds.

For a convex curve in an even-dimensional affine space we introduce a series of convex domains (called Young hulls), describe their structure and give a formulas fo the volume of the biggest of these domains. This paper is an attempt to generalize the classical isoperimetric inequality for the volume of the convex hull…

Study convex hulls of orbits for compact groups, defining new invariants related to polynomial degrees.

We obtain an upper bound for the volume of the convex hull of a simple closed Frenet curve with exactly four vertices, i.e., four points of vanishing torsion, and lying on the boundary of its convex hull. Moreover, we show that the upper bound is attained when the curve intersects every plane in at most four points, a …

Study finds knots with ideal length need not have smallest volume.

Optimal algorithm finds if point is in convex hull of distributions.

Optimum in Convex Hulls (OCH) generalizes clinical trial results to broader populations.

Estimate collapsibility of causal effects in CPDAGs via strong d-convex hulls.

Sketching algorithm finds closest point on convex hull efficiently.

New interpretation of discrete conformality using polyhedral convex hulls.

Study on curvature bounds for specific hypersurfaces in Anti-de Sitter space.

The main result is a direct proof of the implication $(LVKF_{k,3})\Rightarrow( LT_{3k-1,3})$ below. Consider the following statements: ($LVKF_{1,3}$) From any 11 points in $\mathbb{R}^{3}$ one can choose 3 pairwise disjoint triples whose convex hulls have a common point. ($LVKF_{k,3}$) From any $6k + 5$ points in $ \m…

New approach to convex hulls for low-rank problems.

The paper develops mixed-integer formulations for neural networks using partitioning.

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

We show that for any extreme curve in a 3-manifold M, there exist a canonical mean convex hull containing all least area disks spanning the curve. Similar result is true for asymptotic case in hyperbolic 3-space such that for any asymptotic curve, there is a canonical mean convex hull containing all minimal planes span…

We show how to construct the nonstandard hull of certain infinite-dimensional Lie algebras in order to generalize a theorem of Pestov on the enlargeability of Banach-Lie algebras. In the process, we consider a nonstandard smoothness condition on functions between locally convex spaces to ensure that the induced functio…

Dual explanation method using convex hulls and example-based vectors.

The integer hull of a polyhedron is the convex hull of the integer points contained in it. We show that the vertices of the integer hulls of a rational family of polyhedra of size O(n) have quasipolynomial coordinates. As a corollary, we show that the stable commutator length of elements in a surgery family is a ratio …

GraphHull models networks with clear multi-scale explanations of community structure.

We define two non-linear operations with random (not necessarily closed) sets in Banach space: the conditional core and the conditional convex hull. While the first is sublinear, the second one is superlinear (in the reverse set inclusion ordering). Furthermore, we introduce the generalised conditional expectation of r…

The convex hull of a set K in space consists of points which are, in a certain sense, "surrounded" by K. When K is a closed curve, we define its higher hulls, consisting of points which are "multiply surrounded" by the curve. Our main theorem shows that if a curve is knotted then it has a nonempty second hull. This pro…

Paper develops compact formulations for optimization problems with rank-one convex functions and indicator variables.

Traditional nearest points methods use all the samples in an image set to construct a single convex or affine hull model for classification. However, strong artificial features and noisy data may be generated from combinations of training samples when significant intra-class variations and/or noise occur in the image s…

Novel technique reduces Bayesian network complexity while preserving inference accuracy.

New conditions ensure Dantzig-Wolfe relaxation matches rank-constrained optimization problems.

We prove that if an analytic subset $A$ of a linear metric space $X$ is not contained in a $σZ_ω$-subset of $X$ then for every Polish convex set $K$ with dense affine hull in $X$ the sum $A+K$ is non-meager in $X$ and the sets $A+A+K$ and $A-A+K$ have non-empty interior in the completion $\bar X$ of $X$. This implies t…

The paper studies the convex hull of random points in a triangle, focusing on the asymptotic behavior and phase transitions.

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism $φ$, we study the relation between the width of the convex hull of th…

We consider learning a convex combination of basis models, and present some new theoretical and empirical results that demonstrate the effectiveness of a greedy approach. Theoretically, we first consider whether we can use linear, instead of convex, combinations, and obtain generalization results similar to existing on…

A new method solves convex optimization on curved spaces.

Positive weights improve kernel quadrature's accuracy.

We show that there exists a universal constant C>0 such that the convex hull of any N points in the hyperbolic space H^n is of volume smaller than C N, and that for any dimension n there exists a constant C_n > 0 such that for any subset A of H^n, Vol(Conv(A_1)) < C_n Vol(A_1) where A_1 is the set of points of hyperbol…

If $Γ$ is the range of a Jordan curve that bounds a convex set in $\mathbb{R}^2,$ then $\frac{1}{2}(Γ+Γ)=\mathsf{co}(Γ),$ where $+$ is the Minkowski sum and $\mathsf{co}$ is the convex hull. Answering a question of V.N. Ushakov, we construct a simple closed curve in $\mathbb{R}^3$ with range $Γ$ such that $\frac{1}{2}(…

A curve around a sphere must be at least 4π long.

One-harmonic maps from a curved surface to hyperbolic plane have specific interior properties.

We present a numerical algorithm for nonnegative matrix factorization (NMF) problems under noisy separability. An NMF problem under separability can be stated as one of finding all vertices of the convex hull of data points. The research interest of this paper is to find the vectors as close to the vertices as possible…

Soap films hanging from a wire frame are studied in the framework of capillarity theory. Minimizers in the corresponding variational problem are known to consist of positive volume regions with boundaries of constant mean curvature/pressure, possibly connected by "collapsed" minimal surfaces. We prove here that collaps…

Sullivan showed that there exists $K_0$ such that if $Ω\subset \hat{\mathbb{C}}$ is a simply connected hyperbolic domain, then there exists a conformally natural $K_0$-quasiconformal map from $Ω$ to the boundary ${\rm Dome}(Ω)$ of the convex hull of its complement which extends to the identity on $\partialΩ$. Explicit …

Closed surfaces minimize total curvature in curved spaces.

The paper tackles sampling biases by ensuring minority groups are adequately represented in training data.

NNLMs optimize poorly for word probabilities due to embedding space structure.

We develop a new method to price SOFR futures contracts considering convexity, skew, and smile.