The Pearson correlation coefficient is generally not invariant under common marginal transforms, but such an invariance property may hold true for specific models such as independence. A bivariate random vector is said to have an invariant correlation if its Pearson correlation coefficient remains unchanged under any common marginal transforms. We characterize all models of such a random vector via a certain combination of independence and the strongest positive dependence called comonotonicity. In particular, we show that the class of exchangeable copulas with invariant correlation is precisely described by what we call positive Fr\'echet copulas. We then extend the concept of invariant correlation to multi-dimensional models, and characterize the set of all invariant correlation matrices via the clique partition polytope. We also propose a positive regression dependent model which admits any prescribed invariant correlation matrix. Finally, all our characterization results of invariant correlation, except one special case, remain the same if the common marginal transforms are confined to the set of increasing ones.
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In this work we present a new WENO b-spline based quasi-interpolation algorithm. The novelty of this construction resides in the application of the WENO weights to the b-spline functions, that are a partition of unity, instead to the coefficients that multiply the b-spline functions of the spline. The result obtained conserves the smoothness of the original spline and presents adaption to discontinuities in the function. Another new idea that we introduce in this work is the use of different base weight functions from those proposed in classical WENO algorithms. Apart from introducing the construction of the new algorithms, we present theoretical results regarding the order of accuracy obtained at smooth zones and close to the discontinuity, as well as theoretical considerations about how to design the new weight functions. Through a tensor product strategy, we extend our results to several dimensions. In order to check the theoretical results obtained, we present an extended battery of numerical experiments in one, two and tree dimensions that support our conclussions.
A matrix-free parallel two-level deflation preconditioner for the two-dimensional Helmholtz problems
We propose a matrix-free parallel two-level-deflation preconditioner combined with the Complex Shifted Laplacian preconditioner(CSLP) for the two-dimensional Helmholtz problems. The Helmholtz equation is widely studied in seismic exploration, antennas, and medical imaging. It is one of the hardest problems to solve both in terms of accuracy and convergence, due to scalability issues of the numerical solvers. Motivated by the observation that for large wavenumbers, the eigenvalues of the CSLP-preconditioned system shift towards zero, deflation with multigrid vectors, and further high-order vectors were incorporated to obtain wave-number-independent convergence. For large-scale applications, high-performance parallel scalable methods are also indispensable. In our method, we consider the preconditioned Krylov subspace methods for solving the linear system obtained from finite-difference discretization. The CSLP preconditioner is approximated by one parallel geometric multigrid V-cycle. For the two-level deflation, the matrix-free Galerkin coarsening as well as high-order re-discretization approaches on the coarse grid are studied. The results of matrix-vector multiplications in Krylov subspace methods and the interpolation/restriction operators are implemented based on the finite-difference grids without constructing any coefficient matrix. These adjustments lead to direct improvements in terms of memory consumption. Numerical experiments of model problems show that wavenumber independence has been obtained for medium wavenumbers. The matrix-free parallel framework shows satisfactory weak and strong parallel scalability.
Blumer et al. (1987, 1989) showed that any concept class that is learnable by Occam algorithms is PAC learnable. Board and Pitt (1990) showed a partial converse of this theorem: for concept classes that are closed under exception lists, any class that is PAC learnable is learnable by an Occam algorithm. However, their Occam algorithm outputs a hypothesis whose complexity is $\delta$-dependent, which is an important limitation. In this paper, we show that their partial converse applies to Occam algorithms with $\delta$-independent complexities as well. Thus, we provide a posteriori justification of various theoretical results and algorithm design methods which use the partial converse as a basis for their work.
Brain structural networks are often represented as discrete adjacency matrices with elements summarizing the connectivity between pairs of regions of interest (ROIs). These ROIs are typically determined a-priori using a brain atlas. The choice of atlas is often arbitrary and can lead to a loss of important connectivity information at the sub-ROI level. This work introduces an atlas-free framework that overcomes these issues by modeling brain connectivity using smooth random functions. In particular, we assume that the observed pattern of white matter fiber tract endpoints is driven by a latent random function defined over a product manifold domain. To facilitate statistical analysis of these high dimensional functional data objects, we develop a novel algorithm to construct a data-driven reduced-rank function space that offers a desirable trade-off between computational complexity and flexibility. Using real data from the Human Connectome Project, we show that our method outperforms state-of-the-art approaches that use the traditional atlas-based structural connectivity representation on a variety of connectivity analysis tasks. We further demonstrate how our method can be used to detect localized regions and connectivity patterns associated with group differences.
