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Chenyang Xu

Publications and source records attributed to Chenyang Xu.

At least 73 records · Page 4Linked to original sources
arXiv2021

Using Predicted Weights for Ad Delivery

We study the performance of a proportional weights algorithm for online capacitated bipartite matching modeling the delivery of impression ads. The algorithm uses predictions on the advertiser nodes to match arriving impression nodes fractionally in proportion to the weights of its neighbors. This paper gives a thorough empirical study of the performance of the algorithm on a data-set of ad impressions from Yahoo! and shows its superior performance compared to natural baselines such as a greedy water-filling algorithm and the ranking algorithm. The proportional weights algorithm has recently received interest in the theoretical literature where it was shown to have strong guarantees beyond the worst-case model of algorithms augmented with predictions. We extend these results to the case where the advertisers' capacities are no longer stationary over time. Additionally, we show the algorithm has near optimal performance in the random-order arrival model when the number of impressions and the optimal matching are sufficiently large.

cs.DS↗
arXiv2021

Algebraicity of the Metric Tangent Cones and Equivariant K-stability

We prove two new results on the K-polystability of Q-Fano varieties based on purely algebro-geometric arguments. The first one says that any K-semistable log Fano cone has a special degeneration to a uniquely determined K-polystable log Fano cone. As a corollary, we combine it with the differential-geometric results to complete the proof of Donaldson-Sun's Conjecture which says that the metric tangent cone of any close point appearing on a Gromov-Hausdorff limit of Kahler-Einstein Fano manifolds only depends on the algebraic structure of the singularity. The second result says that for any log Fano variety with a torus action, the K-polystability is equivalent to the equivariant K-polystability, that is, to check K-polystability, it is sufficient to check special test configurations which are equivariant under the torus action.

math.AG↗
arXiv2020

On positivity of the CM line bundle on K-moduli spaces

In this paper, we consider the CM line bundle on the K-moduli space, i.e., the moduli space parametrizing K-polystable Fano varieties. We prove it is ample on any proper subspace parametrizing reduced uniformly K-stable Fano varieties which conjecturally should be the entire moduli space. As a corollary, we prove that the moduli space parametrizing smoothable K-polystable Fano varieties is projective. During the course of proof, we develop a new invariant for filtrations which can be used to test various K-stability notions of Fano varieties.

math.AG↗
arXiv2020

Reductivity of the automorphism group of K-polystable Fano varieties

We prove that K-polystable log Fano pairs have reductive automorphism groups. In fact, we deduce this statement by establishing more general results concerning the S-completeness and $Θ$-reductivity of the moduli of K-semistable log Fano pairs. Assuming the conjecture that K-semistability is an open condition, we prove that the Artin stack parametrizing K-semistable Fano varieties admits a separated good moduli space.

math.AG↗
arXiv2020

Uniqueness of the minimizer of the normalized volume function

We confirm a conjecture of Chi Li which says that the minimizer of the normalized volume function for a klt singularity is unique up to rescaling. This is achieved by defining stability thresholds for valuations, and then showing that a valuation is a minimizer if and only if it is K-semistable, and that K-semistable valuation is unique up to rescaling. As applications, we prove a finite degree formula for volumes of klt singularities and an effective bound of the local fundamental group of a klt singularity.

math.AG↗
arXiv2019

A Minimizing Valuation is Quasi-monomial

We prove a version of Jonsson-Mustaţǎ's Conjecture, which says for any graded sequence of ideals, there exists a quasi-monomial valuation computing its log canonical threshold. As a corollary, we confirm Chi Li's conjecture that a minimizer of the normalized volume function is always quasi-monomial. Applying our techniques to a family of klt singularities, we show that the volume of klt singularities is a constructible function. As a corollary, we prove that in a family of klt log Fano pairs, the K-semistable ones form a Zariski open set. Together with [Jia17], we conclude that all K-semistable klt Fano varieties with a fixed dimension and volume are parametrized by an Artin stack of finite type, which then admits a separated good moduli space by [BX18, ABHLX19], whose geometric points parametrize K-polystable klt Fano varieties.

math.AG↗
arXiv2019

Uniqueness of K-polystable degenerations of Fano varieties

We prove that K-polystable degenerations of Q-Fano varieties are unique. Furthermore, we show that the moduli stack of K-stable Q-Fano varieties is separated. Together with [Jia17,BL18], the latter result yields a separated Deligne-Mumford stack parametrizing all uniformly K-stable Q-Fano varieties of fixed dimension and volume. The result also implies that the automorphism group of a K-stable Q-Fano variety is finite.

