GeoAI has emerged as an exciting interdisciplinary research area that combines spatial theories and data with cutting-edge AI models to address geospatial problems in a novel, data-driven manner. While GeoAI research has flourished in the GIScience literature, its reproducibility and replicability (R&R), fundamental principles that determine the reusability, reliability, and scientific rigor of research findings, have rarely been discussed. This paper aims to provide an in-depth analysis of this topic from both computational and spatial perspectives. We first categorize the major goals for reproducing GeoAI research, namely, validation (repeatability), learning and adapting the method for solving a similar or new problem (reproducibility), and examining the generalizability of the research findings (replicability). Each of these goals requires different levels of understanding of GeoAI, as well as different methods to ensure its success. We then discuss the factors that may cause the lack of R&R in GeoAI research, with an emphasis on (1) the selection and use of training data; (2) the uncertainty that resides in the GeoAI model design, training, deployment, and inference processes; and more importantly (3) the inherent spatial heterogeneity of geospatial data and processes. We use a deep learning-based image analysis task as an example to demonstrate the results' uncertainty and spatial variance caused by different factors. The findings reiterate the importance of knowledge sharing, as well as the generation of a "replicability map" that incorporates spatial autocorrelation and spatial heterogeneity into consideration in quantifying the spatial replicability of GeoAI research.
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Quantum computing becomes more of a reality as time passes, bringing several cybersecurity challenges. Modern cryptography is based on the computational complexity of specific mathematical problems, but as new quantum-based computers appear, classical methods might not be enough to secure communications. In this paper, we analyse the state of the Galileo Open Service Navigation Message Authentication (OSNMA) to overcome these new threats. This analysis and its assessment have been performed using OSNMA documentation, reviewing the available Post Quantum Cryptography (PQC) algorithms competing in the National Institute of Standards and Technology (NIST) standardization process, and studying the possibility of its implementation in the Galileo service. The main barrier to adopting the PQC approach is the size of both the signature and the key. The analysis shows that OSNMA is not yet prepared to face the quantum threat, and a significant change would be required. This work concludes by assessing different temporal countermeasures that can be implemented to sustain the system's integrity in the short term.
We study interpolation inequalities between H\"older Integral Probability Metrics (IPMs) in the case where the measures have densities on closed submanifolds. Precisely, it is shown that if two probability measures $\mu$ and $\mu^\star$ have $\beta$-smooth densities with respect to the volume measure of some submanifolds $\mathcal{M}$ and $\mathcal{M}^\star$ respectively, then the H\"older IPMs $d_{\mathcal{H}^\gamma_1}$ of smoothness $\gamma\geq 1$ and $d_{\mathcal{H}^\eta_1}$ of smoothness $\eta>\gamma$, satisfy $d_{ \mathcal{H}_1^{\gamma}}(\mu,\mu^\star)\lesssim d_{ \mathcal{H}_1^{\eta}}(\mu,\mu^\star)^\frac{\beta+\gamma}{\beta+\eta}$, up to logarithmic factors. We provide an application of this result to high-dimensional inference. These functional inequalities turn out to be a key tool for density estimation on unknown submanifold. In particular, it allows to build the first estimator attaining optimal rates of estimation for all the distances $d_{\mathcal{H}_1^\gamma}$, $\gamma \in [1,\infty)$ simultaneously.
Gaussian elimination is the most popular technique for solving a dense linear system. Large errors in this procedure can occur in floating point arithmetic when the matrix's growth factor is large. We study this potential issue and how perturbations can improve the robustness of the Gaussian elimination algorithm. In their 1989 paper, Higham and Higham characterized the complete set of real n by n matrices that achieves the maximum growth factor under partial pivoting. This set of matrices serves as the critical focus of this work. Through theoretical insights and empirical results, we illustrate the high sensitivity of the growth factor of these matrices to perturbations and show how subtle changes can be strategically applied to matrix entries to significantly reduce the growth, thus enhancing computational stability and accuracy.
Many combinatorial optimization problems can be formulated as the search for a subgraph that satisfies certain properties and minimizes the total weight. We assume here that the vertices correspond to points in a metric space and can take any position in given uncertainty sets. Then, the cost function to be minimized is the sum of the distances for the worst positions of the vertices in their uncertainty sets. We propose two types of polynomial-time approximation algorithms. The first one relies on solving a deterministic counterpart of the problem where the uncertain distances are replaced with maximum pairwise distances. We study in details the resulting approximation ratio, which depends on the structure of the feasible subgraphs and whether the metric space is Ptolemaic or not. The second algorithm is a fully-polynomial time approximation scheme for the special case of $s-t$ paths.
