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Yongxin Chen

Publications and source records attributed to Yongxin Chen.

At least 127 records · Page 7Linked to original sources
arXiv2023

Improved dimension dependence of a proximal algorithm for sampling

We propose a sampling algorithm that achieves superior complexity bounds in all the classical settings (strongly log-concave, log-concave, Logarithmic-Sobolev inequality (LSI), Poincaré inequality) as well as more general settings with semi-smooth or composite potentials. Our algorithm is based on the proximal sampler introduced in~\citet{lee2021structured}. The performance of this proximal sampler is determined by that of the restricted Gaussian oracle (RGO), a key step in the proximal sampler. The main contribution of this work is an inexact realization of RGO based on approximate rejection sampling. To bound the inexactness of RGO, we establish a new concentration inequality for semi-smooth functions over Gaussian distributions, extending the well-known concentration inequality for Lipschitz functions. Applying our RGO implementation to the proximal sampler, we achieve state-of-the-art complexity bounds in almost all settings. For instance, for strongly log-concave distributions, our method has complexity bound $\tilde\mathcal{O}(κd^{1/2})$ without warm start, better than the minimax bound for MALA. For distributions satisfying the LSI, our bound is $\tilde \mathcal{O}(\hat κd^{1/2})$ where $\hat κ$ is the ratio between smoothness and the LSI constant, better than all existing bounds.

math.ST↗
arXiv2023

On a Class of Gibbs Sampling over Networks

We consider the sampling problem from a composite distribution whose potential (negative log density) is $\sum_{i=1}^n f_i(x_i)+\sum_{j=1}^m g_j(y_j)+\sum_{i=1}^n\sum_{j=1}^m\frac{σ_{ij}}{2η} \Vert x_i-y_j \Vert^2_2$ where each of $x_i$ and $y_j$ is in $\mathbb{R}^d$, $f_1, f_2, \ldots, f_n, g_1, g_2, \ldots, g_m$ are strongly convex functions, and $\{σ_{ij}\}$ encodes a network structure. % motivated by the task of drawing samples over a network in a distributed manner. Building on the Gibbs sampling method, we develop an efficient sampling framework for this problem when the network is a bipartite graph. More importantly, we establish a non-asymptotic linear convergence rate for it. This work extends earlier works that involve only a graph with two nodes \cite{lee2021structured}. To the best of our knowledge, our result represents the first non-asymptotic analysis of a Gibbs sampler for structured log-concave distributions over networks. Our framework can be potentially used to sample from the distribution $ \propto \exp(-\sum_{i=1}^n f_i(x)-\sum_{j=1}^m g_j(x))$ in a distributed manner.

math.ST↗
arXiv2023

Data-driven optimal control under safety constraints using sparse Koopman approximation

In this work we approach the dual optimal reach-safe control problem using sparse approximations of Koopman operator. Matrix approximation of Koopman operator needs to solve a least-squares (LS) problem in the lifted function space, which is computationally intractable for fine discretizations and high dimensions. The state transitional physical meaning of the Koopman operator leads to a sparse LS problem in this space. Leveraging this sparsity, we propose an efficient method to solve the sparse LS problem where we reduce the problem dimension dramatically by formulating the problem using only the non-zero elements in the approximation matrix with known sparsity pattern. The obtained matrix approximation of the operators is then used in a dual optimal reach-safe problem formulation where a linear program with sparse linear constraints naturally appears. We validate our proposed method on various dynamical systems and show that the computation time for operator approximation is greatly reduced with high precision in the solutions.

math.OC↗
arXiv2023

Mean field type control with species dependent dynamics via structured tensor optimization

In this work we consider mean field type control problems with multiple species that have different dynamics. We formulate the discretized problem using a new type of entropy-regularized multimarginal optimal transport problems where the cost is a decomposable structured tensor. A novel algorithm for solving such problems is derived, using this structure and leveraging recent results in entropy-regularized optimal transport. The algorithm is then demonstrated on a numerical example in robot coordination problem for search and rescue, where three different types of robots are used to cover a given area at minimal cost.