math.AG↗
arXiv2019

Stability of Valuations: Higher Rational Rank

Given a klt singularity $x\in (X, D)$, we show that a quasi-monomial valuation $v$ with a finitely generated associated graded ring is the minimizer of the normalized volume function $\widehat{\rm vol}_{(X,D),x}$, if and only if $v$ induces a degeneration to a K-semistable log Fano cone singularity. Moreover, such a minimizer is unique among all quasi-monomial valuations up to rescaling. As a consequence, we prove that for a klt singularity $x\in X$ on the Gromov-Hausdorff limit of Kähler-Einstein Fano manifolds, the intermediate K-semistable cone associated to its metric tangent cone is uniquely determined by the algebraic structure of $x\in X$, hence confirming a conjecture by Donaldson-Sun.

math.AG↗
arXiv2019

Multiple Constrained Routing Algorithms in Large-Scaled Software Defined Networks

In this paper, we consider the bandwidth-delay-hop constrained routing problem in large-scaled software defined networks. A number of demands, each of which specifies a source vertex and a sink vertex, are required to route in a given network. We are asked to select a subset of demands, and assign a routing path for each selected demand without violating the hop and delay constraints, while assuring that the bandwidth occupied in each edge is not beyond its capacity. The goal is to maximize the throughput (the total bandwidth) of the selected demands. We develop an efficient heuristic algorithm for the problem, which consists of three main steps, namely, computing feasible paths for each demand, sorting the demands with some priority rules, selecting a path for each demand. The algorithm is tested with networks of actual sizes and topologies, generated by Huawei Technologies Company. The experiments show that the proposed approach outperforms existing algorithms both in throughput and in running time. In the experiments, our algorithm achieves more than 90\% of the total bandwidth of the given demands within 10 seconds. Moreover, a large part of our algorithm can run in parallel which largely speeds up the process when using multi-core processors.

cs.NI↗
arXiv2019

The non-archimedean SYZ fibration

We construct non-archimedean SYZ fibrations for maximally degenerate Calabi-Yau varieties, and we show that they are affinoid torus fibrations away from a codimension two subset of the base. This confirms a prediction by Kontsevich and Soibelman. We also give an explicit description of the induced integral affine structure on the base of the SYZ fibration. Our main technical tool is a study of the structure of minimal dlt-models along one-dimensional strata.

math.AG↗
arXiv2019

K-stability of cubic threefolds

We prove the K-moduli space of cubic threefolds is identical to their GIT moduli. More precisely, the K-(semi,poly)-stability of cubic threefolds coincide to the corresponding GIT stabilities, which could be explicitly calculated. In particular, this implies that all smooth cubic threefolds admit Kähler-Einstein metric as well as provides a precise list of singular KE ones. To achieve this, our main new contribution is an estimate in dimension three of the volumes of kawamata log terminal singularities introduced by Chi Li. This is obtained via a detailed study of the classification of three dimensional canonical and terminal singularities, which was established during the study of the explicit three dimensional minimal model program.

math.AG↗
arXiv2019

On the proper moduli spaces of smoothable Kähler-Einstein Fano varieties

In this paper, we investigate the geometry of the orbit space of the closure of the subscheme parametrizing smooth Fano Kähler-Einstein manifolds inside an appropriate Hilbert scheme. In particular, we prove that being K-semistable is a Zariski open condition and establish the uniqueness for the Gromov-Hausdorff limit for a punctured flat family of Fano Kähler-Einstein manifolds. Based on these, we construct a proper scheme parameterizing the S-equivalent classes of $\QQ$-Gorenstein smoothable, K-semistable Fano varieties, and verify various necessary properties to guarantee that it is a good moduli space.

math.AG↗
arXiv2018

Stability of Valuations and Kollár Components

We prove that among all Kollár components obtained by plt blow ups of a klt singularity $o \in (X, D)$, there is at most one that is (log-)K-semistable. We achieve this by showing that if such a Kollár component exists, it uniquely minimizes the normalized volume function introduced in [Li15a] among all divisorial valuations. Conversely, we show any divisorial minimizer of the normalized volume function yields a K-semistable Kollár component. We also prove that for any klt singularity, the infimum of the normalized function is always approximated by the normalized volumes of Kollár components.

math.AG↗