MonoFlow: Rethinking Divergence GANs via the Perspective of Wasserstein Gradient Flows

The conventional understanding of adversarial training in generative adversarial networks (GANs) is that the discriminator is trained to estimate a divergence, and the generator learns to minimize this divergence. We argue that despite the fact that many variants of GANs were developed following this paradigm, the current theoretical understanding of GANs and their practical algorithms are inconsistent. In this paper, we leverage Wasserstein gradient flows which characterize the evolution of particles in the sample space, to gain theoretical insights and algorithmic inspiration of GANs. We introduce a unified generative modeling framework - MonoFlow: the particle evolution is rescaled via a monotonically increasing mapping of the log density ratio. Under our framework, adversarial training can be viewed as a procedure first obtaining MonoFlow's vector field via training the discriminator and the generator learns to draw the particle flow defined by the corresponding vector field. We also reveal the fundamental difference between variational divergence minimization and adversarial training. This analysis helps us to identify what types of generator loss functions can lead to the successful training of GANs and suggest that GANs may have more loss designs beyond the literature (e.g., non-saturated loss), as long as they realize MonoFlow. Consistent empirical studies are included to validate the effectiveness of our framework

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we introduce a hierarchy of integrable higher order equations of Camassa-Holm (CH) type, that is, we present infinitely many nonlinear equations depending on inertia operators which generalize the standard momentum operator A2=∂xx−1 appearing in the Camassa-Holm equation mt=−mxu−2mux, m=A2(u). Our higher order CH-type equations are integrable in the sense that they possess an infinite number of local conservation laws, quadratic pseudo-potentials, and zero curvature formulations. We focus mainly on the fifth order CH-type equation and we show that it admits {\em pseudo-peakons}, this is, bounded solutions with differentiable first derivative and continuous and bounded second derivative, but whose higher order derivatives blow up. Furthermore, we investigate the Cauchy problem of our fifth order equation on the real line and prove local well-posedness for initial conditions u0∈Hs(R), s\u3e7/2. In addition, we discuss conditions for global well-posedness in H4(R) as well as conditions causing local solutions to blow up in a finite time. We finish our paper with some comments on the geometric content of our equations of CH-type

Pseudo-peakons and Cauchy analysis for an integrable fifth-order equation of Camassa-Holm type

In this paper we discuss integrable higher order equations {\em of Camassa-Holm (CH) type}. Our higher order CH-type equations are "geometrically integrable", that is, they describe one-parametric families of pseudo-spherical surfaces, in a sense explained in Section 1, and they are integrable in the sense of zero curvature formulation ($\simeq$ Lax pair) with infinitely many local conservation laws. The major focus of the present paper is on a specific fifth order CH-type equation admitting {\em pseudo-peakons} solutions, that is, weak bounded solutions with differentiable first derivative and continuous and bounded second derivative, but such that any higher order derivative blows up. Furthermore, we investigate the Cauchy problem of this fifth order CH-type equation on the real line and prove local well-posedness under the initial conditions $u_0 \in H^s(\mathbb{R})$, $s > 7/2$. In addition, we study conditions for global well-posedness in $H^4(\mathbb{R})$ as well as conditions causing local solutions to blow up in a finite time. We conclude our paper with some comments on the geometric content of the high order CH-type equations.Comment: 6 figures; 32 page

Analytical Properties for the Fifth Order Camassa-Holm (FOCH) Model

This paper devotes to present analysiswork on the fifth order Camassa-Holm (FOCH) modelwhich recently proposed by Liu and Qiao. Firstly, we establish the local and global existence of the solution to the FOCH model. Secondly, we study the property of the infinite propagation speed. Finally, we discuss the long time behavior of the support of momentum density with a compactly supported initial data

Rogue peakon, well-posedness, ill-posedness and blow-up phenomenon for an integrable Camassa-Holm type equation

In this paper, we study an integrable Camassa-Holm (CH) type equation with quadratic nonlinearity. The CH type equation is shown integrable through a Lax pair, and particularly the equation is found to possess a new kind of peaked soliton (peakon) solution - called {\sf rogue peakon}, that is given in a rational form with some logarithmic function, but not a regular traveling wave. We also provide multi-rogue peakon solutions. Furthermore, we discuss the local well-posedness of the solution in the Besov space $B_{p,r}^{s}$ with $1\leq p,r\leq\infty$, $s>\max \left\{1+1/p,3/2\right\}$ or $B_{2,1}^{3/2}$, and then prove the ill-posedness of the solution in $B_{2,\infty}^{3/2}$. Moreover, we establish the global existence and blow-up phenomenon of the solution, which is, if $m_0(x)=u_0-u_{0xx}\geq(\not\equiv) 0$, then the corresponding solution exists globally, meanwhile, if $m_0(x)\leq(\not\equiv) 0$, then the corresponding solution blows up in a finite time.Comment: 23 pages, 6 figure

On the Cauchy Problem for the b

In this paper, we consider b-family equations with a strong dispersive term. First, we present a criterion on blow-up. Then global existence and persistence property of the solution are also established. Finally, we discuss infinite propagation speed of this equation

Beyond Triplet: Leveraging the Most Data for Multimodal Machine Translation

Multimodal machine translation (MMT) aims to improve translation quality by incorporating information from other modalities, such as vision. Previous MMT systems mainly focus on better access and use of visual information and tend to validate their methods on image-related datasets. These studies face two challenges. First, they can only utilize triple data (bilingual texts with images), which is scarce; second, current benchmarks are relatively restricted and do not correspond to realistic scenarios. Therefore, this paper correspondingly establishes new methods and new datasets for MMT. First, we propose a framework 2/3-Triplet with two new approaches to enhance MMT by utilizing large-scale non-triple data: monolingual image-text data and parallel text-only data. Second, we construct an English-Chinese {e}-commercial {m}ulti{m}odal {t}ranslation dataset (including training and testing), named EMMT, where its test set is carefully selected as some words are ambiguous and shall be translated mistakenly without the help of images. Experiments show that our method is more suitable for real-world scenarios and can significantly improve translation performance by using more non-triple data. In addition, our model also rivals various SOTA models in conventional multimodal translation benchmarks.Comment: 8 pages, ACL 2023 Findin

BigVideo: A Large-scale Video Subtitle Translation Dataset for Multimodal Machine Translation

We present a large-scale video subtitle translation dataset, BigVideo, to facilitate the study of multi-modality machine translation. Compared with the widely used How2 and VaTeX datasets, BigVideo is more than 10 times larger, consisting of 4.5 million sentence pairs and 9,981 hours of videos. We also introduce two deliberately designed test sets to verify the necessity of visual information: Ambiguous with the presence of ambiguous words, and Unambiguous in which the text context is self-contained for translation. To better model the common semantics shared across texts and videos, we introduce a contrastive learning method in the cross-modal encoder. Extensive experiments on the BigVideo show that: a) Visual information consistently improves the NMT model in terms of BLEU, BLEURT, and COMET on both Ambiguous and Unambiguous test sets. b) Visual information helps disambiguation, compared to the strong text baseline on terminology-targeted scores and human evaluation. Dataset and our implementations are available at https://github.com/DeepLearnXMU/BigVideo-VMT.Comment: Accepted to ACL 2023 Finding