We study the problem of verification and synthesis of robust control barrier functions (CBF) for control-affine polynomial systems with bounded additive uncertainty and convex polynomial constraints on the control. We first formulate robust CBF verification and synthesis as multilevel polynomial optimization problems (POP), where verification optimizes -- in three levels -- the uncertainty, control, and state, while synthesis additionally optimizes the parameter of a chosen parametric CBF candidate. We then show that, by invoking the KKT conditions of the inner optimizations over uncertainty and control, the verification problem can be simplified as a single-level POP and the synthesis problem reduces to a min-max POP. This reduction leads to multilevel semidefinite relaxations. For the verification problem, we apply Lasserre's hierarchy of moment relaxations. For the synthesis problem, we draw connections to existing relaxation techniques for robust min-max POP, which first use sum-of-squares programming to find increasingly tight polynomial lower bounds to the unknown value function of the verification POP, and then call Lasserre's hierarchy again to maximize the lower bounds. Both semidefinite relaxations guarantee asymptotic global convergence to optimality. We provide an in-depth study of our framework on the controlled Van der Pol Oscillator, both with and without additive uncertainty.
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We introduce a new debiasing framework for high-dimensional linear regression that bypasses the restrictions on covariate distributions imposed by modern debiasing technology. We study the prevalent setting where the number of features and samples are both large and comparable. In this context, state-of-the-art debiasing technology uses a degrees-of-freedom correction to remove shrinkage bias of regularized estimators and conduct inference. However, this method requires that the observed samples are i.i.d., the covariates follow a mean zero Gaussian distribution, and reliable covariance matrix estimates for observed features are available. This approach struggles when (i) covariates are non-Gaussian with heavy tails or asymmetric distributions, (ii) rows of the design exhibit heterogeneity or dependencies, and (iii) reliable feature covariance estimates are lacking. To address these, we develop a new strategy where the debiasing correction is a rescaled gradient descent step (suitably initialized) with step size determined by the spectrum of the sample covariance matrix. Unlike prior work, we assume that eigenvectors of this matrix are uniform draws from the orthogonal group. We show this assumption remains valid in diverse situations where traditional debiasing fails, including designs with complex row-column dependencies, heavy tails, asymmetric properties, and latent low-rank structures. We establish asymptotic normality of our proposed estimator (centered and scaled) under various convergence notions. Moreover, we develop a consistent estimator for its asymptotic variance. Lastly, we introduce a debiased Principal Component Regression (PCR) technique using our Spectrum-Aware approach. In varied simulations and real data experiments, we observe that our method outperforms degrees-of-freedom debiasing by a margin.
Most of the existing work in one-stage referring expression comprehension (REC) mainly focuses on multi-modal fusion and reasoning, while the influence of other factors in this task lacks in-depth exploration. To fill this gap, we conduct an empirical study in this paper. Concretely, we first build a very simple REC network called SimREC, and ablate 42 candidate designs/settings, which covers the entire process of one-stage REC from network design to model training. Afterwards, we conduct over 100 experimental trials on three benchmark datasets of REC. The extensive experimental results not only show the key factors that affect REC performance in addition to multi-modal fusion, e.g., multi-scale features and data augmentation, but also yield some findings that run counter to conventional understanding. For example, as a vision and language (V&L) task, REC does is less impacted by language prior. In addition, with a proper combination of these findings, we can improve the performance of SimREC by a large margin, e.g., +27.12% on RefCOCO+, which outperforms all existing REC methods. But the most encouraging finding is that with much less training overhead and parameters, SimREC can still achieve better performance than a set of large-scale pre-trained models, e.g., UNITER and VILLA, portraying the special role of REC in existing V&L research.
Distribution-dependent stochastic dynamical systems arise widely in engineering and science. We consider a class of such systems which model the limit behaviors of interacting particles moving in a vector field with random fluctuations. We aim to examine the most likely transition path between equilibrium stable states of the vector field. In the small noise regime, the action functional does not involve the solution of the skeleton equation which describes the unperturbed deterministic flow of the vector field shifted by the interaction at zero distance. As a result, we are led to study the most likely transition path for a stochastic differential equation without distribution dependency. This enables the computation of the most likely transition path for these distribution-dependent stochastic dynamical systems by the adaptive minimum action method and we illustrate our approach in two examples.
This paper investigates the performance of a singleuser fluid antenna system (FAS), by exploiting a class of elliptical copulas to describe the structure of dependency amongst the fluid antenna ports. By expressing Jakes' model in terms of the Gaussian copula, we consider two cases: (i) the general case, i.e., any arbitrary correlated fading distribution; and (ii) the specific case, i.e., correlated Nakagami-m fading. For both scenarios, we first derive analytical expressions for the cumulative distribution function (CDF) and probability density function (PDF) of the equivalent channel in terms of multivariate normal distribution. Then, we obtain the outage probability (OP) and the delay outage rate (DOR) to analyze the performance of the FAS. By employing the popular rank correlation coefficients such as Spearman's \{rho} and Kendall's {\tau}, we measure the degree of dependency in correlated arbitrary fading channels and illustrate how the Gaussian copula can be accurately connected to Jakes' model in FAS without complicated mathematical analysis. Numerical results show that increasing the fluid antenna size provides lower OP and DOR, but the system performance saturates as the number of antenna ports increases. In addition, our results indicate that FAS provides better performance compared to conventional single-fixed antenna systems even when the size of fluid antenna is small.
