arXiv subjects

Jianfeng Zhang

Publications and source records attributed to Jianfeng Zhang.

At least 127 records · Page 7Linked to original sources

arXiv2021

Path Dependent Feynman-Kac Formula for Forward Backward Stochastic Volterra Integral Equations

This paper is concerned with the relationship between forward-backward stochastic Volterra integral equations (FBSVIEs, for short) and a system of (non-local in time) path dependent partial differential equations (PPDEs, for short). Due to the nature of Volterra type equations, the usual flow property (or semigroup property) does not hold. Inspired by Viens-Zhang \cite{Viens-Zhang-2019} and Wang-Yong \cite{Wang-Yong-2019}, auxiliary processes are introduced so that the flow property of adapted solutions to the FBSVIEs is recovered in a suitable sense, and thus the functional Itô's formula is applicable. Having achieved this stage, a natural PPDE is found so that the adapted solution of the backward SVIEs admits a representation in terms of the solution to the forward SVIE via the solution to a PPDE. On the other hand, the solution of the PPDE admits a representation in terms of adapted solution to the (path dependent) FBSVIE, which is referred to as a Feynman-Kac formula. This leads to the existence and uniqueness of a classical solution to the PPDE, under smoothness conditions on the coefficients of the FBSVIEs. Further, when the smoothness conditions are relaxed with the backward component of FBSVIE being one-dimensional, a new (and suitable) notion of viscosity solution is introduced for the PPDE, for which a comparison principle of the viscosity solutions is established, leading to the uniqueness of the viscosity solution. Finally, some results have been extended to coupled FBSVIEs and type-II BSVIEs, and a representation formula for the path derivatives of PPDE solution is obtained by a closer investigation of linear FBSVIEs.

math.PR↗

arXiv2020

RSAVS superconductors: materials with a superconducting state that is robust against large volume shrinkage

TThe transition temperature (TC) between normal and superconducting states usually exhibits a dramatic increase or decrease with increasing applied pressure. Here we present, in contrast, a new kind of superconductor that exhibits the exotic feature that TC is robust against large volume shrinkages induced by applied pressure (here naming them as "RSAVS superconductors"). Extraordinarily, the TC in these materials stays almost constant over a large pressure range, e.g. over 136 GPa in the (TaNb)0.67(HfZrTi)0.33 high entropy alloy and 141 GPa in the NbTi commercial alloy. We show that the RSAVS behavior also exists in another high entropy alloy (ScZrNbTa)0.6(RhPd)0.4, and in superconducting elemental Ta and Nb, indicating that this behavior, which has never previously been identified or predicted by theory, occurs universally in some conventional superconductors. Our electronic structure calculations indicate that although the electronic density of state (DOS) at the Fermi level in the RSAVS state is dominated by the electrons from the degenerate dxy, dxz and dyz orbitals, these electrons decrease in influence with increasing pressure. In contrast, however, the contribution of the degenerate dx2-y2 and dz2 orbital electrons remains almost unchanged at the Fermi level, suggesting that these are the electrons that may play a crucial role in stabilizing the TC in the RSAVS state.

cond-mat.supr-con↗

arXiv2020

Dynamic Set Values for Nonzero Sum Games with Multiple Equilibriums

Nonzero sum games typically have multiple Nash equilibriums (or no equilibrium), and unlike the zero sum case, they may have different values at different equilibriums. Instead of focusing on the existence of individual equilibriums, we study the set of values over all equilibriums, which we call the set value of the game. The set value is unique by nature and always exists (with possible value $\emptyset$). Similar to the standard value function in control literature, it enjoys many nice properties such as regularity, stability, and more importantly the dynamic programming principle. There are two main features in order to obtain the dynamic programming principle: (i) we must use closed-loop controls (instead of open-loop controls); (ii) we must allow for path dependent controls, even if the problem is in a state dependent (Markovian) setting. We shall consider both discrete and continuous time models with finite time horizon. For the latter we will also provide a duality approach through certain standard PDE (or path dependent PDE), which is quite efficient for numerically computing the set value of the game.

math.OC↗

arXiv2020

Inference Stage Optimization for Cross-scenario 3D Human Pose Estimation

Existing 3D human pose estimation models suffer performance drop when applying to new scenarios with unseen poses due to their limited generalizability. In this work, we propose a novel framework, Inference Stage Optimization (ISO), for improving the generalizability of 3D pose models when source and target data come from different pose distributions. Our main insight is that the target data, even though not labeled, carry valuable priors about their underlying distribution. To exploit such information, the proposed ISO performs geometry-aware self-supervised learning (SSL) on each single target instance and updates the 3D pose model before making prediction. In this way, the model can mine distributional knowledge about the target scenario and quickly adapt to it with enhanced generalization performance. In addition, to handle sequential target data, we propose an online mode for implementing our ISO framework via streaming the SSL, which substantially enhances its effectiveness. We systematically analyze why and how our ISO framework works on diverse benchmarks under cross-scenario setup. Remarkably, it yields new state-of-the-art of 83.6% 3D PCK on MPI-INF-3DHP, improving upon the previous best result by 9.7%. Code will be released.

