We propose and analyze a class of meshfree, super-algebraically convergent methods for partial differential equations (PDEs) on surfaces using Fourier extensions minimizing a measure of non-smoothness (such as a Sobolev norm). Current spectral methods for surface PDEs are primarily limited to a small class of surfaces, such as subdomains of spheres. Other high order methods for surface PDEs typically use radial basis functions (RBFs). Many of these methods are not well-understood analytically for surface PDEs and are highly ill-conditioned. Our methods work by extending a surface PDE into a box-shaped domain so that differential operators of the extended function agree with the surface differential operators, as in the Closest Point Method. The methods can be proven to converge super-algebraically for certain well-posed linear PDEs, and spectral convergence to machine error has been observed numerically for a variety of problems. Our approach works on arbitrary smooth surfaces (closed or non-closed) defined by point clouds with minimal conditions.
Knowledge Distillation (KD) for object detection aims to train a compact detector by transferring knowledge from a teacher model. Since the teacher model perceives data in a way different from humans, existing KD methods only distill knowledge that is consistent with labels annotated by human expert while neglecting knowledge that is not consistent with human perception, which results in insufficient distillation and sub-optimal performance. In this paper, we propose inconsistent knowledge distillation (IKD), which aims to distill knowledge inherent in the teacher model's counter-intuitive perceptions. We start by considering the teacher model's counter-intuitive perceptions of frequency and non-robust features. Unlike previous works that exploit fine-grained features or introduce additional regularizations, we extract inconsistent knowledge by providing diverse input using data augmentation. Specifically, we propose a sample-specific data augmentation to transfer the teacher model's ability in capturing distinct frequency components and suggest an adversarial feature augmentation to extract the teacher model's perceptions of non-robust features in the data. Extensive experiments demonstrate the effectiveness of our method which outperforms state-of-the-art KD baselines on one-stage, two-stage and anchor-free object detectors (at most +1.0 mAP). Our codes will be made available at \url{//github.com/JWLiang007/IKD.git}.
The Riemann problem for first-order hyperbolic systems of partial differential equations is of fundamental importance for both theoretical and numerical purposes. Many approximate solvers have been developed for such systems; exact solution algorithms have received less attention because computation of the exact solution typically requires iterative solution of algebraic equations. Iterative algorithms may be less computationally efficient or might fail to converge in some cases. We investigate the achievable efficiency of robust iterative Riemann solvers for relatively simple systems, focusing on the shallow water and Euler equations. We consider a range of initial guesses and iterative schemes applied to an ensemble of test Riemann problems. For the shallow water equations, we find that Newton's method with a simple modification converges quickly and reliably. For the Euler equations we obtain similar results; however, when the required precision is high, a combination of Ostrowski and Newton iterations converges faster. These solvers are slower than standard approximate solvers like Roe and HLLE, but come within a factor of two in speed. We also provide a preliminary comparison of the accuracy of a finite volume discretization using an exact solver versus standard approximate solvers.
Quantum Neural Network (QNN) combines the Deep Learning (DL) principle with the fundamental theory of quantum mechanics to achieve machine learning tasks with quantum acceleration. Recently, QNN systems have been found to manifest robustness issues similar to classical DL systems. There is an urgent need for ways to test their correctness and security. However, QNN systems differ significantly from traditional quantum software and classical DL systems, posing critical challenges for QNN testing. These challenges include the inapplicability of traditional quantum software testing methods, the dependence of quantum test sample generation on perturbation operators, and the absence of effective information in quantum neurons. In this paper, we propose QuanTest, a quantum entanglement-guided adversarial testing framework to uncover potential erroneous behaviors in QNN systems. We design a quantum entanglement adequacy criterion to quantify the entanglement acquired by the input quantum states from the QNN system, along with two similarity metrics to measure the proximity of generated quantum adversarial examples to the original inputs. Subsequently, QuanTest formulates the problem of generating test inputs that maximize the quantum entanglement sufficiency and capture incorrect behaviors of the QNN system as a joint optimization problem and solves it in a gradient-based manner to generate quantum adversarial examples. Experimental results demonstrate that QuanTest possesses the capability to capture erroneous behaviors in QNN systems (generating 67.48%-96.05% more test samples than the random noise under the same perturbation size constraints). The entanglement-guided approach proves effective in adversarial testing, generating more adversarial examples (maximum increase reached 21.32%).
Bilevel optimization problems, which are problems where two optimization problems are nested, have more and more applications in machine learning. In many practical cases, the upper and the lower objectives correspond to empirical risk minimization problems and therefore have a sum structure. In this context, we propose a bilevel extension of the celebrated SARAH algorithm. We demonstrate that the algorithm requires $\mathcal{O}((n+m)^{\frac12}\varepsilon^{-1})$ oracle calls to achieve $\varepsilon$-stationarity with $n+m$ the total number of samples, which improves over all previous bilevel algorithms. Moreover, we provide a lower bound on the number of oracle calls required to get an approximate stationary point of the objective function of the bilevel problem. This lower bound is attained by our algorithm, making it optimal in terms of sample complexity.