Pseudo-labeling is significant for semi-supervised instance segmentation, which generates instance masks and classes from unannotated images for subsequent training. However, in existing pipelines, pseudo-labels that contain valuable information may be directly filtered out due to mismatches in class and mask quality. To address this issue, we propose a novel framework, called pseudo-label aligning instance segmentation (PAIS), in this paper. In PAIS, we devise a dynamic aligning loss (DALoss) that adjusts the weights of semi-supervised loss terms with varying class and mask score pairs. Through extensive experiments conducted on the COCO and Cityscapes datasets, we demonstrate that PAIS is a promising framework for semi-supervised instance segmentation, particularly in cases where labeled data is severely limited. Notably, with just 1\% labeled data, PAIS achieves 21.2 mAP (based on Mask-RCNN) and 19.9 mAP (based on K-Net) on the COCO dataset, outperforming the current state-of-the-art model, \ie, NoisyBoundary with 7.7 mAP, by a margin of over 12 points. Code is available at: \url{//github.com/hujiecpp/PAIS}.
By combining a logarithm transformation with a corrected Milstein-type method, the present article proposes an explicit, unconditional boundary and dynamics preserving scheme for the stochastic susceptible-infected-susceptible (SIS) epidemic model that takes value in (0,N). The scheme applied to the model is first proved to have a strong convergence rate of order one. Further, the dynamic behaviors are analyzed for the numerical approximations and it is shown that the scheme can unconditionally preserve both the domain and the dynamics of the model. More precisely, the proposed scheme gives numerical approximations living in the domain (0,N) and reproducing the extinction and persistence properties of the original model for any time discretization step-size h > 0, without any additional requirements on the model parameters. Numerical experiments are presented to verify our theoretical results.
This article re-examines Lawvere's abstract, category-theoretic proof of the fixed-point theorem whose contrapositive is a `universal' diagonal argument. The main result is that the necessary axioms for both the fixed-point theorem and the diagonal argument can be stripped back further, to a semantic analogue of a weak substructural logic lacking weakening or exchange.
Very distinct strategies can be deployed to recognize and characterize an unknown environment or a shape. A recent and promising approach, especially in robotics, is to reduce the complexity of the exploratory units to a minimum. Here, we show that this frugal strategy can be taken to the extreme by exploiting the power of statistical geometry and introducing new invariant features. We show that an elementary robot devoid of any orientation or observation system, exploring randomly, can access global information about an environment such as the values of the explored area and perimeter. The explored shapes are of arbitrary geometry and may even non-connected. From a dictionary, this most simple robot can thus identify various shapes such as famous monuments and even read a text.
Ghost, or fictitious points allow to capture boundary conditions that are not located on the finite difference grid discretization. We explore in this paper the impact of ghost points on the stability of the explicit Euler finite difference scheme in the context of the diffusion equation. In particular, we consider the case of a one-touch option under the Black-Scholes model. The observations and results are however valid for a much wider range of financial contracts and models.
Internet of Things (IoT) systems require highly scalable infrastructure to adaptively provide services to meet various performance requirements. Combining Software-Defined Networking (SDN) with Mobile Edge Cloud (MEC) technology brings more flexibility for IoT systems. We present a four-tier task processing architecture for MEC and vehicular networks, which includes processing tasks locally within a vehicle, on neighboring vehicles, on an edge cloud, and on a remote cloud. The flexible network connection is controlled by SDN. We propose a CPU resource allocation algorithm, called Partial Idle Resource Strategy (PIRS) with Vehicle to Vehicle (V2V) communications, based on Asymmetric Nash Bargaining Solution (ANBS) in Game Theory. PIRS encourages vehicles in the same location to cooperate by sharing part of their spare CPU resources. In our simulations, we adopt four applications running on the vehicles to generate workload. We compare the proposed algorithm with Non-Cooperation Strategy (NCS) and All Idle Resource Strategy (AIRS). In NCS, the vehicles execute tasks generated by the applications in their own On-Board Units (OBU), while in AIRS vehicles provide all their CPU resources to help other vehicles offloading requests. Our simulation results show that our PIRS strategy can execute more tasks on the V2V layer and lead to fewer number of task (and their length) to be offloaded to the cloud, reaching up to 28% improvement compared to NCS and up to 10% improvement compared to AIRS.
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