Efficient computation of the optimal transport distance between two distributions serves as an algorithm subroutine that empowers various applications. This paper develops a scalable first-order optimization-based method that computes optimal transport to within $\varepsilon$ additive accuracy with runtime $\widetilde{O}( n^2/\varepsilon)$, where $n$ denotes the dimension of the probability distributions of interest. Our algorithm achieves the state-of-the-art computational guarantees among all first-order methods, while exhibiting favorable numerical performance compared to classical algorithms like Sinkhorn and Greenkhorn. Underlying our algorithm designs are two key elements: (a) converting the original problem into a bilinear minimax problem over probability distributions; (b) exploiting the extragradient idea -- in conjunction with entropy regularization and adaptive learning rates -- to accelerate convergence.
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Estimating the entropy rate of discrete time series is a challenging problem with important applications in numerous areas including neuroscience, genomics, image processing and natural language processing. A number of approaches have been developed for this task, typically based either on universal data compression algorithms, or on statistical estimators of the underlying process distribution. In this work, we propose a fully-Bayesian approach for entropy estimation. Building on the recently introduced Bayesian Context Trees (BCT) framework for modelling discrete time series as variable-memory Markov chains, we show that it is possible to sample directly from the induced posterior on the entropy rate. This can be used to estimate the entire posterior distribution, providing much richer information than point estimates. We develop theoretical results for the posterior distribution of the entropy rate, including proofs of consistency and asymptotic normality. The practical utility of the method is illustrated on both simulated and real-world data, where it is found to outperform state-of-the-art alternatives.
Post-training quantization (PTQ) is widely regarded as one of the most efficient compression methods practically, benefitting from its data privacy and low computation costs. We argue that an overlooked problem of oscillation is in the PTQ methods. In this paper, we take the initiative to explore and present a theoretical proof to explain why such a problem is essential in PTQ. And then, we try to solve this problem by introducing a principled and generalized framework theoretically. In particular, we first formulate the oscillation in PTQ and prove the problem is caused by the difference in module capacity. To this end, we define the module capacity (ModCap) under data-dependent and data-free scenarios, where the differentials between adjacent modules are used to measure the degree of oscillation. The problem is then solved by selecting top-k differentials, in which the corresponding modules are jointly optimized and quantized. Extensive experiments demonstrate that our method successfully reduces the performance drop and is generalized to different neural networks and PTQ methods. For example, with 2/4 bit ResNet-50 quantization, our method surpasses the previous state-of-the-art method by 1.9%. It becomes more significant on small model quantization, e.g. surpasses BRECQ method by 6.61% on MobileNetV2*0.5.
The paper focuses on a new error analysis of a class of mixed FEMs for stationary incompressible magnetohydrodynamics with the standard inf-sup stable velocity-pressure space pairs to Navier-Stokes equations and the N\'ed\'elec's edge element for the magnetic field. The methods have been widely used in various numerical simulations in the last several decades, while the existing analysis is not optimal due to the strong coupling of system and the pollution of the lower-order N\'ed\'elec's edge approximation in analysis. In terms of a newly modified Maxwell projection we establish new and optimal error estimates. In particular, we prove that the method based on the commonly-used Taylor-Hood/lowest-order N\'ed\'elec's edge element is efficient and the method provides the second-order accuracy for numerical velocity. Two numerical examples for the problem in both convex and nonconvex polygonal domains are presented. Numerical results confirm our theoretical analysis.
We propose new tools for the geometric exploration of data objects taking values in a general separable metric space $(\Omega, d)$. Given a probability measure on $\Omega$, we introduce depth profiles, where the depth profile of an element $\omega\in\Omega$ refers to the distribution of the distances between $\omega$ and the other elements of $\Omega$. Depth profiles can be harnessed to define transport ranks, which capture the centrality of each element in $\Omega$ with respect to the entire data cloud based on optimal transport maps between depth profiles. We study the properties of transport ranks and show that they provide an effective device for detecting and visualizing patterns in samples of random objects and also entail notions of transport medians, modes, level sets and quantiles for data in general separable metric spaces. Specifically, we study estimates of depth profiles and transport ranks based on samples of random objects and establish the convergence of the empirical estimates to the population targets using empirical process theory. We demonstrate the usefulness of depth profiles and associated transport ranks and visualizations for distributional data through a sample of age-at-death distributions for various countries, for compositional data through energy usage for U.S. states and for network data through New York taxi trips.
The extensive-form game has been studied considerably in recent years. It can represent games with multiple decision points and incomplete information, and hence it is helpful in formulating games with uncertain inputs, such as poker. We consider an extended-form game with two players and zero-sum, i.e., the sum of their payoffs is always zero. In such games, the problem of finding the optimal strategy can be formulated as a bilinear saddle-point problem. This formulation grows huge depending on the size of the game, since it has variables representing the strategies at all decision points for each player. To solve such large-scale bilinear saddle-point problems, the excessive gap technique (EGT), a smoothing method, has been studied. This method generates a sequence of approximate solutions whose error is guaranteed to converge at $\mathcal{O}(1/k)$, where $k$ is the number of iterations. However, it has the disadvantage of having poor theoretical bounds on the error related to the game size. This makes it inapplicable to large games. Our goal is to improve the smoothing method for solving extensive-form games so that it can be applied to large-scale games. To this end, we make two contributions in this work. First, we slightly modify the strongly convex function used in the smoothing method in order to improve the theoretical bounds related to the game size. Second, we propose a heuristic called centering trick, which allows the smoothing method to be combined with other methods and consequently accelerates the convergence in practice. As a result, we combine EGT with CFR+, a state-of-the-art method for extensive-form games, to achieve good performance in games where conventional smoothing methods do not perform well. The proposed smoothing method is shown to have the potential to solve large games in practice.
