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Protein structure prediction is pivotal for understanding the structure-function relationship of proteins, advancing biological research, and facilitating pharmaceutical development and experimental design. While deep learning methods and the expanded availability of experimental 3D protein structures have accelerated structure prediction, the dynamic nature of protein structures has received limited attention. This study introduces an innovative 4D diffusion model incorporating molecular dynamics (MD) simulation data to learn dynamic protein structures. Our approach is distinguished by the following components: (1) a unified diffusion model capable of generating dynamic protein structures, including both the backbone and side chains, utilizing atomic grouping and side-chain dihedral angle predictions; (2) a reference network that enhances structural consistency by integrating the latent embeddings of the initial 3D protein structures; and (3) a motion alignment module aimed at improving temporal structural coherence across multiple time steps. To our knowledge, this is the first diffusion-based model aimed at predicting protein trajectories across multiple time steps simultaneously. Validation on benchmark datasets demonstrates that our model exhibits high accuracy in predicting dynamic 3D structures of proteins containing up to 256 amino acids over 32 time steps, effectively capturing both local flexibility in stable states and significant conformational changes.

Despite significant progress in static protein structure collection and prediction, the dynamic behavior of proteins, one of their most vital characteristics, has been largely overlooked in prior research. This oversight can be attributed to the limited availability, diversity, and heterogeneity of dynamic protein datasets. To address this gap, we propose to enhance existing prestigious static 3D protein structural databases, such as the Protein Data Bank (PDB), by integrating dynamic data and additional physical properties. Specifically, we introduce a large-scale dataset, Dynamic PDB, encompassing approximately 12.6K proteins, each subjected to all-atom molecular dynamics (MD) simulations lasting 1 microsecond to capture conformational changes. Furthermore, we provide a comprehensive suite of physical properties, including atomic velocities and forces, potential and kinetic energies of proteins, and the temperature of the simulation environment, recorded at 1 picosecond intervals throughout the simulations. For benchmarking purposes, we evaluate state-of-the-art methods on the proposed dataset for the task of trajectory prediction. To demonstrate the value of integrating richer physical properties in the study of protein dynamics and related model design, we base our approach on the SE(3) diffusion model and incorporate these physical properties into the trajectory prediction process. Preliminary results indicate that this straightforward extension of the SE(3) model yields improved accuracy, as measured by MAE and RMSD, when the proposed physical properties are taken into consideration. https://fudan-generative-vision.github.io/dynamicPDB/ .

The field of portrait image animation, driven by speech audio input, has experienced significant advancements in the generation of realistic and dynamic portraits. This research delves into the complexities of synchronizing facial movements and creating visually appealing, temporally consistent animations within the framework of diffusion-based methodologies. Moving away from traditional paradigms that rely on parametric models for intermediate facial representations, our innovative approach embraces the end-to-end diffusion paradigm and introduces a hierarchical audio-driven visual synthesis module to enhance the precision of alignment between audio inputs and visual outputs, encompassing lip, expression, and pose motion. Our proposed network architecture seamlessly integrates diffusion-based generative models, a UNet-based denoiser, temporal alignment techniques, and a reference network. The proposed hierarchical audio-driven visual synthesis offers adaptive control over expression and pose diversity, enabling more effective personalization tailored to different identities. Through a comprehensive evaluation that incorporates both qualitative and quantitative analyses, our approach demonstrates obvious enhancements in image and video quality, lip synchronization precision, and motion diversity. Further visualization and access to the source code can be found at: https://fudan-generative-vision.github.io/hallo., Comment: 20 pages

This paper is devoted to studying the stability properties of the Karush-Kuhn-Tucker (KKT) solution mapping $S_{\rm KKT}$ for Nash equilibrium problems (NEPs) with canonical perturbations. Firstly, we obtain an exact characterization of the strong regularity of $S_{\rm KKT}$ and a sufficient condition that is easy to verify. Secondly, we propose equivalent conditions for the continuously differentiable single-valued localization of $S_{\rm KKT}$. Thirdly, the isolated calmness of $S_{\rm KKT}$ is studied based on two conditions: Property A and Property B, and Property B proves to be sufficient for the robustness of both $E(p)$ and $S_{\rm KKT}$ under the convex assumptions, where $E(p)$ denotes the Nash equilibria at perturbation $p$. Furthermore, we establish that studying the stability properties of the NEP with canonical perturbations is equivalent to studying those of the NEP with only tilt perturbations based on the prior discussions. Finally, we provide detailed characterizations of stability for NEPs whose each individual player solves a quadratic programming (QP) problem., Comment: 34 pages, 1 figure