math.OC↗
arXiv2023

Data-Driven Convex Approach to Off-road Navigation via Linear Transfer Operators

We consider the problem of optimal navigation control design for navigation on off-road terrain. We use traversability measure to characterize the degree of difficulty of navigation on the off-road terrain. The traversability measure captures the property of terrain essential for navigation, such as elevation map, terrain roughness, slope, and terrain texture. The terrain with the presence or absence of obstacles becomes a particular case of the proposed traversability measure. We provide a convex formulation to the off-road navigation problem by lifting the problem to the density space using the linear Perron-Frobenius (P-F) operator. The convex formulation leads to an infinite-dimensional optimal navigation problem for control synthesis. The finite-dimensional approximation of the infinite-dimensional convex problem is constructed using data. We use a computational framework involving the Koopman operator and the duality between the Koopman and P-F operator for the data-driven approximation. This makes our proposed approach data-driven and can be applied in cases where an explicit system model is unavailable. Finally, we demonstrate the application of the developed framework for the navigation of vehicle dynamics with Dubin's car model.

eess.SY↗
arXiv2023

DiffCollage: Parallel Generation of Large Content with Diffusion Models

We present DiffCollage, a compositional diffusion model that can generate large content by leveraging diffusion models trained on generating pieces of the large content. Our approach is based on a factor graph representation where each factor node represents a portion of the content and a variable node represents their overlap. This representation allows us to aggregate intermediate outputs from diffusion models defined on individual nodes to generate content of arbitrary size and shape in parallel without resorting to an autoregressive generation procedure. We apply DiffCollage to various tasks, including infinite image generation, panorama image generation, and long-duration text-guided motion generation. Extensive experimental results with a comparison to strong autoregressive baselines verify the effectiveness of our approach.

cs.CV↗
arXiv2023

Covariance Steering for Nonlinear Control-affine Systems

We consider the covariance steering problem for nonlinear control-affine systems. Our objective is to find an optimal control strategy to steer the state of a system from an initial distribution to a target one whose mean and covariance are given. Due to the nonlinearity, the existing techniques for linear covariance steering problems are not directly applicable. By leveraging the celebrated Girsanov theorem, we formulate the problem into an optimization over the space path distributions. We then adopt a generalized proximal gradient algorithm to solve this optimization, where each update requires solving a linear covariance steering problem. Our algorithm is guaranteed to converge to a local optimal solution with a sublinear rate. In addition, each iteration of the algorithm can be achieved in closed form, and thus the computational complexity of it is insensitive to the resolution of time-discretization. In the examples, our method achieves 1000 times speedup over an existing algorithm.

math.OC↗
arXiv2023

gDDIM: Generalized denoising diffusion implicit models

Our goal is to extend the denoising diffusion implicit model (DDIM) to general diffusion models~(DMs) besides isotropic diffusions. Instead of constructing a non-Markov noising process as in the original DDIM, we examine the mechanism of DDIM from a numerical perspective. We discover that the DDIM can be obtained by using some specific approximations of the score when solving the corresponding stochastic differential equation. We present an interpretation of the accelerating effects of DDIM that also explains the advantages of a deterministic sampling scheme over the stochastic one for fast sampling. Building on this insight, we extend DDIM to general DMs, coined generalized DDIM (gDDIM), with a small but delicate modification in parameterizing the score network. We validate gDDIM in two non-isotropic DMs: Blurring diffusion model (BDM) and Critically-damped Langevin diffusion model (CLD). We observe more than 20 times acceleration in BDM. In the CLD, a diffusion model by augmenting the diffusion process with velocity, our algorithm achieves an FID score of 2.26, on CIFAR10, with only 50 number of score function evaluations~(NFEs) and an FID score of 2.86 with only 27 NFEs. Code is available at https://github.com/qsh-zh/gDDIM

cs.LG↗
arXiv2023

Fast Sampling of Diffusion Models with Exponential Integrator

The past few years have witnessed the great success of Diffusion models~(DMs) in generating high-fidelity samples in generative modeling tasks. A major limitation of the DM is its notoriously slow sampling procedure which normally requires hundreds to thousands of time discretization steps of the learned diffusion process to reach the desired accuracy. Our goal is to develop a fast sampling method for DMs with a much less number of steps while retaining high sample quality. To this end, we systematically analyze the sampling procedure in DMs and identify key factors that affect the sample quality, among which the method of discretization is most crucial. By carefully examining the learned diffusion process, we propose Diffusion Exponential Integrator Sampler~(DEIS). It is based on the Exponential Integrator designed for discretizing ordinary differential equations (ODEs) and leverages a semilinear structure of the learned diffusion process to reduce the discretization error. The proposed method can be applied to any DMs and can generate high-fidelity samples in as few as 10 steps. In our experiments, it takes about 3 minutes on one A6000 GPU to generate $50k$ images from CIFAR10. Moreover, by directly using pre-trained DMs, we achieve the state-of-art sampling performance when the number of score function evaluation~(NFE) is limited, e.g., 4.17 FID with 10 NFEs, 3.37 FID, and 9.74 IS with only 15 NFEs on CIFAR10. Code is available at https://github.com/qsh-zh/deis