Symplectic integrators are widely implemented numerical integrators for Hamiltonian mechanics, which preserve the Hamiltonian structure (symplecticity) of the system. Although the symplectic integrator does not conserve the energy of the system, it is well known that there exists a conserving modified Hamiltonian, called the shadow Hamiltonian. For the Nambu mechanics, which is a kind of generalized Hamiltonian mechanics, we can also construct structure-preserving integrators by the same procedure used to construct the symplectic integrators. In the structure-preserving integrator, however, the existence of shadow Hamiltonians is nontrivial. This is because the Nambu mechanics is driven by multiple Hamiltonians and it is nontrivial whether the time evolution by the integrator can be cast into the Nambu mechanical time evolution driven by multiple shadow Hamiltonians. In this paper we present a general procedure to calculate the shadow Hamiltonians of structure-preserving integrators for Nambu mechanics, and give an example where the shadow Hamiltonians exist. This is the first attempt to determine the concrete forms of the shadow Hamiltonians for a Nambu mechanical system. We show that the fundamental identity, which corresponds to the Jacobi identity in Hamiltonian mechanics, plays an important role in calculating the shadow Hamiltonians using the Baker-Campbell-Hausdorff formula. It turns out that the resulting shadow Hamiltonians have indefinite forms depending on how the fundamental identities are used. This is not a technical artifact, because the exact shadow Hamiltonians obtained independently have the same indefiniteness.
Optimization constrained by high-fidelity computational models has potential for transformative impact. However, such optimization is frequently unattainable in practice due to the complexity and computational intensity of the model. An alternative is to optimize a low-fidelity model and use limited evaluations of the high-fidelity model to assess the quality of the solution. This article develops a framework to use limited high-fidelity simulations to update the optimization solution computed using the low-fidelity model. Building off a previous article [22], which introduced hyper-differential sensitivity analysis with respect to model discrepancy, this article provides novel extensions of the algorithm to enable uncertainty quantification of the optimal solution update via a Bayesian framework. Specifically, we formulate a Bayesian inverse problem to estimate the model discrepancy and propagate the posterior model discrepancy distribution through the post-optimality sensitivity operator for the low-fidelity optimization problem. We provide a rigorous treatment of the Bayesian formulation, a computationally efficient algorithm to compute posterior samples, a guide to specify and interpret the algorithm hyper-parameters, and a demonstration of the approach on three examples which highlight various types of discrepancy between low and high-fidelity models.
離散數學
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We study three systems of equations, together with a way to count the number of solutions. One of the results was used in the recent computation of D(9), the others have potential to speed up existing techniques in the future.
The denoising diffusion model has recently emerged as a powerful generative technique that converts noise into data. While there are many studies providing theoretical guarantees for diffusion processes based on discretized stochastic differential equation (D-SDE), many generative samplers in real applications directly employ a discrete-time (DT) diffusion process. However, there are very few studies analyzing these DT processes, e.g., convergence for DT diffusion processes has been obtained only for distributions with bounded support. In this paper, we establish the convergence guarantee for substantially larger classes of distributions under DT diffusion processes and further improve the convergence rate for distributions with bounded support. In particular, we first establish the convergence rates for both smooth and general (possibly non-smooth) distributions having a finite second moment. We then specialize our results to a number of interesting classes of distributions with explicit parameter dependencies, including distributions with Lipschitz scores, Gaussian mixture distributions, and any distributions with early-stopping. We further propose a novel accelerated sampler and show that it improves the convergence rates of the corresponding regular sampler by orders of magnitude with respect to all system parameters. Our study features a novel analytical technique that constructs a tilting factor representation of the convergence error and exploits Tweedie's formula for handling Taylor expansion power terms.
We consider the application of the generalized Convolution Quadrature (gCQ) to approximate the solution of an important class of sectorial problems. The gCQ is a generalization of Lubich's Convolution Quadrature (CQ) that allows for variable steps. The available stability and convergence theory for the gCQ requires non realistic regularity assumptions on the data, which do not hold in many applications of interest, such as the approximation of subdiffusion equations. It is well known that for non smooth enough data the original CQ, with uniform steps, presents an order reduction close to the singularity. We generalize the analysis of the gCQ to data satisfying realistic regularity assumptions and provide sufficient conditions for stability and convergence on arbitrary sequences of time points. We consider the particular case of graded meshes and show how to choose them optimally, according to the behaviour of the data. An important advantage of the gCQ method is that it allows for a fast and memory reduced implementation. We describe how the fast and oblivious gCQ can be implemented and illustrate our theoretical results with several numerical experiments.
Hashing has been widely used in approximate nearest search for large-scale database retrieval for its computation and storage efficiency. Deep hashing, which devises convolutional neural network architecture to exploit and extract the semantic information or feature of images, has received increasing attention recently. In this survey, several deep supervised hashing methods for image retrieval are evaluated and I conclude three main different directions for deep supervised hashing methods. Several comments are made at the end. Moreover, to break through the bottleneck of the existing hashing methods, I propose a Shadow Recurrent Hashing(SRH) method as a try. Specifically, I devise a CNN architecture to extract the semantic features of images and design a loss function to encourage similar images projected close. To this end, I propose a concept: shadow of the CNN output. During optimization process, the CNN output and its shadow are guiding each other so as to achieve the optimal solution as much as possible. Several experiments on dataset CIFAR-10 show the satisfying performance of SRH.
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