Dynamical low-rank approximation, as has been demonstrated recently, can be extremely efficient in solving kinetic equations. However, a major deficiency is that they do not preserve the structure of the underlying physical problem. For example, the classic dynamical low-rank methods violate mass, momentum, and energy conservation. In [L. Einkemmer, I. Joseph, J. Comput. Phys. 443:110495, 2021] a conservative dynamical low-rank approach has been proposed. However, directly integrating the resulting equations of motion, similar to the classic dynamical low-rank approach, results in an ill-posed scheme. In this work we propose a robust, i.e. well-posed, integrator for the conservative dynamical low-rank approach that conserves mass and momentum (up to machine precision) and significantly improves energy conservation. We also report improved qualitative results for some problems and show how the approach can be combined with a rank adaptive scheme.
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Empirical studies of the loss landscape of deep networks have revealed that many local minima are connected through low-loss valleys. Yet, little is known about the theoretical origin of such valleys. We present a general framework for finding continuous symmetries in the parameter space, which carve out low-loss valleys. Our framework uses equivariances of the activation functions and can be applied to different layer architectures. To generalize this framework to nonlinear neural networks, we introduce a novel set of nonlinear, data-dependent symmetries. These symmetries can transform a trained model such that it performs similarly on new samples, which allows ensemble building that improves robustness under certain adversarial attacks. We then show that conserved quantities associated with linear symmetries can be used to define coordinates along low-loss valleys. The conserved quantities help reveal that using common initialization methods, gradient flow only explores a small part of the global minimum. By relating conserved quantities to convergence rate and sharpness of the minimum, we provide insights on how initialization impacts convergence and generalizability.
Three asymptotic limits exist for the Euler equations at low Mach number - purely convective, purely acoustic, and mixed convective-acoustic. Standard collocated density-based numerical schemes for compressible flow are known to fail at low Mach number due to the incorrect asymptotic scaling of the artificial diffusion. Previous studies of this class of schemes have shown a variety of behaviours across the different limits and proposed guidelines for the design of low-Mach schemes. However, these studies have primarily focused on specific discretisations and/or only the convective limit. In this paper, we review the low-Mach behaviour using the modified equations - the continuous Euler equations augmented with artificial diffusion terms - which are representative of a wide range of schemes in this class. By considering both convective and acoustic effects, we show that three diffusion scalings naturally arise. Single- and multiple-scale asymptotic analysis of these scalings shows that many of the important low-Mach features of this class of schemes can be reproduced in a straightforward manner in the continuous setting. As an example, we show that many existing low-Mach Roe-type finite-volume schemes match one of these three scalings. Our analysis corroborates previous analysis of these schemes, and we are able to refine previous guidelines on the design of low-Mach schemes by including both convective and acoustic effects. Discrete analysis and numerical examples demonstrate the behaviour of minimal Roe-type schemes with each of the three scalings for convective, acoustic, and mixed flows.
Various contrastive learning approaches have been proposed in recent years and achieve significant empirical success. While effective and prevalent, contrastive learning has been less explored for time series data. A key component of contrastive learning is to select appropriate augmentations imposing some priors to construct feasible positive samples, such that an encoder can be trained to learn robust and discriminative representations. Unlike image and language domains where ``desired'' augmented samples can be generated with the rule of thumb guided by prefabricated human priors, the ad-hoc manual selection of time series augmentations is hindered by their diverse and human-unrecognizable temporal structures. How to find the desired augmentations of time series data that are meaningful for given contrastive learning tasks and datasets remains an open question. In this work, we address the problem by encouraging both high \textit{fidelity} and \textit{variety} based upon information theory. A theoretical analysis leads to the criteria for selecting feasible data augmentations. On top of that, we propose a new contrastive learning approach with information-aware augmentations, InfoTS, that adaptively selects optimal augmentations for time series representation learning. Experiments on various datasets show highly competitive performance with up to 12.0\% reduction in MSE on forecasting tasks and up to 3.7\% relative improvement in accuracy on classification tasks over the leading baselines.
Typical cooperative multi-agent systems (MASs) exchange information to coordinate their motion in proximity-based control consensus schemes to complete a common objective. However, in the event of faults or cyber attacks to on-board positioning sensors of agents, global control performance may be compromised resulting in a hijacking of the entire MAS. For systems that operate in unknown or landmark-free environments (e.g., open terrain, sea, or air) and also beyond range/proximity sensing of nearby agents, compromised agents lose localization capabilities. To maintain resilience in these scenarios, we propose a method to recover compromised agents by utilizing Received Signal Strength Indication (RSSI) from nearby agents (i.e., mobile landmarks) to provide reliable position measurements for localization. To minimize estimation error: i) a multilateration scheme is proposed to leverage RSSI and position information received from neighboring agents as mobile landmarks and ii) a Kalman filtering method adaptively updates the unknown RSSI-based position measurement covariance matrix at runtime that is robust to unreliable state estimates. The proposed framework is demonstrated with simulations on MAS formations in the presence of faults and cyber attacks to on-board position sensors.
