arXiv research

A fast algorithm for $K$-means clustering using subsampled SDP.

The nearest-centroid classifier is a simple linear-time classifier based on computing the centroids of the data classes in the training phase, and then assigning a new datum to the class corresponding to its nearest centroid. Thanks to its very low computational cost, the nearest-centroid classifier is still widely use…

A new classification method using disjoint centroids and normalized distance.

Multilayer bootstrap network builds a gradually narrowed multilayer nonlinear network from bottom up for unsupervised nonlinear dimensionality reduction. Each layer of the network is a nonparametric density estimator. It consists of a group of k-centroids clusterings. Each clustering randomly selects data points with r…

Gradient descent with biased rounding errors converges faster under certain conditions.

Round surgery diagrams represent 3-manifolds in $S^3$.

Contact round surgeries on $(\mathbb{S}^3,ξ_{st})$ help in constructing and understanding contact 3-manifolds.

In this article, we extend Huisken's theorem that convex surfaces flow to round points by mean curvature flow. We construct certain classes of mean convex and non-mean convex hypersurfaces that shrink to round points and use these constructions to create pathological examples of flows. We find a sequence of flows that …

Optimizes sample and round complexity in adaptive sampling from multiple distributions.

Individual's semantics have been used for guiding the learning process of Genetic Programming solving supervised learning problems. The semantics has been used to proposed novel genetic operators as well as different ways of performing parent selection. The latter is the focus of this contribution by proposing three he…

Consider an analytic map of a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has r…

New findings show infinitely many knots cannot be smoothly round handle slices.

We discuss the integrability of orthogonal almost complex structures on Riemannian products of even-dimensional round spheres and give a partial answer to the question raised by E. Calabi concerning the existence of complex structures on a product manifold of a round 2-sphere and a round 4-sphere.

This work investigates how multi-round reasoning improves LLM performance.

New findings on $κ$-solutions with round cylinder as asymptotic shrinker.

Round balls minimize liquid drop model volumes ≤ 1.

Gradient descent stagnates in low-precision, but unbiased rounding schemes improve convergence.

Round cylinders are rigid in Ricci shrinkers close to the standard product.

Deep neural networks have been shown to suffer from a surprising weakness: their classification outputs can be changed by small, non-random perturbations of their inputs. This adversarial example phenomenon has been explained as originating from deep networks being "too linear" (Goodfellow et al., 2014). We show here t…

A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.

We show that if the entropy of any closed hypersurface is close to that of a round hyper-sphere, then it is close to a round sphere in Hausdorff distance. Generalizing the result of \cite{BW1} to higher dimensions.

In this paper, we study the limiting behavior of the Brown-York mass and Hawking mass along nearly round surfaces at infinity of an asymptotically flat manifold. Nearly round surfaces can be defined in an intrinsic way. Our results show that the ADM mass of an asymptotically flat 3-manifold can be approximated by some …

The study characterizes round spheres in Euclidean space based on r-mean curvature conditions.

Contact round surgery of contact 3-manifolds is introduced in this paper. By using this method, an alternative proof of the existence of a contact structure on any closed orientable 3-manifold is given. It is also proved that any contact structure on any closed orientable 3-manifold is constructed from the standard con…

In recent work, the notion of Double Convexity for a foliation of a conical null hypersurface was introduced to give a proof, if satisfied, of the Null Penrose Inequality. Double Convexity constrains the geometry of a Marginally Outer Trapped Surface (MOTS), called a quasi-round MOTS. In the first part of this paper, f…

Study cohomology rings of 3D manifolds with round fold maps into the plane.

Research explores real algebraic realization of round fold maps of codimension -1.

We classify the radially symmetric connections in vector bundles over round spheres by proving that they are all parallel.

New Einstein metrics found on a 10-dimensional sphere.

New compact mean convex hypersurfaces found for positive λ.

We prove that the well-rounded retract of SO_n\SL_n(R) is a minimal SL_n(Z)-invariant spine.

We present the Round Handle Problem, proposed by Freedman and Krushkal. It asks whether a collection of links, which contains the Generalised Borromean Rings, are slice in a 4-manifold R constructed from adding round handles to the four ball. A negative answer would contradict the union of the surgery conjecture and th…

High-frequency traders can act as either small informed traders or round-trippers, affecting price discovery and liquidity.

One-round FL method improves robustness and reduces communication rounds.

Eigenvalues on spheres are compared to the unit round sphere, proving a sharp bound and equality condition.

This paper studies a deformation retraction of Teichmüller space and its analogy with well-rounded retractions.

We consider the problem of learning a general graph $G=(V,E)$ using edge-detecting queries, where the number of vertices $|V|=n$ is given to the learner. The information theoretic lower bound gives $m\log n$ for the number of queries, where $m=|E|$ is the number of edges. In case the number of edges $m$ is also given t…

Proposes rounding method for precise treatment effect estimation under budget constraints.

AdaRound improves post-training quantization of neural networks.

In this paper, we construct round fold maps or stable fold maps with concentric singular value sets introduced by the author on smooth bundles over spheres or bundles over more general manifolds. The class of round fold maps includes special generic maps on spheres and such maps have been constructed on smooth bundles …

CRA improves UL-based CO solvers by dynamically smoothing and enforcing discreteness.

Proposes a flexible tournament design combining knockout and round-robin.

A new algorithm reduces communication rounds for distributed convex optimization.

Authors construct hypertori with constant negative mean curvature in a sphere.

3D manifolds can map to a plane with specific curve patterns.

In this short note, we review the well-known result that there is no orthogonal complex structure on the 6-sphere with respect to the round metric.

We introduce cosymplectic circles and cosymplectic spheres, which are the analogues in the cosymplectic setting of contact circles and contact spheres. We provide a complete classification of compact 3-manifolds that admit a cosymplectic circle. The properties of tautness and roundness for a cosymplectic $p$-sphere are…

In low dimensions, minimizers for the second conformal eigenvalue do not exist near the round sphere.