On ergodic optimization for unimodal maps
In this article, we show that for a typical non-uniformly expanding unimodal map, the unique maximizing measure of a generic Lipschitz function is supported on a periodic orbit.
math.DS↗
arXiv subjects
Rui Gao
Publications and source records attributed to Rui Gao.
Transitive fusion systems over a class of finite p-groups
Let $p$ be an odd prime and $S$ a nonabelian finite $p$-group. In [9, 10], they proposed the following conjecture: if $\mathcal{F}$ be a transitive fusion system over a finite $p$-group $S$, then $S$ is either extraspecial of order $p^{3}$ or elementary abelian. In this note, we use an easy method to prove that this conjecture holds when the $p$-rank of $S$ is 2.
math.GR↗
H-SIREN: Improving implicit neural representations with hyperbolic periodic functions
Implicit neural representations (INR) have been recently adopted in various applications ranging from computer vision tasks to physics simulations by solving partial differential equations. Among existing INR-based works, multi-layer perceptrons with sinusoidal activation functions find widespread applications and are also frequently treated as a baseline for the development of better activation functions for INR applications. Recent investigations claim that the use of sinusoidal activation functions could be sub-optimal due to their limited supported frequency set as well as their tendency to generate over-smoothed solutions. We provide a simple solution to mitigate such an issue by changing the activation function at the first layer from $\sin(x)$ to $\sin(\sinh(2x))$. We demonstrate H-SIREN in various computer vision and fluid flow problems, where it surpasses the performance of several state-of-the-art INRs.
cs.CV↗
Regularization for Adversarial Robust Learning
Despite the growing prevalence of artificial neural networks in real-world applications, their vulnerability to adversarial attacks remains a significant concern, which motivates us to investigate the robustness of machine learning models. While various heuristics aim to optimize the distributionally robust risk using the $\infty$-Wasserstein metric, such a notion of robustness frequently encounters computation intractability. To tackle the computational challenge, we develop a novel approach to adversarial training that integrates $ϕ$-divergence regularization into the distributionally robust risk function. This regularization brings a notable improvement in computation compared with the original formulation. We develop stochastic gradient methods with biased oracles to solve this problem efficiently, achieving the near-optimal sample complexity. Moreover, we establish its regularization effects and demonstrate it is asymptotic equivalence to a regularized empirical risk minimization framework, by considering various scaling regimes of the regularization parameter and robustness level. These regimes yield gradient norm regularization, variance regularization, or a smoothed gradient norm regularization that interpolates between these extremes. We numerically validate our proposed method in supervised learning, reinforcement learning, and contextual learning and showcase its state-of-the-art performance against various adversarial attacks.
cs.LG↗
Data-driven Multistage Distributionally Robust Linear Optimization with Nested Distance
We study multistage distributionally robust linear optimization, where the uncertainty set is defined as a ball of distribution centered at a scenario tree using the nested distance. The resulting minimax problem is notoriously difficult to solve due to its inherent non-convexity. In this paper, we demonstrate that, under mild conditions, the robust risk evaluation of a given policy can be expressed in an equivalent recursive form. Furthermore, assuming stagewise independence, we derive equivalent dynamic programming reformulations to find an optimal robust policy that is time-consistent and well-defined on unseen sample paths. Our reformulations reconcile two modeling frameworks: the multistage-static formulation (with nested distance) and the multistage-dynamic formulation (with one-period Wasserstein distance). Moreover, we identify tractable cases when the value functions can be computed efficiently using convex optimization techniques.
math.OC↗
Min-max theory and existence of H-spheres with arbitrary codimensions
We demonstrate the existence of branched immersed 2-spheres with prescribed mean curvature, with controlled Morse index and with arbitrary codimensions in closed Riemannian manifold $N$ admitting finite fundamental group, where $π_k(N) \neq 0$ and $k \geq 2$, for certain generic choice of prescribed mean curvature vector. Moreover, we enhance this existence result to encompass all possible choices of prescribed mean curvatures under certain Ricci curvature condition on $N$ when $\dim{N} = 3$. When $\dim{N} \geq 4$, we establish a Morse index lower bound while $N$ satisfies some isotropic curvature condition. As a consequence, we can leverage latter strengthened result to construct 2-spheres with parallel mean curvature when $N$ has positive isotropic curvature and $\dim{N} \geq 4$. At last, we partially resolve the homotopy problem concerning the existence of a representative surface with prescribed mean curvature type vector field in some given homotopy classes.