We develop an optimization-based algorithm for parametric model order reduction (PMOR) of linear time-invariant dynamical systems. Our method aims at minimizing the $\mathcal{H}_\infty \otimes \mathcal{L}_\infty$ approximation error in the frequency and parameter domain by an optimization of the reduced order model (ROM) matrices. State-of-the-art PMOR methods often compute several nonparametric ROMs for different parameter samples, which are then combined to a single parametric ROM. However, these parametric ROMs can have a low accuracy between the utilized sample points. In contrast, our optimization-based PMOR method minimizes the approximation error across the entire parameter domain. Moreover, due to our flexible approach of optimizing the system matrices directly, we can enforce favorable features such as a port-Hamiltonian structure in our ROMs across the entire parameter domain. Our method is an extension of the recently developed SOBMOR-algorithm to parametric systems. We extend both the ROM parameterization and the adaptive sampling procedure to the parametric case. Several numerical examples demonstrate the effectiveness and high accuracy of our method in a comparison with other PMOR methods.
Finding meaningful ways to measure the statistical dependency between random variables $\xi$ and $\zeta$ is a timeless statistical endeavor. In recent years, several novel concepts, like the distance covariance, have extended classical notions of dependency to more general settings. In this article, we propose and study an alternative framework that is based on optimal transport. The transport dependency $\tau \ge 0$ applies to general Polish spaces and intrinsically respects metric properties. For suitable ground costs, independence is fully characterized by $\tau = 0$. Via proper normalization of $\tau$, three transport correlations $\rho_\alpha$, $\rho_\infty$, and $\rho_*$ with values in $[0, 1]$ are defined. They attain the value $1$ if and only if $\zeta = \varphi(\xi)$, where $\varphi$ is an $\alpha$-Lipschitz function for $\rho_\alpha$, a measurable function for $\rho_\infty$, or a multiple of an isometry for $\rho_*$. The transport dependency can be estimated consistently by an empirical plug-in approach, but alternative estimators with the same convergence rate but significantly reduced computational costs are also proposed. Numerical results suggest that $\tau$ robustly recovers dependency between data sets with different internal metric structures. The usage for inferential tasks, like transport dependency based independence testing, is illustrated on a data set from a cancer study.
Flexible Bayesian models are typically constructed using limits of large parametric models with a multitude of parameters that are often uninterpretable. In this article, we offer a novel alternative by constructing an exponentially tilted empirical likelihood carefully designed to concentrate near a parametric family of distributions of choice with respect to a novel variant of the Wasserstein metric, which is then combined with a prior distribution on model parameters to obtain a robustified posterior. The proposed approach finds applications in a wide variety of robust inference problems, where we intend to perform inference on the parameters associated with the centering distribution in presence of outliers. Our proposed transport metric enjoys great computational simplicity, exploiting the Sinkhorn regularization for discrete optimal transport problems, and being inherently parallelizable. We demonstrate superior performance of our methodology when compared against state-of-the-art robust Bayesian inference methods. We also demonstrate equivalence of our approach with a nonparametric Bayesian formulation under a suitable asymptotic framework, testifying to its flexibility. The constrained entropy maximization that sits at the heart of our likelihood formulation finds its utility beyond robust Bayesian inference; an illustration is provided in a trustworthy machine learning application.
Novel view synthesis using neural radiance fields (NeRF) is the state-of-the-art technique for generating high-quality images from novel viewpoints. Existing methods require a priori knowledge about extrinsic and intrinsic camera parameters. This limits their applicability to synthetic scenes, or real-world scenarios with the necessity of a preprocessing step. Current research on the joint optimization of camera parameters and NeRF focuses on refining noisy extrinsic camera parameters and often relies on the preprocessing of intrinsic camera parameters. Further approaches are limited to cover only one single camera intrinsic. To address these limitations, we propose a novel end-to-end trainable approach called NeRFtrinsic Four. We utilize Gaussian Fourier features to estimate extrinsic camera parameters and dynamically predict varying intrinsic camera parameters through the supervision of the projection error. Our approach outperforms existing joint optimization methods on LLFF and BLEFF. In addition to these existing datasets, we introduce a new dataset called iFF with varying intrinsic camera parameters. NeRFtrinsic Four is a step forward in joint optimization NeRF-based view synthesis and enables more realistic and flexible rendering in real-world scenarios with varying camera parameters.
Topology Optimization seeks to find the best design that satisfies a set of constraints while maximizing system performance. Traditional iterative optimization methods like SIMP can be computationally expensive and get stuck in local minima, limiting their applicability to complex or large-scale problems. Learning-based approaches have been developed to accelerate the topology optimization process, but these methods can generate designs with floating material and low performance when challenged with out-of-distribution constraint configurations. Recently, deep generative models, such as Generative Adversarial Networks and Diffusion Models, conditioned on constraints and physics fields have shown promise, but they require extensive pre-processing and surrogate models for improving performance. To address these issues, we propose a Generative Optimization method that integrates classic optimization like SIMP as a refining mechanism for the topology generated by a deep generative model. We also remove the need for conditioning on physical fields using a computationally inexpensive approximation inspired by classic ODE solutions and reduce the number of steps needed to generate a feasible and performant topology. Our method allows us to efficiently generate good topologies and explicitly guide them to regions with high manufacturability and high performance, without the need for external auxiliary models or additional labeled data. We believe that our method can lead to significant advancements in the design and optimization of structures in engineering applications, and can be applied to a broader spectrum of performance-aware engineering design problems.
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