We propose a new joint mean and correlation regression model for correlated multivariate discrete responses, that simultaneously regresses the mean of each response against a set of covariates, and the correlations between responses against a set of similarity/distance measures. A set of joint estimating equations are formulated to construct an estimator of both the mean regression coefficients and the correlation regression parameters. Under a general setting where the number of responses can tend to infinity, the joint estimator is demonstrated to be consistent and asymptotically normally distributed, with differing rates of convergence due to the mean regression coefficients being heterogeneous across responses. An iterative estimation procedure is developed to obtain parameter estimates in the required, constrained parameter space. We apply the proposed model to a multivariate abundance dataset comprising overdispersed counts of 38 Carabidae ground beetle species sampled throughout Scotland, along with information about the environmental conditions of each site and the traits of each species. Results show in particular that the relationships between the mean abundances of various beetle species and environmental covariates are different and that beetle total length has statistically important effect in driving the correlations between the species. Simulations demonstrate the strong finite sample performance of the proposed estimator in terms of point estimation and inference.
The problem of minimizing the maximum of $N$ convex, Lipschitz functions plays significant roles in optimization and machine learning. It has a series of results, with the most recent one requiring $O(N\epsilon^{-2/3} + \epsilon^{-8/3})$ queries to a first-order oracle to compute an $\epsilon$-suboptimal point. On the other hand, quantum algorithms for optimization are rapidly advancing with speedups shown on many important optimization problems. In this paper, we conduct a systematic study for quantum algorithms and lower bounds for minimizing the maximum of $N$ convex, Lipschitz functions. On one hand, we develop quantum algorithms with an improved complexity bound of $\tilde{O}(\sqrt{N}\epsilon^{-5/3} + \epsilon^{-8/3})$. On the other hand, we prove that quantum algorithms must take $\tilde{\Omega}(\sqrt{N}\epsilon^{-2/3})$ queries to a first order quantum oracle, showing that our dependence on $N$ is optimal up to poly-logarithmic factors.
Geometry problem solving (GPS) is a challenging mathematical reasoning task requiring multi-modal understanding, fusion, and reasoning. Existing neural solvers take GPS as a vision-language task but are short in the representation of geometry diagrams that carry rich and complex layout information. In this paper, we propose a layout-aware neural solver named LANS, integrated with two new modules: multimodal layout-aware pre-trained language module (MLA-PLM) and layout-aware fusion attention (LA-FA). MLA-PLM adopts structural-semantic pre-training (SSP) to implement global relationship modeling, and point-match pre-training (PMP) to achieve alignment between visual points and textual points. LA-FA employs a layout-aware attention mask to realize point-guided cross-modal fusion for further boosting layout awareness of LANS. Extensive experiments on datasets Geometry3K and PGPS9K validate the effectiveness of the layout-aware modules and superior problem-solving performance of our LANS solver, over existing symbolic and neural solvers. The code will be made public available soon.
This paper introduces PDEformer, a neural solver for partial differential equations (PDEs) capable of simultaneously addressing various types of PDEs. We advocate representing the PDE in the form of a computational graph, facilitating the seamless integration of both symbolic and numerical information inherent in a PDE. A graph Transformer and an implicit neural representation (INR) are employed to generate mesh-free predicted solutions. Following pretraining on data exhibiting a certain level of diversity, our model achieves zero-shot accuracies on benchmark datasets that surpass those of adequately trained expert models. Additionally, PDEformer demonstrates promising results in the inverse problem of PDE coefficient recovery.
We present combinatorial and parallelizable algorithms for maximization of a submodular function, not necessarily monotone, with respect to a size constraint. We improve the best approximation factor achieved by an algorithm that has optimal adaptivity and nearly optimal query complexity to $0.193 - \varepsilon$. The conference version of this work mistakenly employed a subroutine that does not work for non-monotone, submodular functions. In this version, we propose a fixed and improved subroutine to add a set with high average marginal gain, ThreshSeq, which returns a solution in $O( \log(n) )$ adaptive rounds with high probability. Moreover, we provide two approximation algorithms. The first has approximation ratio $1/6 - \varepsilon$, adaptivity $O( \log (n) )$, and query complexity $O( n \log (k) )$, while the second has approximation ratio $0.193 - \varepsilon$, adaptivity $O( \log^2 (n) )$, and query complexity $O(n \log (k))$. Our algorithms are empirically validated to use a low number of adaptive rounds and total queries while obtaining solutions with high objective value in comparison with state-of-the-art approximation algorithms, including continuous algorithms that use the multilinear extension.
Cold-start problems are long-standing challenges for practical recommendations. Most existing recommendation algorithms rely on extensive observed data and are brittle to recommendation scenarios with few interactions. This paper addresses such problems using few-shot learning and meta learning. Our approach is based on the insight that having a good generalization from a few examples relies on both a generic model initialization and an effective strategy for adapting this model to newly arising tasks. To accomplish this, we combine the scenario-specific learning with a model-agnostic sequential meta-learning and unify them into an integrated end-to-end framework, namely Scenario-specific Sequential Meta learner (or s^2 meta). By doing so, our meta-learner produces a generic initial model through aggregating contextual information from a variety of prediction tasks while effectively adapting to specific tasks by leveraging learning-to-learn knowledge. Extensive experiments on various real-world datasets demonstrate that our proposed model can achieve significant gains over the state-of-the-arts for cold-start problems in online recommendation. Deployment is at the Guess You Like session, the front page of the Mobile Taobao.