cs.LG↗
arXiv2023

A Proximal Algorithm for Sampling from Non-convex Potentials

We study sampling problems associated with non-convex potentials that meanwhile lack smoothness. In particular, we consider target distributions that satisfy either logarithmic-Sobolev inequality or Poincaré inequality. Rather than smooth, the potentials are assumed to be semi-smooth or the summation of multiple semi-smooth functions. We develop a sampling algorithm that resembles proximal algorithms in optimization for this challenging sampling task. Our algorithm is based on a special case of Gibbs sampling known as the alternating sampling framework (ASF). The key contribution of this work is a practical realization of the ASF based on rejection sampling in the non-convex and semi-smooth setting. This work extends the recent algorithm in \cite{LiaChe21,LiaChe22} for non-smooth/semi-smooth log-concave distribution to the setting with non-convex potentials. In almost all the cases of sampling considered in this work, our proximal sampling algorithm achieves better complexity than all existing methods.

cs.LG↗
arXiv2023

A Gaussian variational inference approach to motion planning

We propose a Gaussian variational inference framework for the motion planning problem. In this framework, motion planning is formulated as an optimization over the distribution of the trajectories to approximate the desired trajectory distribution by a tractable Gaussian distribution. Equivalently, the proposed framework can be viewed as a standard motion planning with an entropy regularization. Thus, the solution obtained is a transition from an optimal deterministic solution to a stochastic one, and the proposed framework can recover the deterministic solution by controlling the level of stochasticity. To solve this optimization, we adopt the natural gradient descent scheme. The sparsity structure of the proposed formulation induced by factorized objective functions is further leveraged to improve the scalability of the algorithm. We evaluate our method on several robot systems in simulated environments, and show that it achieves collision avoidance with smooth trajectories, and meanwhile brings robustness to the deterministic baseline results, especially in challenging environments and tasks.

cs.RO↗
arXiv2022

Geometry of finite-time thermodynamic cycles with anisotropic thermal fluctuations

In contrast to the classical concept of a Carnot engine that alternates contact between heat baths of different temperatures, naturally occurring processes usually harvest energy from anisotropy, being exposed simultaneously to chemical and thermal fluctuations of different intensities. In these cases, the enabling mechanism responsible for transduction of energy is typically the presence of a non-equilibrium steady state (NESS). A suitable stochastic model for such a phenomenon is the Brownian gyrator -- a two-degree of freedom stochastically driven system that exchanges energy and heat with the environment. In the context of such a model we present, from a stochastic control perspective, a geometric view of the energy harvesting mechanism that entails a forced periodic trajectory of the system state on the thermodynamic manifold. Dissipation and work output are expressed accordingly as path integrals of a controlled process, and fundamental limitations on power and efficiency are expressed in geometric terms via a relationship to an isoperimetric problem. The theory is presented for high-order systems far from equilibrium and beyond the linear response regime.

eess.SY↗
arXiv2022

An optimal control approach to particle filtering on Lie groups

We study the filtering problem over a Lie group that plays an important role in robotics and aerospace applications. We present a new particle filtering algorithm based on stochastic control. In particular, our algorithm is based on a duality between smoothing and optimal control. Leveraging this duality, we reformulate the smoothing problem into an optimal control problem, and by approximately solving it (using, e.g., iLQR) we establish a superior proposal for particle smoothing. Combining it with a suitably designed sliding window mechanism, we obtain a particle filtering algorithm that suffers less from sample degeneracy compared with existing methods. The efficacy of our algorithm is illustrated by a filtering problem over SO(3) for satellite attitude estimation.