cs.CV↗

arXiv2020

Time-consistent conditional expectation under probability distortion

We introduce a new notion of conditional nonlinear expectation under probability distortion. Such a distorted nonlinear expectation is not sub-additive in general, so it is beyond the scope of Peng's framework of nonlinear expectations. A more fundamental problem when extending the distorted expectation to a dynamic setting is time-inconsistency, that is, the usual "tower property" fails. By localizing the probability distortion and restricting to a smaller class of random variables, we introduce a so-called distorted probability and construct a conditional expectation in such a way that it coincides with the original nonlinear expectation at time zero, but has a time-consistent dynamics in the sense that the tower property remains valid. Furthermore, we show that in the continuous time model this conditional expectation corresponds to a parabolic differential equation whose coefficient involves the law of the underlying diffusion. This work is the first step towards a new understanding of nonlinear expectations under probability distortion, and will potentially be a helpful tool for solving time-inconsistent stochastic optimization problems.

q-fin.MF↗

arXiv2019

Hierarchical Graph Pooling with Structure Learning

Graph Neural Networks (GNNs), which generalize deep neural networks to graph-structured data, have drawn considerable attention and achieved state-of-the-art performance in numerous graph related tasks. However, existing GNN models mainly focus on designing graph convolution operations. The graph pooling (or downsampling) operations, that play an important role in learning hierarchical representations, are usually overlooked. In this paper, we propose a novel graph pooling operator, called Hierarchical Graph Pooling with Structure Learning (HGP-SL), which can be integrated into various graph neural network architectures. HGP-SL incorporates graph pooling and structure learning into a unified module to generate hierarchical representations of graphs. More specifically, the graph pooling operation adaptively selects a subset of nodes to form an induced subgraph for the subsequent layers. To preserve the integrity of graph's topological information, we further introduce a structure learning mechanism to learn a refined graph structure for the pooled graph at each layer. By combining HGP-SL operator with graph neural networks, we perform graph level representation learning with focus on graph classification task. Experimental results on six widely used benchmarks demonstrate the effectiveness of our proposed model.

cs.LG↗

arXiv2019

Single-Stage Multi-Person Pose Machines

Multi-person pose estimation is a challenging problem. Existing methods are mostly two-stage based--one stage for proposal generation and the other for allocating poses to corresponding persons. However, such two-stage methods generally suffer low efficiency. In this work, we present the first single-stage model, Single-stage multi-person Pose Machine (SPM), to simplify the pipeline and lift the efficiency for multi-person pose estimation. To achieve this, we propose a novel Structured Pose Representation (SPR) that unifies person instance and body joint position representations. Based on SPR, we develop the SPM model that can directly predict structured poses for multiple persons in a single stage, and thus offer a more compact pipeline and attractive efficiency advantage over two-stage methods. In particular, SPR introduces the root joints to indicate different person instances and human body joint positions are encoded into their displacements w.r.t. the roots. To better predict long-range displacements for some joints, SPR is further extended to hierarchical representations. Based on SPR, SPM can efficiently perform multi-person poses estimation by simultaneously predicting root joints (location of instances) and body joint displacements via CNNs. Moreover, to demonstrate the generality of SPM, we also apply it to multi-person 3D pose estimation. Comprehensive experiments on benchmarks MPII, extended PASCAL-Person-Part, MSCOCO and CMU Panoptic clearly demonstrate the state-of-the-art efficiency of SPM for multi-person 2D/3D pose estimation, together with outstanding accuracy.

cs.CV↗

arXiv2019

Predicting Path Failure In Time-Evolving Graphs

In this paper we use a time-evolving graph which consists of a sequence of graph snapshots over time to model many real-world networks. We study the path classification problem in a time-evolving graph, which has many applications in real-world scenarios, for example, predicting path failure in a telecommunication network and predicting path congestion in a traffic network in the near future. In order to capture the temporal dependency and graph structure dynamics, we design a novel deep neural network named Long Short-Term Memory R-GCN (LRGCN). LRGCN considers temporal dependency between time-adjacent graph snapshots as a special relation with memory, and uses relational GCN to jointly process both intra-time and inter-time relations. We also propose a new path representation method named self-attentive path embedding (SAPE), to embed paths of arbitrary length into fixed-length vectors. Through experiments on a real-world telecommunication network and a traffic network in California, we demonstrate the superiority of LRGCN to other competing methods in path failure prediction, and prove the effectiveness of SAPE on path representation.