math.DG↗
SFC: Achieve Accurate Fast Convolution under Low-precision Arithmetic
Fast convolution algorithms, including Winograd and FFT, can efficiently accelerate convolution operations in deep models. However, these algorithms depend on high-precision arithmetic to maintain inference accuracy, which conflicts with the model quantization. To resolve this conflict and further improve the efficiency of quantized convolution, we proposes SFC, a new algebra transform for fast convolution by extending the Discrete Fourier Transform (DFT) with symbolic computing, in which only additions are required to perform the transformation at specific transform points, avoiding the calculation of irrational number and reducing the requirement for precision. Additionally, we enhance convolution efficiency by introducing correction terms to convert invalid circular convolution outputs of the Fourier method into effective ones. The numerical error analysis is presented for the first time in this type of work and proves that our algorithms can provide a 3.68x multiplication reduction for 3x3 convolution, while the Winograd algorithm only achieves a 2.25x reduction with similarly low numerical errors. Experiments carried out on benchmarks and FPGA show that our new algorithms can further improve the computation efficiency of quantized models while maintaining accuracy, surpassing both the quantization-alone method and existing works on fast convolution quantization.
cs.LG↗
A Short and General Duality Proof for Wasserstein Distributionally Robust Optimization
We present a general duality result for Wasserstein distributionally robust optimization that holds for any Kantorovich transport cost, measurable loss function, and nominal probability distribution. Assuming an interchangeability principle inherent in existing duality results, our proof only uses one-dimensional convex analysis. Furthermore, we demonstrate that the interchangeability principle holds if and only if certain measurable projection and weak measurable selection conditions are satisfied. To illustrate the broader applicability of our approach, we provide a rigorous treatment of duality results in distributionally robust Markov decision processes and distributionally robust multistage stochastic programming. Additionally, we extend our analysis to other problems such as infinity-Wasserstein distributionally robust optimization, risk-averse optimization, and globalized distributionally robust counterpart.
math.OC↗
Relating-Up: Advancing Graph Neural Networks through Inter-Graph Relationships
Graph Neural Networks (GNNs) have excelled in learning from graph-structured data, especially in understanding the relationships within a single graph, i.e., intra-graph relationships. Despite their successes, GNNs are limited by neglecting the context of relationships across graphs, i.e., inter-graph relationships. Recognizing the potential to extend this capability, we introduce Relating-Up, a plug-and-play module that enhances GNNs by exploiting inter-graph relationships. This module incorporates a relation-aware encoder and a feedback training strategy. The former enables GNNs to capture relationships across graphs, enriching relation-aware graph representation through collective context. The latter utilizes a feedback loop mechanism for the recursively refinement of these representations, leveraging insights from refining inter-graph dynamics to conduct feedback loop. The synergy between these two innovations results in a robust and versatile module. Relating-Up enhances the expressiveness of GNNs, enabling them to encapsulate a wider spectrum of graph relationships with greater precision. Our evaluations across 16 benchmark datasets demonstrate that integrating Relating-Up into GNN architectures substantially improves performance, positioning Relating-Up as a formidable choice for a broad spectrum of graph representation learning tasks.
cs.LG↗
Aleatoric and Epistemic Discrimination: Fundamental Limits of Fairness Interventions
Machine learning (ML) models can underperform on certain population groups due to choices made during model development and bias inherent in the data. We categorize sources of discrimination in the ML pipeline into two classes: aleatoric discrimination, which is inherent in the data distribution, and epistemic discrimination, which is due to decisions made during model development. We quantify aleatoric discrimination by determining the performance limits of a model under fairness constraints, assuming perfect knowledge of the data distribution. We demonstrate how to characterize aleatoric discrimination by applying Blackwell's results on comparing statistical experiments. We then quantify epistemic discrimination as the gap between a model's accuracy when fairness constraints are applied and the limit posed by aleatoric discrimination. We apply this approach to benchmark existing fairness interventions and investigate fairness risks in data with missing values. Our results indicate that state-of-the-art fairness interventions are effective at removing epistemic discrimination on standard (overused) tabular datasets. However, when data has missing values, there is still significant room for improvement in handling aleatoric discrimination.