This paper presents a stochastic approximation proximal subgradient (SAPS) method for stochastic convex-concave minimax optimization. By accessing unbiased and variance bounded approximate subgradients, we show that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate of the minimax optimality measure if the parameters in the algorithm are properly chosen, where $N$ denotes the number of iterations. Moreover, we show that the algorithm has ${\rm O}(\log(N)N^{-1/2})$ minimax optimality measure bound with high probability. Further we study a specific stochastic convex-concave minimax optimization problems arising from stochastic convex conic optimization problems, which the the bounded subgradient condition is fail. To overcome the lack of the bounded subgradient conditions in convex-concave minimax problems, we propose a linearized stochastic approximation augmented Lagrange (LSAAL) method and prove that this algorithm exhibits ${\rm O}(N^{-1/2})$ expected convergence rate for the minimax optimality measure and ${\rm O}(\log^2(N)N^{-1/2})$ minimax optimality measure bound with high probability as well. Preliminary numerical results demonstrate the effect of the SAPS and LSAAL methods.

This paper focuses on a distributed coupled constrained convex optimization problem over directed unbalanced and time-varying multi-agent networks, where the global objective function is the sum of all agents' private local objective functions, and decisions of all agents are subject to coupled equality and inequality constraints and a compact convex subset. In the multi-agent networks, each agent exchanges information with other neighboring agents. Finally, all agents reach a consensus on decisions, meanwhile achieving the goal of minimizing the global objective function under the given constraint conditions. For the purpose of protecting the information privacy of each agent, we first establish the saddle point problem of the constrained convex optimization problem considered in this article, then based on the push-pull method, develop a distributed primal-dual algorithm to solve the dual problem. Under Slater's condition, we will show that the sequence of points generated by the proposed algorithm converges to a saddle point of the Lagrange function. Moreover, we analyze the iteration complexity of the algorithm.

This paper focuses on a class of inclusion problems of maximal monotone operators in a multi-agent network, where each agent is characterized by an operator that is not available to any other agents, but the agents can cooperate by exchanging information with their neighbors according to a given communication topology. All agents aim at finding a common decision vector that is the solution to the sum of agents' operators. This class of problems is motivated by distributed convex optimization with coupled constraints. In this paper, we propose a distributed proximal point method with a cumulative correction term (named Proximal-Correction Algorithm) for this class of inclusion problems of operators. It's proved that the Proximal-Correction Algorithm converges for any value of a constant penalty parameter. In order to make the Proximal-Correction ALgorithm computationally implementable for a wide variety of distributed optimization problems, we adopt two inexact criteria for calculating the proximal steps of the algorithm. Under each of these two criteria, the convergence of Proximal-Correction Algorithm can be guaranteed, and the linear convergence rate is established when the stronger one is satisfied. In numerical simulations, both exact and inexact versions of Proximal-Correction Algorithm are executed for a distributed convex optimization problem with coupled constraints. Compared with several alternative algorithms in the literature, the exact and inexact versions of Proximal-Correction both exhibit good numerical performance.

In this paper, a decentralized proximal method of multipliers (DPMM) is proposed to solve constrained convex optimization problems over multi-agent networks, where the local objective of each agent is a general closed convex function, and the constraints are coupled equalities and inequalities. This algorithm strategically integrates the dual decomposition method and the proximal point algorithm. One advantage of DPMM is that subproblems can be solved inexactly and in parallel by agents at each iteration, which relaxes the restriction of requiring exact solutions to subproblems in many distributed constrained optimization algorithms. We show that the first-order optimality residual of the proposed algorithm decays to $0$ at a rate of $o(1/k)$ under general convexity. Furthermore, if a structural assumption for the considered optimization problem is satisfied, the sequence generated by DPMM converges linearly to an optimal solution. In numerical simulations, we compare DPMM with several existing algorithms using two examples to demonstrate its effectiveness.

