Discrete Quasi-Einstein Metrics and Combinatorial Curvature Flows in 3-Dimension

We define Discrete Quasi-Einstein metrics (DQE-metrics) as the critical points of discrete total curvature functional on triangulated 3-manifolds. We study DQE-metrics by introducing some combinatorial curvature flows. We prove that these flows produce solutions which converge to discrete quasi-Einstein metrics when the initial energy is small enough. The proof relies on a careful analysis of discrete dual-Laplacians which we interpret as the Jacobian matrix of the curvature map. As a consequence, combinatorial curvature flow provides an algorithm to compute discrete sphere packing metrics with prescribed curvatures.Comment: 20 pages, 1 figure

2-Dimensional Combinatorial Calabi Flow in Hyperbolic Background Geometry

For triangulated surfaces locally embedded in the standard hyperbolic space, we introduce combinatorial Calabi flow as the negative gradient flow of combinatorial Calabi energy. We prove that the flow produces solutions which converge to ZCCP-metric (zero curvature circle packing metric) if the initial energy is small enough. Assuming the curvature has a uniform upper bound less than $2\pi$, we prove that combinatorial Calabi flow exists for all time. Moreover, it converges to ZCCP-metric if and only if ZCCP-metric exists.Comment: 15 page

A combinatorial Yamabe problem on two and three dimensional manifolds

In this paper, we introduce a new combinatorial curvature on two and three dimensional triangulated manifolds, which transforms in the same way as that of the smooth scalar curvature under scaling of the metric and could be used to approximate the Gauss curvature on two dimensional manifolds. Then we use the flow method to study the corresponding constant curvature problem, which is called combinatorial Yamabe problem.Comment: We add a proof of the discrete maximal principle in this versio

3-Dimensional Discrete curvature flows and discrete Einstein metric

We introduce the discrete Einstein metrics as critical points of discrete energy on triangulated 3-manifolds, and study them by discrete curvature flow of second (fourth) order. We also study the convergence of the discrete curvature flow. Discrete curvature flow of second order is an analogue of smooth Ricci flow.Comment: 15 pages, 1 figure

Energy Optimal Interpolation in Quantum Evolution

We introduce the concept of interpolation in quantum evolution and present a general framework to find the energy optimal Hamiltonian for a quantum system evolving among a given set of middle states using variational and geometric methods. A few special cases are carefully studied. The quantum brachistochrone problem is proved as a special case.Comment: 8 pages, 0 figure

Parallel Data Augmentation for Formality Style Transfer

The main barrier to progress in the task of Formality Style Transfer is the inadequacy of training data. In this paper, we study how to augment parallel data and propose novel and simple data augmentation methods for this task to obtain useful sentence pairs with easily accessible models and systems. Experiments demonstrate that our augmented parallel data largely helps improve formality style transfer when it is used to pre-train the model, leading to the state-of-the-art results in the GYAFC benchmark dataset.Comment: Accepted by ACL 2020. arXiv admin note: text overlap with arXiv:1909.0600

Some extremal results on hypergraph Tur\'{a}n problems

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. In this paper, using the random algebraic method, we prove that if $s$ is sufficiently larger than $t$, then $$\text{ex}(n,\mathcal{T},K_{s,t}^{(r)})=\Omega(n^{v-\frac{e}{t}}),$$ where $\mathcal{T}$ is an $r$-graph with $v$ vertices and $e$ edges. In particular, when $\mathcal{T}$ is an edge or a complete bipartite $r$-graph, we can determine their asymptotics. We show that if $s$ is sufficiently larger than $t$, then $$\text{ex}(n,K_{s,t}^{(r)})=\Theta(n^{r-\frac{1}{t}}),$$ and $$\text{ex}(n,K_{a,b}^{(r)},K_{s,t}^{(r)})=\Theta(n^{a+b(r-1)-\frac{ab}{t}}),$$ where $a<s$ and $b<t$. Meanwhile, we provide an explicit construction giving $$\text{ex}(n,K_{2,2,7}^{(3)})\geqslant\frac{1}{27}n^{\frac{19}{7}}+o(n^{\frac{19}{7}}).$$ This improves the previous best known lower bound $\Omega(n^{\frac{73}{27}})$ obtained by probabilistic method

New Theoretical Bounds and Constructions of Permutation Codes under Block Permutation Metric

Permutation codes under different metrics have been extensively studied due to their potentials in various applications. Generalized Cayley metric is introduced to correct generalized transposition errors, including previously studied metrics such as Kendall's $\tau$-metric, Ulam metric and Cayley metric as special cases. Since the generalized Cayley distance between two permutations is not easily computable, Yang et al. introduced a related metric of the same order, named the block permutation metric. Given positive integers $n$ and $d$, let $\mathcal{C}_{B}(n,d)$ denote the maximum size of a permutation code in $S_n$ with minimum block permutation distance $d$. In this paper, we focus on the theoretical bounds of $\mathcal{C}_{B}(n,d)$ and the constructions of permutation codes under block permutation metric. Using a graph theoretic approach, we improve the Gilbert-Varshamov type bound by a factor of $\Omega(\log{n})$, when $d$ is fixed and $n$ goes into infinity. We also propose a new encoding scheme based on binary constant weight codes. Moreover, an upper bound beating the sphere-packing type bound is given when $d$ is relatively close to $n$

Wyner's Common Information: Generalizations and A New Lossy Source Coding Interpretation

Wyner's common information was originally defined for a pair of dependent discrete random variables. Its significance is largely reflected in, hence also confined to, several existing interpretations in various source coding problems. This paper attempts to both generalize its definition and to expand its practical significance by providing a new operational interpretation. The generalization is two-folded: the number of dependent variables can be arbitrary, so are the alphabet of those random variables. New properties are determined for the generalized Wyner's common information of N dependent variables. More importantly, a lossy source coding interpretation of Wyner's common information is developed using the Gray-Wyner network. In particular, it is established that the common information equals to the smallest common message rate when the total rate is arbitrarily close to the rate distortion function with joint decoding. A surprising observation is that such equality holds independent of the values of distortion constraints as long as the distortions are within some distortion region. Examples about the computation of common information are given, including that of a pair of dependent Gaussian random variables.Comment: 31 pages, 5 figures. Submitted to IEEE Transactions on Information Theor

Formality Style Transfer with Hybrid Textual Annotations

Formality style transformation is the task of modifying the formality of a given sentence without changing its content. Its challenge is the lack of large-scale sentence-aligned parallel data. In this paper, we propose an omnivorous model that takes parallel data and formality-classified data jointly to alleviate the data sparsity issue. We empirically demonstrate the effectiveness of our approach by achieving the state-of-art performance on a recently proposed benchmark dataset of formality transfer. Furthermore, our model can be readily adapted to other unsupervised text style transfer tasks like unsupervised sentiment transfer and achieve competitive results on three widely recognized benchmarks