cs.LG↗
Two-sample Test using Projected Wasserstein Distance
We develop a projected Wasserstein distance for the two-sample test, a fundamental problem in statistics and machine learning: given two sets of samples, to determine whether they are from the same distribution. In particular, we aim to circumvent the curse of dimensionality in Wasserstein distance: when the dimension is high, it has diminishing testing power, which is inherently due to the slow concentration property of Wasserstein metrics in the high dimension space. A key contribution is to couple optimal projection to find the low dimensional linear mapping to maximize the Wasserstein distance between projected probability distributions. We characterize the theoretical property of the finite-sample convergence rate on IPMs and present practical algorithms for computing this metric. Numerical examples validate our theoretical results.
stat.ML↗
Non-Convex Robust Hypothesis Testing using Sinkhorn Uncertainty Sets
We present a new framework to address the non-convex robust hypothesis testing problem, wherein the goal is to seek the optimal detector that minimizes the maximum of worst-case type-I and type-II risk functions. The distributional uncertainty sets are constructed to center around the empirical distribution derived from samples based on Sinkhorn discrepancy. Given that the objective involves non-convex, non-smooth probabilistic functions that are often intractable to optimize, existing methods resort to approximations rather than exact solutions. To tackle the challenge, we introduce an exact mixed-integer exponential conic reformulation of the problem, which can be solved into a global optimum with a moderate amount of input data. Subsequently, we propose a convex approximation, demonstrating its superiority over current state-of-the-art methodologies in literature. Furthermore, we establish connections between robust hypothesis testing and regularized formulations of non-robust risk functions, offering insightful interpretations. Our numerical study highlights the satisfactory testing performance and computational efficiency of the proposed framework.
stat.ML↗
Sentinel-Guided Zero-Shot Learning: A Collaborative Paradigm without Real Data Exposure
With increasing concerns over data privacy and model copyrights, especially in the context of collaborations between AI service providers and data owners, an innovative SG-ZSL paradigm is proposed in this work. SG-ZSL is designed to foster efficient collaboration without the need to exchange models or sensitive data. It consists of a teacher model, a student model and a generator that links both model entities. The teacher model serves as a sentinel on behalf of the data owner, replacing real data, to guide the student model at the AI service provider's end during training. Considering the disparity of knowledge space between the teacher and student, we introduce two variants of the teacher model: the omniscient and the quasi-omniscient teachers. Under these teachers' guidance, the student model seeks to match the teacher model's performance and explores domains that the teacher has not covered. To trade off between privacy and performance, we further introduce two distinct security-level training protocols: white-box and black-box, enhancing the paradigm's adaptability. Despite the inherent challenges of real data absence in the SG-ZSL paradigm, it consistently outperforms in ZSL and GZSL tasks, notably in the white-box protocol. Our comprehensive evaluation further attests to its robustness and efficiency across various setups, including stringent black-box training protocol.
cs.CV↗
Point cloud-based registration and image fusion between cardiac SPECT MPI and CTA
A method was proposed for the point cloud-based registration and image fusion between cardiac single photon emission computed tomography (SPECT) myocardial perfusion images (MPI) and cardiac computed tomography angiograms (CTA). Firstly, the left ventricle (LV) epicardial regions (LVERs) in SPECT and CTA images were segmented by using different U-Net neural networks trained to generate the point clouds of the LV epicardial contours (LVECs). Secondly, according to the characteristics of cardiac anatomy, the special points of anterior and posterior interventricular grooves (APIGs) were manually marked in both SPECT and CTA image volumes. Thirdly, we developed an in-house program for coarsely registering the special points of APIGs to ensure a correct cardiac orientation alignment between SPECT and CTA images. Fourthly, we employed ICP, SICP or CPD algorithm to achieve a fine registration for the point clouds (together with the special points of APIGs) of the LV epicardial surfaces (LVERs) in SPECT and CTA images. Finally, the image fusion between SPECT and CTA was realized after the fine registration. The experimental results showed that the cardiac orientation was aligned well and the mean distance error of the optimal registration method (CPD with affine transform) was consistently less than 3 mm. The proposed method could effectively fuse the structures from cardiac CTA and SPECT functional images, and demonstrated a potential in assisting in accurate diagnosis of cardiac diseases by combining complementary advantages of the two imaging modalities.
