Fractional vector calculus is the building block of the fractional partial differential equations that model non-local or long-range phenomena, e.g., anomalous diffusion, fractional electromagnetism, and fractional advection-dispersion. In this work, we reformulate a type of fractional vector calculus that uses Caputo fractional partial derivatives and discretize this reformulation using discrete exterior calculus on a cubical complex in the structure-preserving way, meaning that the continuous-level properties $\operatorname{curl}^\alpha \operatorname{grad}^\alpha = \mathbf{0}$ and $\operatorname{div}^\alpha \operatorname{curl}^\alpha = 0$ hold exactly on the discrete level. We discuss important properties of our fractional discrete exterior derivatives and verify their second-order convergence in the root mean square error numerically. Our proposed discretization has the potential to provide accurate and stable numerical solutions to fractional partial differential equations and exactly preserve fundamental physics laws on the discrete level regardless of the mesh size.
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Consider the semigroup of random walk on a complete graph, which we call the Potts semigroup. Diaconis and Saloff-Coste computed the maximum of the ratio of the relative entropy and the Dirichlet form obtaining the constant $\alpha_2$ in the $2$-log-Sobolev inequality ($2$-LSI). In this paper, we obtain the best possible non-linear inequality relating entropy and the Dirichlet form (i.e., $p$-NLSI, $p\ge1$). As an example, we show $\alpha_1 = 1+\frac{1+o(1)}{\log k}$. By integrating the $1$-NLSI we obtain the new strong data processing inequality (SDPI), which in turn allows us to improve results of Mossel and Peres on reconstruction thresholds for Potts models on trees. A special case is the problem of reconstructing color of the root of a $k$-colored tree given knowledge of colors of all the leaves. We show that to have a non-trivial reconstruction probability the branching number of the tree should be at least $$\frac{\log k}{\log k - \log(k-1)} = (1-o(1))k\log k.$$ This recovers previous results (of Sly and Bhatnagar et al.) in (slightly) more generality, but more importantly avoids the need for any coloring-specialized arguments. Similarly, we improve the state-of-the-art on the weak recovery threshold for the stochastic block model with $k$ balanced groups, for all $k\ge 3$. To further show the power of our method, we prove optimal non-reconstruction results for a broadcasting on trees model with Gaussian kernels, closing a gap left open by Eldan et al. These improvements advocate information-theoretic methods as a useful complement to the conventional techniques originating from the statistical physics.
Many calculi exist for modelling various features of object-oriented languages. Many of them are based on $\lambda$-calculus and focus either on statically typed class-based languages or dynamic prototype-based languages. We formalize untyped calculus of decorated objects, informally presented by Bugayenko, which is defined in terms of objects and relies on decoration as a primary mechanism of object extension. It is not based on $\lambda$-calculus, yet with only four basic syntactic constructions is just as complete. We prove the calculus is confluent (i.e. possesses Church-Rosser property), and introduce an abstract machine for call-by-name evaluation. Finally, we provide a sound translation to $\lambda$-calculus with records.
Recent advances in technology have allowed an automation system to recognize its errors and repair trust more actively than ever. While previous research has called for further studies of different human factors and design features, their effect on human-automation trust repair scenarios remains unknown, especially concerning emotions. This paper seeks to fill such gaps by investigating the impact of anthropomorphism, users' individual differences, and emotional responses on human-automation trust repair. Our experiment manipulated various types of trust violations and apology messages with different emotionally expressive anthropomorphic cues. While no significant effect from the different apology representations was found, our participants displayed polarizing attitudes toward the anthropomorphic cues. We also found that (1). some personality traits, such as openness and conscientiousness, negatively correlate with the effectiveness of the apology messages, and (2). a person's emotional response toward a trust violation positively correlates with the effectiveness of the apology messages.
Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.
In many applications, we want to influence the decisions of independent agents by designing incentives for their actions. We revisit a fundamental problem in this area, called GAME IMPLEMENTATION: Given a game in standard form and a set of desired strategies, can we design a set of payment promises such that if the players take the payment promises into account, then all undominated strategies are desired? Furthermore, we aim to minimize the cost, that is, the worst-case amount of payments. We study the tractability of computing such payment promises and determine more closely what obstructions we may have to overcome in doing so. We show that GAME IMPLEMENTATION is NP-hard even for two players, solving in particular a long open question (Eidenbenz et al. 2011) and suggesting more restrictions are necessary to obtain tractability results. We thus study the regime in which players have only a small constant number of strategies and obtain the following. First, this case remains NP-hard even if each player's utility depends only on three others. Second, we repair a flawed efficient algorithm for the case of both small number of strategies and small number of players. Among further results, we characterize sets of desired strategies that can be implemented at zero cost as a kind of stable core of the game.
