arXiv research

The paper develops algorithms for finding metrics with prescribed combinatorial curvature on polyhedral surfaces.

The paper introduces combinatorial curvature and flow for polyhedral surfaces, proving rigidity and solving the Yamabe problem.

The paper studies rigid sphere packings on 3D manifolds with boundary.

The paper introduces combinatorial Calabi flows to find hyperbolic metrics on surfaces with boundary.

The paper finds hyperbolic metrics on surfaces with boundary using combinatorial curvature flows.

We give a combinatorial characterization of generic minimal rigidity for planar periodic frameworks. The characterization is a true analogue of the Maxwell-Laman Theorem from rigidity theory: it is stated in terms of a finite combinatorial object and the conditions are checkable by polynomial time combinatorial algorit…

A framework for generating 3D shapes by sequentially assembling primitives.

In this article we give combinatorial criteria to decide whether a transitive cyclic combinatorial d-manifold can be generalized to an infinite family of such complexes, together with an explicit construction in the case that such a family exists. In addition, we substantially extend the classification of combinatorial…

In this paper, we introduce two discrete curvature flows, which are called $α$-flows on two and three dimensional triangulated manifolds. For triangulated surface $M$, we introduce a new normalization of combinatorial Ricci flow (first introduced by Bennett Chow and Feng Luo \cite{CL1}), aiming at evolving $α$ order di…

Paper introduces new flows to find circle packings with specific curvature.

Combining models in appropriate ways to achieve high performance is commonly seen in machine learning fields today. Although a large amount of combinatorial models have been created, little attention is drawn to the commons in different models and their connections. A general modelling technique is thus worth studying …

In this paper, we introduce a new combinatorial curvature on triangulated surfaces with inversive distance circle packing metrics. Then we prove that this combinatorial curvature has global rigidity. To study the Yamabe problem of the new curvature, we introduce a combinatorial Ricci flow, along which the curvature evo…

Fractional combinatorial flow improves surface conformal structures.

Every closed oriented PL 4-manifold is a branched cover of the 4-sphere branched over a PL-surface with finitely many singularities by Piergallini [Topology 34(3):497-508, 1995]. This generalizes a long standing result by Hilden and Montesinos to dimension four. Izmestiev and Joswig [Adv. Geom. 3(2):191-225, 2003] gave…

The paper solves fractional combinatorial flows for prescribed hyperbolic bordered surfaces.

New method finds metrics on surfaces with prescribed curvatures using circle packings and surgery.

A new method uses heat diffusion to efficiently solve combinatorial optimization problems.

Generalizes cohomology ring result for combinatorial line arrangements.

Proves existence of circle patterns on surfaces with cusps.

For triangulated surfaces and any $p>1$, we introduce the combinatorial $p$-th Calabi flow which precisely equals the combinatorial Calabi flows first introduced in H. Ge's thesis when $p=2$. The difficulties for the generalizations come from the nonlinearity of the $p$-th flow equation when $p\neq 2$. Adopting differe…

A new deep learning framework for topological data.

Abstract graph networks combine modular meta-learning for flexible combinatorial generalization.

Floer constructs homology from flow lines in generalized dynamical systems and combinatorial vector fields.

New combinatorial type helps distinguish plane curve topologies.

The paper studies deformation of discrete conformal structures on surfaces using combinatorial curvature flows.

Improved statistical efficiency of Thompson Sampling for combinatorial semi-bandits.

A stochastic combinatorial semi-bandit is an online learning problem where at each step a learning agent chooses a subset of ground items subject to combinatorial constraints, and then observes stochastic weights of these items and receives their sum as a payoff. In this paper, we consider efficient learning in large-s…

New combinatorial method for sparse PCA works beyond spiked identity model.

We develop a tighter implementation of basic PL topology, which keeps track of some combinatorial structure beyond PL homeomorphism type. With this technique we clarify some aspects of PL transversality and give combinatorial proofs of a number of known results. New results include a combinatorial characterization of c…

Oracle-efficient algorithms reduce combinatorial semi-bandit regret to logarithmic time.

Novel theory combines combinatorial and topological elements.

A new algorithm for top-k combinatorial bandits with full-bandit feedback.

The Waldhausen construction of Mayer-Vietoris splittings of chain complexes over an injective generalized free product of group rings is extended to a combinatorial construction of Seifert-van Kampen splittings of CW complexes with fundamental group an injective generalized free product.

New discrete conformal structures on surfaces with boundary, proving global rigidity and constructing hyperbolic metrics.

This is the first installment of a book on combinatorial and geometric group theory from the topological point of view. This is a classical subject. The installment contains Chapters 1, 3 and 4, and there are nine chapters in total: 1. Combinatorial Complexes 2. Topological Invariants 3. Coverings 4. Galois Theory 5. G…

Following the general strategy proposed by G.Rybnikov, we present a proof of his well-known result, that is, the existence of two arrangements of lines having the same combinatorial type, but non-isomorphic fundamental groups. To do so, the Alexander Invariant and certain invariants of combinatorial line arrangements a…

New algorithm detects changes in combinatorial semi-bandit rewards.

The paper studies rigidity of sphere packings on 3D manifolds with boundary.

This paper surveys RL for combinatorial optimization, focusing on TSP.

Differentiable losses for combinatorial optimization problems in sequence modeling.

New algorithms for neural bandits learn from context and arm features.

New lower bounds for combinatorial multi-armed bandits for general reward functions.

New method uses diffusion models for unsupervised combinatorial optimization.

For an integer $m\geq 1$, a combinatorial manifold $\widetilde{M}$ is defined to be a geometrical object $\widetilde{M}$ such that for $\forall p\in\widetilde{M}$, there is a local chart $(U_p,φ_p)$ enable $φ_p:U_p\to B^{n_{i_1}}\bigcup B^{n_{i_2}}\bigcup...\bigcup B^{n_{i_{s(p)}}}$ with $B^{n_{i_1}}\bigcap B^{n_{i_2}}…

We consider the homogeneous and the non-homogeneous convex relaxations for combinatorial penalty functions defined on support sets. Our study identifies key differences in the tightness of the resulting relaxations through the notion of the lower combinatorial envelope of a set-function along with new necessary conditi…

Paper uses RL to solve constrained combinatorial optimization problems.

New algorithm eliminates arms to minimize regret in complex bandit problems.

For triangulated surfaces, we introduce the combinatorial Calabi flow which is an analogue of smooth Calabi flow. We prove that the solution of combinatorial Calabi flow exists for all time. Moreover, the solution converges if and only if Thurston's circle packing exists. As a consequence, combinatorial Calabi flow pro…