Second-order optimality conditions of the bilevel programming problems are dependent on the second-order directional derivatives of the value functions or the solution mappings of the lower level problems under some regular conditions, which can not be calculated or evaluated. To overcome this difficulty, we propose the notion of the bi-local solution. Under the Jacobian uniqueness conditions for the lower level problem, we prove that the bi-local solution is a local minimizer of some one-level minimization problem. Basing on this property, the first-order necessary optimality conditions and second-order necessary and sufficient optimality conditions for the bi-local optimal solution of a given bilevel program are established. The second-order optimality conditions proposed here only involve second-order derivatives of the defining functions of the bilevel problem. The second-order sufficient optimality conditions are used to derive the Q-linear convergence rate of the classical augmented Lagrangian method.

A large number matrix optimization problems are described by orthogonally invariant norms. This paper is devoted to the study of variational analysis of the orthogonally invariant norm cone of symmetric matrices. For a general orthogonally invariant norm cone of symmetric matrices, formulas for the tangent cone, normal cone and second-order tangent set are established. The differentiability properties of the projection operator onto the orthogonally invariant norm cone are developed, including formulas for the directional derivative and the B-subdifferential. Importantly, the directional derivative is characterized by the second-order derivative of the corresponding symmetric function, which is convenient for computation. All these results are specified to the Schatten $p$-norm cone, especially to the second-order cone of symmetric matrices.

In this paper, we consider online convex optimization (OCO) with time-varying loss and constraint functions. Specifically, the decision maker chooses sequential decisions based only on past information, meantime the loss and constraint functions are revealed over time. We first develop a class of model-based augmented Lagrangian methods (MALM) for time-varying functional constrained OCO (without feedback delay). Under standard assumptions, we establish sublinear regret and sublinear constraint violation of MALM. Furthermore, we extend MALM to deal with time-varying functional constrained OCO with delayed feedback, in which the feedback information of loss and constraint functions is revealed to decision maker with delays. Without additional assumptions, we also establish sublinear regret and sublinear constraint violation for the delayed version of MALM. Finally, numerical results for several examples of constrained OCO including online network resource allocation, online logistic regression and online quadratically constrained quadratical program are presented to demonstrate the efficiency of the proposed algorithms.

The online optimization problem with non-convex loss functions over a closed convex set, coupled with a set of inequality (possibly non-convex) constraints is a challenging online learning problem. A proximal method of multipliers with quadratic approximations (named as OPMM) is presented to solve this online non-convex optimization with long term constraints. Regrets of the violation of Karush-Kuhn-Tucker conditions of OPMM for solving online non-convex optimization problems are analyzed. Under mild conditions, it is shown that this algorithm exhibits ${\cO}(T^{-1/8})$ Lagrangian gradient violation regret, ${\cO}(T^{-1/8})$ constraint violation regret and ${\cO}(T^{-1/4})$ complementarity residual regret if parameters in the algorithm are properly chosen, where $T$ denotes the number of time periods. For the case that the objective is a convex quadratic function, we demonstrate that the regret of the objective reduction can be established even the feasible set is non-convex. For the case when the constraint functions are convex, if the solution of the subproblem in OPMM is obtained by solving its dual, OPMM is proved to be an implementable projection method for solving the online non-convex optimization problem.

It is well known that there have been many numerical algorithms for solving nonsmooth minimax problems, numerical algorithms for nonsmooth minimax problems with joint linear constraints are very rare. This paper aims to discuss optimality conditions and develop practical numerical algorithms for minimax problems with joint linear constraints. First of all, we use the properties of proximal mapping and KKT system to establish optimality conditions. Secondly, we propose a framework of alternating coordinate algorithm for the minimax problem and analyze its convergence properties. Thirdly, we develop a proximal gradient multi-step ascent decent method (PGmsAD) as a numerical algorithm and demonstrate that the method can find an $\epsilon$-stationary point for this kind of nonsmooth nonconvex-nonconcave problem in ${\cal O}(\epsilon^{-2}\log\epsilon^{-1})$ iterations. Finally, we apply PGmsAD to generalized absolute value equations, generalized linear projection equations and linear regression problems and report the efficiency of PGmsAD on large-scale optimization.