math.OC↗
arXiv2022

Neural Monge Map estimation and its applications

Monge map refers to the optimal transport map between two probability distributions and provides a principled approach to transform one distribution to another. Neural network based optimal transport map solver has gained great attention in recent years. Along this line, we present a scalable algorithm for computing the neural Monge map between two probability distributions. Our algorithm is based on a weak form of the optimal transport problem, thus it only requires samples from the marginals instead of their analytic expressions, and can accommodate optimal transport between two distributions with different dimensions. Our algorithm is suitable for general cost functions, compared with other existing methods for estimating Monge maps using samples, which are usually for quadratic costs. The performance of our algorithms is demonstrated through a series of experiments with both synthetic and realistic data, including text-to-image generation and image inpainting tasks.

cs.LG↗
arXiv2022

Signed Graph Neural Networks: A Frequency Perspective

Graph convolutional networks (GCNs) and its variants are designed for unsigned graphs containing only positive links. Many existing GCNs have been derived from the spectral domain analysis of signals lying over (unsigned) graphs and in each convolution layer they perform low-pass filtering of the input features followed by a learnable linear transformation. Their extension to signed graphs with positive as well as negative links imposes multiple issues including computational irregularities and ambiguous frequency interpretation, making the design of computationally efficient low pass filters challenging. In this paper, we address these issues via spectral analysis of signed graphs and propose two different signed graph neural networks, one keeps only low-frequency information and one also retains high-frequency information. We further introduce magnetic signed Laplacian and use its eigendecomposition for spectral analysis of directed signed graphs. We test our methods for node classification and link sign prediction tasks on signed graphs and achieve state-of-the-art performances.

cs.LG↗
arXiv2022

Inertialess Gyrating Engines

A typical model for a gyrating engine consists of an inertial wheel powered by an energy source that generates an angle-dependent torque. Examples of such engines include a pendulum with an externally applied torque, Stirling engines, and the Brownian gyrating engine. Variations in the torque are averaged out by the inertia of the system to produce limit cycle oscillations. While torque generating mechanisms are also ubiquitous in the biological world, where they typically feed on chemical gradients, inertia is not a property that one naturally associates with such processes. In the present work, seeking ways to dispense of the need for inertial effects, we study an inertia-less concept where the combined effect of coupled torque-producing components averages out variations in the ambient potential and helps overcome dissipative forces to allow sustained operation for vanishingly small inertia. We exemplify this inertia-less concept through analysis of two of the aforementioned engines, the Stirling engine and the Brownian gyrating engine. An analogous principle may be sought in biomolecular processes as well as in modern-day technological engines, where for the latter, the coupled torque-producing components reduce vibrations that stem from the variability of the generated torque.

cond-mat.stat-mech↗
arXiv2022

Variational Wasserstein gradient flow

Wasserstein gradient flow has emerged as a promising approach to solve optimization problems over the space of probability distributions. A recent trend is to use the well-known JKO scheme in combination with input convex neural networks to numerically implement the proximal step. The most challenging step, in this setup, is to evaluate functions involving density explicitly, such as entropy, in terms of samples. This paper builds on the recent works with a slight but crucial difference: we propose to utilize a variational formulation of the objective function formulated as maximization over a parametric class of functions. Theoretically, the proposed variational formulation allows the construction of gradient flows directly for empirical distributions with a well-defined and meaningful objective function. Computationally, this approach replaces the computationally expensive step in existing methods, to handle objective functions involving density, with inner loop updates that only require a small batch of samples and scale well with the dimension. The performance and scalability of the proposed method are illustrated with the aid of several numerical experiments involving high-dimensional synthetic and real datasets.

cs.LG↗
arXiv2022

Stochastic thermodynamic engines under time-varying temperature profile

In the present paper, we study the power output and efficiency of overdamped stochastic thermodynamic engines that are in contact with a heat bath having a temperature that varies periodically with time. This is in contrast to most of the existing literature that considers the Carnot paradigm of alternating contact with heat baths having different fixed temperatures, hot and cold. Specifically, we consider a periodic and bounded but otherwise arbitrary temperature profile and derive explicit bounds on the power and efficiency achievable by a suitable controlling potential that couples the thermodynamic engine to the external world. Standing assumptions in our analysis are bounds on the norm of the gradient of effective potentials -- in the absence of any such constraint, the physically questionable conclusion of arbitrarily large power can be drawn.

cond-mat.stat-mech↗