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Jinhua Zhang

Publications and source records attributed to Jinhua Zhang.

31 records · Page 2Linked to original sources
arXiv2023

Twin support vector quantile regression

We propose a twin support vector quantile regression (TSVQR) to capture the heterogeneous and asymmetric information in modern data. Using a quantile parameter, TSVQR effectively depicts the heterogeneous distribution information with respect to all portions of data points. Correspondingly, TSVQR constructs two smaller sized quadratic programming problems (QPPs) to generate two nonparallel planes to measure the distributional asymmetry between the lower and upper bounds at each quantile level. The QPPs in TSVQR are smaller and easier to solve than those in previous quantile regression methods. Moreover, the dual coordinate descent algorithm for TSVQR also accelerates the training speed. Experimental results on six artiffcial data sets, ffve benchmark data sets, two large scale data sets, two time-series data sets, and two imbalanced data sets indicate that the TSVQR outperforms previous quantile regression methods in terms of the effectiveness of completely capturing the heterogeneous and asymmetric information and the efffciency of the learning process.

stat.ML↗
arXiv2020

Ergodic optimization for some dynamical systems beyond uniform hyperbolicity

In this paper, we show that for several interesting systems beyond uniform hyperbolicity, any generic continuous function has a unique maximizing measure with zero entropy. In some cases, we also know that the maximizing measure has full support. These interesting systems including singular hyperbolic attractors, $C^\infty$ surface diffeomorphisms and diffeomorphisms away from homoclinic tangencies.

math.DS↗
arXiv2020

On the notions of singular domination and (multi-)singular hyperbolicity

The properties of uniform hyperbolicity and dominated splitting have been introduced to study the stability of the dynamics of diffeomorphisms. One meets difficulties when one tries to extend these definitions to vector fields and Shantao Liao has shown that it is more relevant to consider the linear Poincaré flow rather than the tangent flow in order to study the properties of the derivative. In this paper we define the notion of singular domination, an analog of the dominated splitting for the linear Poincaré flow which is robust under perturbations. Based on this, we give a new definition of multi-singular hyperbolicity which is equivalent to the one recently introduced by Bonatti-da Luz. The novelty of our definition is that it does not involve the blowup of the singular set and the renormalization cocycle of the linear flows.

math.DS↗
arXiv2019

Transitive partially hyperbolic diffeomorphisms with one-dimensional neutral center

In this paper, we study transitive partially hyperbolic diffeomorphisms with one-dimensional topologically neutral center, meaning that the length of the iterate of small center segments remains small. Such systems are dynamically coherent. We show that there exists a continuous metric along the center foliation which is invariant under the dynamics. As an application, we classify the transitive partially hyperbolic diffeomorphisms on 3-manifolds with topologically neutral center.

math.DS↗
arXiv2018

Empirical measures of partially hyperbolic attractors

In this paper, we study the limit measures of the empirical measures of Lebesgue almost every point in the basin of a partially hyperbolic attractor. They are strongly related to a notion named Gibbs u-state, which can be defined in a large class of diffeomorphisms with less regularity and which is the same as Pesin-Sinai's notion for partially hyperbolic attractors of $C^{1+α}$ diffeomorphisms. In particular, we prove that for partially hyperbolic $C^{1+α}$ diffeomorphisms with one-dimensional center, and for Lebesgue almost every point: (1) the center Lyapunov exponent is well defined, but (2) the sequence of empirical measures may not converge. We also give some consequences on SRB measures and large deviations.

math.DS↗
arXiv2017

Partially hyperbolic diffeomorphisms with one-dimensional neutral center on 3-manifolds

We prove that for any partially hyperbolic diffeomorphism with one dimensional neutral center on a 3-manifold, the center stable and center unstable foliations are complete; moreover, each leaf of center stable and center unstable foliations is a cylinder, a M$\ddot{o}$bius band or a plane. Further properties of the Bonatti-Parwani-Potrie type of partially hyperbolic diffeomorphisms are studied. Such examples are obtained by composing the time $m$-map (for $m>0$ large) of a non-transitive Anosov flow $ϕ_t$ on an orientable 3-manifold with Dehn twists along some transverse tori, and the examples are partially hyperbolic with one-dimensional neutral center. We prove that the center foliation gives a topologically Anosov flow which is topologically equivalent to $ϕ_t$. We also prove that for the precise example constructed by Bonatti-Parwani-Potrie, the center stable and center unstable foliations are robustly complete.