cs.LG↗

arXiv2019

Viscosity Solutions to Parabolic Master Equations and McKean-Vlasov SDEs with Closed-loop Controls

The master equation is a type of PDE whose state variable involves the distribution of certain underlying state process. It is a powerful tool for studying the limit behavior of large interacting systems, including mean field games and systemic risk. It also appears naturally in stochastic control problems with partial information and in time inconsistent problems. In this paper we propose a novel notion of viscosity solution for parabolic master equations, arising mainly from control problems, and establish its wellposedness. Our main innovation is to restrict the involved measures to certain set of semimartingale measures which satisfy the desired compactness. As an important example, we study the HJB master equation associated with the control problems for McKean-Vlasov SDEs. Due to practical considerations, we consider closed-loop controls. It turns out that the regularity of the value function becomes much more involved in this framework than the counterpart in the standard control problems. Finally, we build the whole theory in the path dependent setting, which is often seen in applications. The main result in this part is an extension of Dupire \cite{Dupire}'s functional Itô formula. This Itô formula requires a special structure of the derivatives with respect to the measures, which was originally due to Lions \cite{Lions4} in the state dependent case. We provided an elementary proof for this well known result in the short note \cite{WZ}, and the same arguments work in the path dependent setting here.

math.PR↗

arXiv2018

Scanning tunneling microscopic evidence of interface enhanced high-Tc superconductivity in Pb islands grown on SrTiO3

The discovery of the interface enhanced superconductivity in the single layer film of FeSe epitaxially grown on SrTiO3 substrates has triggered a flurry of activity in the field of superconductivity. It raised the hope to find more conventional high-Tc superconductors which are purely driven by the electron-phonon interaction at ambient pressure. Here we report the experimental evidence from the measurement of scanning tunneling spectroscopy for the interface enhanced high-Tc superconductivity in the Pb thin film islands grown on SrTiO3 substrates. The superconducting energy gap of the Pb film is found to depend on both the thickness and the volume of the islands. The largest superconducting energy gap is found to be about 10 meV, which is 7 times larger than that in the bulk Pb. The corresponding superconducting transition temperature, estimated by fitting the temperature dependence of the gap values using the BCS formula, is found to be 47 K, again 7 times higher than that of the bulk Pb.

cond-mat.supr-con↗

arXiv2018

A Martingale Approach for Fractional Brownian Motions and Related Path Dependent PDEs

In this paper we study dynamic backward problems, with the computation of conditional expectations as a main objective, in a framework where the (forward) state process satisfies a Volterra type SDE, with fractional Brownian motion as a typical example. Such processes are neither Markov processes nor semimartingales, and most notably, they feature a certain time inconsistency which makes any direct application of Markovian ideas, such as flow properties, impossible without passing to a path-dependent framework. Our main result is a functional Itô formula, extending the seminal work of Dupire \cite{Dupire} to our more general framework. In particular, unlike in \cite{Dupire} where one needs only to consider the stopped paths, here we need to concatenate the observed path up to the current time with a certain smooth observable curve derived from the distribution of the future paths. This new feature is due to the time inconsistency involved in this paper. We then derive the path dependent PDEs for the backward problems. Finally, an application to option pricing in a financial market with rough volatility is presented.

math.PR↗

arXiv2018

Fully nonlinear stochastic and rough PDEs: Classical and viscosity solutions

We study fully nonlinear second-order (forward) stochastic partial differential equations (SPDEs). They can also be viewed as forward path-dependent PDEs (PPDEs) and will be treated as rough PDEs (RPDEs) under a unified framework. We develop first a local theory of classical solutions and define then viscosity solutions through smooth test functions. Our notion of viscosity solutions is equivalent to the alternative one using semi-jets. Next, we prove basic properties such as consistency, stability, and a partial comparison principle in the general setting. When the diffusion coefficient is semi-linear (but the drift can be fully nonlinear), we establish a complete theory, including global existence and comparison principle. Our methodology relies heavily on the method of characteristics.