eess.IV↗
Timed-Elastic-Band Based Variable Splitting for Autonomous Trajectory Planning
Existing trajectory planning methods are struggling to handle the issue of autonomous track swinging during navigation, resulting in significant errors when reaching the destination. In this article, we address autonomous trajectory planning problems, which aims at developing innovative solutions to enhance the adaptability and robustness of unmanned systems in navigating complex and dynamic environments. We first introduce the variable splitting (VS) method as a constrained optimization method to reimagine the renowned Timed-Elastic-Band (TEB) algorithm, resulting in a novel collision avoidance approach named Timed-Elastic-Band based variable splitting (TEB-VS). The proposed TEB-VS demonstrates superior navigation stability, while maintaining nearly identical resource consumption to TEB. We then analyze the convergence of the proposed TEB-VS method. To evaluate the effectiveness and efficiency of TEB-VS, extensive experiments have been conducted using TurtleBot2 in both simulated environments and real-world datasets.
eess.SY↗
Predicting fluid-structure interaction with graph neural networks
We present a rotation equivariant, quasi-monolithic graph neural network framework for the reduced-order modeling of fluid-structure interaction systems. With the aid of an arbitrary Lagrangian-Eulerian formulation, the system states are evolved temporally with two sub-networks. The movement of the mesh is reduced to the evolution of several coefficients via complex-valued proper orthogonal decomposition, and the prediction of these coefficients over time is handled by a single multi-layer perceptron. A finite element-inspired hypergraph neural network is employed to predict the evolution of the fluid state based on the state of the whole system. The structural state is implicitly modeled by the movement of the mesh on the solid-fluid interface; hence it makes the proposed framework quasi-monolithic. The effectiveness of the proposed framework is assessed on two prototypical fluid-structure systems, namely the flow around an elastically-mounted cylinder, and the flow around a hyperelastic plate attached to a fixed cylinder. The proposed framework tracks the interface description and provides stable and accurate system state predictions during roll-out for at least 2000 time steps, and even demonstrates some capability in self-correcting erroneous predictions. The proposed framework also enables direct calculation of the lift and drag forces using the predicted fluid and mesh states, in contrast to existing convolution-based architectures. The proposed reduced-order model via graph neural network has implications for the development of physics-based digital twins concerning moving boundaries and fluid-structure interactions.
physics.flu-dyn↗
Axion-meson mixing in light of recent lattice $η$-$η'$ simulations and their two-photon couplings within $U(3)$ chiral theory
We study the mixing of the QCD/QCD-like axion and light-flavor mesons $π^0, η, η'$ within the framework of $U(3)$ chiral perturbation theory up to next-to-leading order in this work. The axion-meson mixing formulas are calculated order by order in the $U(3)$ $δ$-expansion scheme, namely the joint expansions of the momentum, light-quark masses and $1/N_C$. We provide axion-meson mixing relations in terms of the $π^0$-$η$-$η'$ mixing parameters and their masses. The recent lattice simulations on the $η$-$η'$ systems turn out to be able to offer valuable inputs to constrain the unknown low-energy constants. The relation of the mass and decay constant of the axion is then further explored based on our updated calculations. The two-photon couplings of the light-flavor mesons, together with the axion, are also investigated in the $U(3)$ chiral theory up to next-to-leading order in the $δ$-counting scheme.
hep-ph↗
A Finite Element-Inspired Hypergraph Neural Network: Application to Fluid Dynamics Simulations
An emerging trend in deep learning research focuses on the applications of graph neural networks (GNNs) for mesh-based continuum mechanics simulations. Most of these learning frameworks operate on graphs wherein each edge connects two nodes. Inspired by the data connectivity in the finite element method, we present a method to construct a hypergraph by connecting the nodes by elements rather than edges. A hypergraph message-passing network is defined on such a node-element hypergraph that mimics the calculation process of local stiffness matrices. We term this method a finite element-inspired hypergraph neural network, in short FEIH($ϕ$)-GNN. We further equip the proposed network with rotation equivariance, and explore its capability for modeling unsteady fluid flow systems. The effectiveness of the network is demonstrated on two common benchmark problems, namely the fluid flow around a circular cylinder and airfoil configurations. Stabilized and accurate temporal roll-out predictions can be obtained using the $ϕ$-GNN framework within the interpolation Reynolds number range. The network is also able to extrapolate moderately towards higher Reynolds number domain out of the training range.
physics.flu-dyn↗