math.DS↗
arXiv2016

Periodic measures and partially hyperbolic homoclinic classes

In this paper, we give a precise meaning to the following fact, and we prove it: $C^1$-open and densely, all the non-hyperbolic ergodic measures generated by a robust cycle are approximated by periodic measures. We apply our technique to the global setting of partially hyperbolic diffeomorphisms with one dimensional center. When both strong stable and unstable foliations are minimal, we get that the closure of the set of ergodic measures is the union of two convex sets corresponding to the two possible $s$-indices; these two convex sets intersect along the closure of the set of non-hyperbolic ergodic measures. That is the case for robustly transitive perturbation of the time one map of a transitive Anosov flow, or of the skew product of an Anosov torus diffeomorphism by rotation of the circle.

math.DS↗
arXiv2016

On the existence of non-hyperbolic ergodic measures as the limit of periodic measures

[GIKN] and [BBD1] propose two very different ways for building non hyperbolic measures, [GIKN] building such a measure as the limit of periodic measures and [BBD1] as the $ω$-limit set of a single orbit, with a uniformly vanishing Lyapunov exponent. The technique in [GIKN] was essentially used in a generic setting, as the periodic orbits were built by small perturbations. It is not known if the measures obtained by the technique in [BBD1] are accumulated by periodic measures. In this paper we use a shadowing lemma from [G]: $\bullet$for getting the periodic orbits in [GIKN] without perturbing the dynamics, $\bullet$for recovering the compact set in [BBD1] with a uniformly vanishing Lyapunov exponent by considering the limit of periodic orbits. As a consequence, we prove that there exists an open and dense subset $\mathcal{U}$ of the set of robustly transitive non-hyperbolic diffeomorphisms far from homoclinic tangencies, such that for any $f\in\mathcal{U}$, there exists a non-hyperbolic ergodic measure with full support and approximated by hyperbolic periodic measures. We also prove that there exists an open and dense subset $\mathcal{V}$ of the set of diffeomorphisms exhibiting a robust cycle, such that for any $f\in\mathcal{V}$, there exists a non-hyperbolic ergodic measure approximated by hyperbolic periodic measures.

math.DS↗
arXiv2016

Ergodic measures with multi-zero Lyapunov exponents inside homoclinic classes

We prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ contains two hyperbolic periodic orbits of indices $i$ and $i+k$ respectively and $H(P)$ has no domination of index $j$ for any $j\in\{i+1,\cdots,i+k-1\}$, then there exists a non-hyperbolic ergodic measure whose $(i+l)^{th}$ Lyapunov exponent vanishes for any $l\in\{1,\cdots, k\}$, and whose support is the whole homoclinic class. We also prove that for $C^1$ generic diffeomorphisms, if a homoclinic class $H(P)$ has a dominated splitting of the form $E\oplus F\oplus G$, such that the center bundle $F$ has no finer dominated splitting, and $H(p)$ contains a hyperbolic periodic orbit $Q_1$ of index $\dim(E)$ and a hyperbolic periodic orbit $Q_2$ whose absolute Jacobian along the bundle $F$ is strictly less than $1$, then there exists a non-hyperbolic ergodic measure whose Lyapunov exponents along the center bundle $F$ all vanish and whose support is the whole homoclinic class.

math.DS↗
arXiv2016

Transverse foliations on the torus $\T^2$ and partially hyperbolic diffeomorphisms on 3-manifolds

In this paper, we prove that given two $C^1$ foliations $\mathcal{F}$ and $\mathcal{G}$ on $\mathbb{T}^2$ which are transverse, there exists a non-null homotopic loop $\{Φ_t\}_{t\in[0,1]}$ in $\diff^{1}(\T^2)$ such that $Φ_t(\calF)\pitchfork \calG$ for every $t\in[0,1]$, and $Φ_0=Φ_1= Id$. As a direct consequence, we get a general process for building new partially hyperbolic diffeomorphisms on closed $3$-manifolds. \cite{BPP} built a new example of dynamically coherent non-transitive partially hyperbolic diffeomorphism on a closed $3$-manifold, the example in \cite{BPP} is obtained by composing the time $t$ map, $t>0$ large enough, of a very specific non-transitive Anosov flow by a Dehn twist along a transverse torus. Our result shows that the same construction holds starting with any non-transitive Anosov flow on an oriented $3$-manifold. Moreover, for a given transverse torus, our result explains which type of Dehn twists lead to partially hyperbolic diffeomorphisms.

math.DS↗