math.PR↗

arXiv2018

Interactive Binary Image Segmentation with Edge Preservation

Binary image segmentation plays an important role in computer vision and has been widely used in many applications such as image and video editing, object extraction, and photo composition. In this paper, we propose a novel interactive binary image segmentation method based on the Markov Random Field (MRF) framework and the fast bilateral solver (FBS) technique. Specifically, we employ the geodesic distance component to build the unary term. To ensure both computation efficiency and effective responsiveness for interactive segmentation, superpixels are used in computing geodesic distances instead of pixels. Furthermore, we take a bilateral affinity approach for the pairwise term in order to preserve edge information and denoise. Through the alternating direction strategy, the MRF energy minimization problem is divided into two subproblems, which then can be easily solved by steepest gradient descent (SGD) and FBS respectively. Experimental results on the VGG interactive image segmentation dataset show that the proposed algorithm outperforms several state-of-the-art ones, and in particular, it can achieve satisfactory edge-smooth segmentation results even when the foreground and background color appearances are quite indistinctive.

cs.CV↗

arXiv2018

Stochastic Control with Delayed Information and Related Nonlinear Master Equation

In this paper we study stochastic control problems with delayed information, that is, the control at time $t$ can depend only on the information observed before time $t-H$ for some delay parameter $H$. Such delay occurs frequently in practice and can be viewed as a special case of partial observation. When the time duration $T$ is smaller than $H$, the problem becomes a deterministic control problem in stochastic setting. While seemingly simple, the problem involves certain time inconsistency issue, and the value function naturally relies on the distribution of the state process and thus is a solution to a nonlinear master equation. Consequently, the optimal state process solves a McKean-Vlasov SDE. In the general case that $T$ is larger than $H$, the master equation becomes path-dependent and the corresponding McKean-Vlasov SDE involves the conditional distribution of the state process. We shall build these connections rigorously, and obtain the existence of classical solution of these nonlinear (path-dependent) master equations in a special case.

math.PR↗

arXiv2018

Zero-sum path-dependent stochastic differential games in weak formulation

We consider zero-sum stochastic differential games with possibly path-dependent controlled state. Unlike the previous literature, we allow for weak solutions of the state equation so that the players' controls are automatically of feedback type. Under some restrictions, needed for the a priori regularity of the upper and lower value functions of the game, we show that the game value exists when both the appropriate path-dependent Isaacs condition, and the uniqueness of viscosity solutions of the corresponding path-dependent Isaacs-HJB equation hold. We also provide a general verification argument and a characterisation of saddle-points by means of an appropriate notion of second-order backward SDEs.

math.PR↗

arXiv2018

An Elementary Proof for the Structure of Wasserstein Derivatives

Let $F: \mathbb{L}^2(Ω, \mathbb{R}) \to \mathbb{R}$ be a law invariant and continuously Fréchet differentiable mapping. Based on Lions \cite{Lions}, Cardaliaguet \cite{Cardaliaguet} (Theorem 6.2 and 6.5) proved that: \bea \label{Derivative} D F (ξ) = g(ξ), \eea where $g: \mathbb{R} \to \mathbb{R}$ is a deterministic function which depends only on the law of $ξ$. See also Carmona \& Delarue \cite{CD} Section 5.2. In this short note we provide an elementary proof for this well known result. This note is part of our accompanying paper \cite{WZ}, which deals with a more general situation.

math.PR↗

arXiv2018

Learning for Disparity Estimation through Feature Constancy

Stereo matching algorithms usually consist of four steps, including matching cost calculation, matching cost aggregation, disparity calculation, and disparity refinement. Existing CNN-based methods only adopt CNN to solve parts of the four steps, or use different networks to deal with different steps, making them difficult to obtain the overall optimal solution. In this paper, we propose a network architecture to incorporate all steps of stereo matching. The network consists of three parts. The first part calculates the multi-scale shared features. The second part performs matching cost calculation, matching cost aggregation and disparity calculation to estimate the initial disparity using shared features. The initial disparity and the shared features are used to calculate the feature constancy that measures correctness of the correspondence between two input images. The initial disparity and the feature constancy are then fed to a sub-network to refine the initial disparity. The proposed method has been evaluated on the Scene Flow and KITTI datasets. It achieves the state-of-the-art performance on the KITTI 2012 and KITTI 2015 benchmarks while maintaining a very fast running time.

cs.CV↗

arXiv2017

Forward Backward SDEs in Weak Formulation

Although having been developed for more than two decades, the theory of forward backward stochastic differential equations is still far from complete. In this paper, we take one step back and investigate the formulation of FBSDEs. Motivated from several considerations, both in theory and in applications, we propose to study FBSDEs in weak formulation, rather than the strong formulation in the standard literature. That is, the backward SDE is driven by the forward component, instead of by the Brownian motion noise. We establish the Feyman-Kac formula for FBSDEs in weak formulation, both in classical and in viscosity sense. Our new framework is efficient especially when the diffusion part of the forward equation involves the $Z$-component of the backward equation.

math.PR↗