We study the corrupted bandit problem, i.e. a stochastic multi-armed bandit problem with $k$ unknown reward distributions, which are heavy-tailed and corrupted by a history-independent adversary or Nature. To be specific, the reward obtained by playing an arm comes from corresponding heavy-tailed reward distribution with probability $1-\varepsilon \in (0.5,1]$ and an arbitrary corruption distribution of unbounded support with probability $\varepsilon \in [0,0.5)$. First, we provide $\textit{a problem-dependent lower bound on the regret}$ of any corrupted bandit algorithm. The lower bounds indicate that the corrupted bandit problem is harder than the classical stochastic bandit problem with sub-Gaussian or heavy-tail rewards. Following that, we propose a novel UCB-type algorithm for corrupted bandits, namely HubUCB, that builds on Huber's estimator for robust mean estimation. Leveraging a novel concentration inequality of Huber's estimator, we prove that HubUCB achieves a near-optimal regret upper bound. Since computing Huber's estimator has quadratic complexity, we further introduce a sequential version of Huber's estimator that exhibits linear complexity. We leverage this sequential estimator to design SeqHubUCB that enjoys similar regret guarantees while reducing the computational burden. Finally, we experimentally illustrate the efficiency of HubUCB and SeqHubUCB in solving corrupted bandits for different reward distributions and different levels of corruptions.
It is a widely observed phenomenon in nonparametric statistics that rate-optimal estimators balance bias and stochastic error. The recent work on overparametrization raises the question whether rate-optimal estimators exist that do not obey this trade-off. In this work we consider pointwise estimation in the Gaussian white noise model with $\beta$-H\"older smooth regression function f. It is shown that an estimator with worst-case bias $\lesssim n^{-\beta/(2\beta+1)}=: \psi_n$ must necessarily also have a worst-case mean absolute deviation that is lower bounded by $\gtrsim \psi_n.$ This proves that any estimator achieving the minimax optimal pointwise estimation rate $\psi_n$ must necessarily balance worst-case bias and worst-case mean absolute deviation. To derive the result, we establish an abstract inequality relating the change of expectation for two probability measures to the mean absolute deviation.
Instrumental variable (IV) strategies are widely used in political science to establish causal relationships. However, the identifying assumptions required by an IV design are demanding, and it remains challenging for researchers to assess their validity. In this paper, we replicate 67 papers published in three top journals in political science during 2010-2022 and identify several troubling patterns. First, researchers often overestimate the strength of their IVs due to non-i.i.d. errors, such as a clustering structure. Second, the most commonly used t-test for the two-stage-least-squares (2SLS) estimates often severely underestimates uncertainty. Using more robust inferential methods, we find that around 19-30% of the 2SLS estimates in our sample are underpowered. Third, in the majority of the replicated studies, the 2SLS estimates are much larger than the ordinary-least-squares estimates, and their ratio is negatively correlated with the strength of the IVs in studies where the IVs are not experimentally generated, suggesting potential violations of unconfoundedness or the exclusion restriction. To help researchers avoid these pitfalls, we provide a checklist for better practice.
We consider the problem of discovering $K$ related Gaussian directed acyclic graphs (DAGs), where the involved graph structures share a consistent causal order and sparse unions of supports. Under the multi-task learning setting, we propose a $l_1/l_2$-regularized maximum likelihood estimator (MLE) for learning $K$ linear structural equation models. We theoretically show that the joint estimator, by leveraging data across related tasks, can achieve a better sample complexity for recovering the causal order (or topological order) than separate estimations. Moreover, the joint estimator is able to recover non-identifiable DAGs, by estimating them together with some identifiable DAGs. Lastly, our analysis also shows the consistency of union support recovery of the structures. To allow practical implementation, we design a continuous optimization problem whose optimizer is the same as the joint estimator and can be approximated efficiently by an iterative algorithm. We validate the theoretical analysis and the effectiveness of the joint estimator in experiments.
Self-supervised learning has been widely used to obtain transferrable representations from unlabeled images. Especially, recent contrastive learning methods have shown impressive performances on downstream image classification tasks. While these contrastive methods mainly focus on generating invariant global representations at the image-level under semantic-preserving transformations, they are prone to overlook spatial consistency of local representations and therefore have a limitation in pretraining for localization tasks such as object detection and instance segmentation. Moreover, aggressively cropped views used in existing contrastive methods can minimize representation distances between the semantically different regions of a single image. In this paper, we propose a spatially consistent representation learning algorithm (SCRL) for multi-object and location-specific tasks. In particular, we devise a novel self-supervised objective that tries to produce coherent spatial representations of a randomly cropped local region according to geometric translations and zooming operations. On various downstream localization tasks with benchmark datasets, the proposed SCRL shows significant performance improvements over the image-level supervised pretraining as well as the state-of-the-art self-supervised learning methods.
Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.
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