We propose a conservative energy method based on neural networks with subdomains for solving variational problems (CENN), where the admissible function satisfying the essential boundary condition without boundary penalty is constructed by the radial basis function (RBF), particular solution neural network, and general neural network. The loss term is the potential energy, optimized based on the principle of minimum potential energy. The loss term at the interfaces has the lower order derivative compared to the strong form PINN with subdomains. The advantage of the proposed method is higher efficiency, more accurate, and less hyperparameters than the strong form PINN with subdomains. Another advantage of the proposed method is that it can apply to complex geometries based on the special construction of the admissible function. To analyze its performance, the proposed method CENN is used to model representative PDEs, the examples include strong discontinuity, singularity, complex boundary, non-linear, and heterogeneous problems. Furthermore, it outperforms other methods when dealing with heterogeneous problems.
This paper proposes embedded Gaussian Process Barrier States (GP-BaS), a methodology to safely control unmodeled dynamics of nonlinear system using Bayesian learning. Gaussian Processes (GPs) are used to model the dynamics of the safety-critical system, which is subsequently used in the GP-BaS model. We derive the barrier state dynamics utilizing the GP posterior, which is used to construct a safety embedded Gaussian process dynamical model (GPDM). We show that the safety-critical system can be controlled to remain inside the safe region as long as we can design a controller that renders the BaS-GPDM's trajectories bounded (or asymptotically stable). The proposed approach overcomes various limitations in early attempts at combining GPs with barrier functions due to the abstention of restrictive assumptions such as linearity of the system with respect to control, relative degree of the constraints and number or nature of constraints. This work is implemented on various examples for trajectory optimization and control including optimal stabilization of unstable linear system and safe trajectory optimization of a Dubins vehicle navigating through an obstacle course and on a quadrotor in an obstacle avoidance task using GP differentiable dynamic programming (GP-DDP). The proposed framework is capable of maintaining safe optimization and control of unmodeled dynamics and is purely data driven.
This paper presents an approach that reconstructs a hand-held object from a monocular video. In contrast to many recent methods that directly predict object geometry by a trained network, the proposed approach does not require any learned prior about the object and is able to recover more accurate and detailed object geometry. The key idea is that the hand motion naturally provides multiple views of the object and the motion can be reliably estimated by a hand pose tracker. Then, the object geometry can be recovered by solving a multi-view reconstruction problem. We devise an implicit neural representation-based method to solve the reconstruction problem and address the issues of imprecise hand pose estimation, relative hand-object motion, and insufficient geometry optimization for small objects. We also provide a newly collected dataset with 3D ground truth to validate the proposed approach.
This paper presents a subsampling-task paradigm for data-driven task-specific experiment design (ED) and a novel method in populationwide supervised feature selection (FS). Optimal ED, the choice of sampling points under constraints of limited acquisition-time, arises in a wide variety of scientific and engineering contexts. However the continuous optimization used in classical approaches depend on a-priori parameter choices and challenging non-convex optimization landscapes. This paper proposes to replace this strategy with a subsampling-task paradigm, analogous to populationwide supervised FS. In particular, we introduce JOFSTO, which performs JOint Feature Selection and Task Optimization. JOFSTO jointly optimizes two coupled networks: one for feature scoring, which provides the ED, the other for execution of a downstream task or process. Unlike most FS problems, e.g. selecting protein expressions for classification, ED problems typically select from highly correlated globally informative candidates rather than seeking a small number of highly informative features among many uninformative features. JOFSTO's construction efficiently identifies potentially correlated, but effective subsets and returns a trained task network. We demonstrate the approach using parameter estimation and mapping problems in clinically-relevant applications in quantitative MRI and in hyperspectral imaging. Results from simulations and empirical data show the subsampling-task paradigm strongly outperforms classical ED, and within our paradigm, JOFSTO outperforms state-of-the-art supervised FS techniques. JOFSTO extends immediately to wider image-based ED problems and other scenarios where the design must be specified globally across large numbers of acquisitions. Code will be released.
With the rapid increase of large-scale, real-world datasets, it becomes critical to address the problem of long-tailed data distribution (i.e., a few classes account for most of the data, while most classes are under-represented). Existing solutions typically adopt class re-balancing strategies such as re-sampling and re-weighting based on the number of observations for each class. In this work, we argue that as the number of samples increases, the additional benefit of a newly added data point will diminish. We introduce a novel theoretical framework to measure data overlap by associating with each sample a small neighboring region rather than a single point. The effective number of samples is defined as the volume of samples and can be calculated by a simple formula $(1-\beta^{n})/(1-\beta)$, where $n$ is the number of samples and $\beta \in [0,1)$ is a hyperparameter. We design a re-weighting scheme that uses the effective number of samples for each class to re-balance the loss, thereby yielding a class-balanced loss. Comprehensive experiments are conducted on artificially induced long-tailed CIFAR datasets and large-scale datasets including ImageNet and iNaturalist. Our results show that when trained with the proposed class-balanced loss, the network is able to achieve significant performance gains on long-tailed datasets.
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