arXiv subjects

Xiaokai Liu

Publications and source records attributed to Xiaokai Liu.

5 recordsLinked to original sources

arXiv2026

The Ergodicity of Geodesic Flows on Rank One Manifolds of Nonpositive Curvature

In this paper, we study the ergodicity of the geodesic flows on closed rank one manifolds of nonpositive sectional curvature. We introduce a subset of the manifold defined by the infinite-order vanishing of the Gaussian curvature in dimension two, or of the fiberwise second moment of the reduced Jacobi determinant in arbitrary dimensions, and derive bounds on the Liouville measure and Hausdorff dimension of the singular set in terms of this subset. In particular, if this subset has zero volume, then the singular set has zero Liouville measure, and the geodesic flow is ergodic with respect to Liouville measure. By extending this method to real analytic metrics, we characterize the singular set as the set of zeros of a nontrivial real analytic determinant constructed from curvature operators, thereby establishing ergodicity in this setting without any additional assumptions.

math.DS↗

arXiv2023

The Hopf-Tsuji-Sullivan Dichotomy on Visibility Manifolds Without Conjugate Points

In this article, we establish the Hopf-Tsuji-Sullivan dichotomy for geodesic flows on certain manifolds with no conjugate points: either the geodesic flow is conservative and ergodic, or it is completely dissipative and non-ergodic. We also show several equivalent conditions to the conservativity, such the Poincaré series diverges at the critical exponent, the conical limit set has full Patterson-Sullivan measure, etc.

math.DS↗

arXiv2022

MATT: A Multiple-instance Attention Mechanism for Long-tail Music Genre Classification

Imbalanced music genre classification is a crucial task in the Music Information Retrieval (MIR) field for identifying the long-tail, data-poor genre based on the related music audio segments, which is very prevalent in real-world scenarios. Most of the existing models are designed for class-balanced music datasets, resulting in poor performance in accuracy and generalization when identifying the music genres at the tail of the distribution. Inspired by the success of introducing Multi-instance Learning (MIL) in various classification tasks, we propose a novel mechanism named Multi-instance Attention (MATT) to boost the performance for identifying tail classes. Specifically, we first construct the bag-level datasets by generating the album-artist pair bags. Second, we leverage neural networks to encode the music audio segments. Finally, under the guidance of a multi-instance attention mechanism, the neural network-based models could select the most informative genre to match the given music segment. Comprehensive experimental results on a large-scale music genre benchmark dataset with long-tail distribution demonstrate MATT significantly outperforms other state-of-the-art baselines.

cs.SD↗

arXiv2021

Some Dynamical Properties on Manifolds with no Conjugate Points

In this article, we study the dynamics of geodesic flows on Riemannian (not necessarily compact) manifolds with no conjugate points. We prove the Anosov Closing Lemma, the local product structure, and the transitivity of the geodesic flows on $Ω_1$ under the conditions of bounded asymptote and uniform visibility. As an application, we further discuss about some generic properties of the set of invariant probability measures

math.DS↗

arXiv2018

On the mixing and Bernoulli properties for geodesic flows on rank 1 manifolds without focal points

If $(M,g)$ is a smooth compact rank $1$ Riemannian manifold without focal points, it is shown that the measure $μ_{\max}$ of maximal entropy for the geodesic flow is unique. In this article, we study the statistic properties and prove that this unique measure $μ_{\max}$ is mixing. Stronger conclusion that the geodesic flow on the unit tangent bundle $SM$ with respect to $μ_{\max}$ is Bernoulli is acquired provided $M$ is a compact surface with genus greater than one and no focal points.

math.DS↗