The myriad complex systems with multiway interactions motivate the extension of graph-based pairwise connections to higher-order relations. In particular, the simplicial complex has inspired generalizations of graph neural networks (GNNs) to simplicial complex-based models. Learning on such systems requires large amounts of data, which can be expensive or impossible to obtain. We propose data augmentation of simplicial complexes through both linear and nonlinear mixup mechanisms that return mixtures of existing labeled samples. In addition to traditional pairwise mixup, we present a convex clustering mixup approach for a data-driven relationship among several simplicial complexes. We theoretically demonstrate that the resultant synthetic simplicial complexes interpolate among existing data with respect to homomorphism densities. Our method is demonstrated on both synthetic and real-world datasets for simplicial complex classification.
Markov processes are widely used mathematical models for describing dynamic systems in various fields. However, accurately simulating large-scale systems at long time scales is computationally expensive due to the short time steps required for accurate integration. In this paper, we introduce an inference process that maps complex systems into a simplified representational space and models large jumps in time. To achieve this, we propose Time-lagged Information Bottleneck (T-IB), a principled objective rooted in information theory, which aims to capture relevant temporal features while discarding high-frequency information to simplify the simulation task and minimize the inference error. Our experiments demonstrate that T-IB learns information-optimal representations for accurately modeling the statistical properties and dynamics of the original process at a selected time lag, outperforming existing time-lagged dimensionality reduction methods.
Research on dynamics of robotic manipulators provides promising support for model-based control. In general, rigorous first-principles-based dynamics modeling and accurate identification of mechanism parameters are critical to achieving high precision in model-based control, while data-driven model reconstruction provides alternative approaches of the above process. Taking the level of activation of data as an indicator, this paper classifies the collected robotic manipulator data by means of K-means clustering algorithm. With the fundamental prior knowledge, we find the corresponding dynamical properties behind the classified data separately. Afterwards, the sparse identification of nonlinear dynamics (SINDy) method is used to reconstruct the dynamics model of the robotic manipulator step by step according to the activation level of the classified data. The simulation results show that the proposed method not only reduces the complexity of the basis function library, enabling the application of SINDy method to multi-degree-of-freedom robotic manipulators, but also decreases the influence of data noise on the regression results. Finally, the dynamic control based on the reconfigured model is deployed on the experimental platform, and the experimental results prove the effectiveness of the proposed method.
We propose a new paradigm to automatically generate training data with accurate labels at scale using the text-to-image synthesis frameworks (e.g., DALL-E, Stable Diffusion, etc.). The proposed approach1 decouples training data generation into foreground object generation, and contextually coherent background generation. To generate foreground objects, we employ a straightforward textual template, incorporating the object class name as input prompts. This is fed into a text-to-image synthesis framework, producing various foreground images set against isolated backgrounds. A foreground-background segmentation algorithm is then used to generate foreground object masks. To generate context images, we begin by creating language descriptions of the context. This is achieved by applying an image captioning method to a small set of images representing the desired context. These textual descriptions are then transformed into a diverse array of context images via a text-to-image synthesis framework. Subsequently, we composite these with the foreground object masks produced in the initial step, utilizing a cut-and-paste method, to formulate the training data. We demonstrate the advantages of our approach on five object detection and segmentation datasets, including Pascal VOC and COCO. We found that detectors trained solely on synthetic data produced by our method achieve performance comparable to those trained on real data (Fig. 1). Moreover, a combination of real and synthetic data yields even much better results. Further analysis indicates that the synthetic data distribution complements the real data distribution effectively. Additionally, we emphasize the compositional nature of our data generation approach in out-of-distribution and zero-shot data generation scenarios. We open-source our code at //github.com/gyhandy/Text2Image-for-Detection
This work considers the question of how convenient access to copious data impacts our ability to learn causal effects and relations. In what ways is learning causality in the era of big data different from -- or the same as -- the traditional one? To answer this question, this survey provides a comprehensive and structured review of both traditional and frontier methods in learning causality and relations along with the connections between causality and machine learning. This work points out on a case-by-case basis how big data facilitates, complicates, or motivates each approach.
We introduce a multi-task setup of identifying and classifying entities, relations, and coreference clusters in scientific articles. We create SciERC, a dataset that includes annotations for all three tasks and develop a unified framework called Scientific Information Extractor (SciIE) for with shared span representations. The multi-task setup reduces cascading errors between tasks and leverages cross-sentence relations through coreference links. Experiments show that our multi-task model outperforms previous models in scientific information extraction without using any domain-specific features. We further show that the framework supports construction of a scientific knowledge graph, which we use to analyze information in scientific literature.
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