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Convolutional neural networks have been widely applied to hyperspectral image classification. However, traditional convolutions can not effectively extract features for objects with irregular distributions. Recent methods attempt to address this issue by performing graph convolutions on spatial topologies, but fixed graph structures and local perceptions limit their performances. To tackle these problems, in this paper, different from previous approaches, we perform the superpixel generation on intermediate features during network training to adaptively produce homogeneous regions, obtain graph structures, and further generate spatial descriptors, which are served as graph nodes. Besides spatial objects, we also explore the graph relationships between channels by reasonably aggregating channels to generate spectral descriptors. The adjacent matrices in these graph convolutions are obtained by considering the relationships among all descriptors to realize global perceptions. By combining the extracted spatial and spectral graph features, we finally obtain a spectral-spatial graph reasoning network (SSGRN). The spatial and spectral parts of SSGRN are separately called spatial and spectral graph reasoning subnetworks. Comprehensive experiments on four public datasets demonstrate the competitiveness of the proposed methods compared with other state-of-the-art graph convolution-based approaches.

Given a Hilbert space $\mathcal H$ and a finite measure space $\Omega$, the approximation of a vector-valued function $f: \Omega \to \mathcal H$ by a $k$-dimensional subspace $\mathcal U \subset \mathcal H$ plays an important role in dimension reduction techniques, such as reduced basis methods for solving parameter-dependent partial differential equations. For functions in the Lebesgue--Bochner space $L^2(\Omega;\mathcal H)$, the best possible subspace approximation error $d_k^{(2)}$ is characterized by the singular values of $f$. However, for practical reasons, $\mathcal U$ is often restricted to be spanned by point samples of $f$. We show that this restriction only has a mild impact on the attainable error; there always exist $k$ samples such that the resulting error is not larger than $\sqrt{k+1} \cdot d_k^{(2)}$. Our work extends existing results by Binev at al. (SIAM J. Math. Anal., 43(3):1457--1472, 2011) on approximation in supremum norm and by Deshpande et al. (Theory Comput., 2:225--247, 2006) on column subset selection for matrices.

There has recently been much interest in Gaussian processes on linear networks and more generally on compact metric graphs. One proposed strategy for defining such processes on a metric graph $\Gamma$ is through a covariance function that is isotropic in a metric on the graph. Another is through a fractional order differential equation $L^\alpha (\tau u) = \mathcal{W}$ on $\Gamma$, where $L = \kappa^2 - \nabla(a\nabla)$ for (sufficiently nice) functions $\kappa, a$, and $\mathcal{W}$ is Gaussian white noise. We study Markov properties of these two types of fields. We first show that there are no Gaussian random fields on general metric graphs that are both isotropic and Markov. We then show that the second type of fields, the generalized Whittle--Mat\'ern fields, are Markov if and only if $\alpha\in\mathbb{N}$, and if $\alpha\in\mathbb{N}$, the field is Markov of order $\alpha$, which essentially means that the process in one region $S\subset\Gamma$ is conditionally independent the process in $\Gamma\setminus S$ given the values of the process and its $\alpha-1$ derivatives on $\partial S$. Finally, we show that the Markov property implies an explicit characterization of the process on a fixed edge $e$, which in particular shows that the conditional distribution of the process on $e$ given the values at the two vertices connected to $e$ is independent of the geometry of $\Gamma$.

We study the balanced $k$-way hypergraph partitioning problem, with a special focus on its practical applications to manycore scheduling. Given a hypergraph on $n$ nodes, our goal is to partition the node set into $k$ parts of size at most $(1+\epsilon)\cdot \frac{n}{k}$ each, while minimizing the cost of the partitioning, defined as the number of cut hyperedges, possibly also weighted by the number of partitions they intersect. We show that this problem cannot be approximated to within a $n^{1/\text{poly} \log\log n}$ factor of the optimal solution in polynomial time if the Exponential Time Hypothesis holds, even for hypergraphs of maximal degree 2. We also study the hardness of the partitioning problem from a parameterized complexity perspective, and in the more general case when we have multiple balance constraints. Furthermore, we consider two extensions of the partitioning problem that are motivated from practical considerations. Firstly, we introduce the concept of hyperDAGs to model precedence-constrained computations as hypergraphs, and we analyze the adaptation of the balanced partitioning problem to this case. Secondly, we study the hierarchical partitioning problem to model hierarchical NUMA (non-uniform memory access) effects in modern computer architectures, and we show that ignoring this hierarchical aspect of the communication cost can yield significantly weaker solutions.

We consider the classic 1-center problem: Given a set $P$ of $n$ points in a metric space find the point in $P$ that minimizes the maximum distance to the other points of $P$. We study the complexity of this problem in $d$-dimensional $\ell_p$-metrics and in edit and Ulam metrics over strings of length $d$. Our results for the 1-center problem may be classified based on $d$ as follows. $\bullet$ Small $d$: Assuming the hitting set conjecture (HSC), we show that when $d=\omega(\log n)$, no subquadratic algorithm can solve 1-center problem in any of the $\ell_p$-metrics, or in edit or Ulam metrics. $\bullet$ Large $d$: When $d=\Omega(n)$, we extend our conditional lower bound to rule out subquartic algorithms for 1-center problem in edit metric (assuming Quantified SETH). On the other hand, we give a $(1+\epsilon)$-approximation for 1-center in Ulam metric with running time $\tilde{O_{\varepsilon}}(nd+n^2\sqrt{d})$. We also strengthen some of the above lower bounds by allowing approximations or by reducing the dimension $d$, but only against a weaker class of algorithms which list all requisite solutions. Moreover, we extend one of our hardness results to rule out subquartic algorithms for the well-studied 1-median problem in the edit metric, where given a set of $n$ strings each of length $n$, the goal is to find a string in the set that minimizes the sum of the edit distances to the rest of the strings in the set.

Random functional-linked types of neural networks (RFLNNs), e.g., the extreme learning machine (ELM) and broad learning system (BLS), which avoid suffering from a time-consuming training process, offer an alternative way of learning in deep structure. The RFLNNs have achieved excellent performance in various classification and regression tasks, however, the properties and explanations of these networks are ignored in previous research. This paper gives some insights into the properties of RFLNNs from the viewpoints of frequency domain, and discovers the presence of frequency principle in these networks, that is, they preferentially capture low-frequencies quickly and then fit the high frequency components during the training process. These findings are valuable for understanding the RFLNNs and expanding their applications. Guided by the frequency principle, we propose a method to generate a BLS network with better performance, and design an efficient algorithm for solving Poison's equation in view of the different frequency principle presenting in the Jacobi iterative method and BLS network.

Repairing inconsistent knowledge bases is a task that has been assessed, with great advances over several decades, from within the knowledge representation and reasoning and the database theory communities. As information becomes more complex and interconnected, new types of repositories, representation languages and semantics are developed in order to be able to query and reason about it. Graph databases provide an effective way to represent relationships among data, and allow processing and querying these connections efficiently. In this work, we focus on the problem of computing preferred (subset and superset) repairs for graph databases with data values, using a notion of consistency based on a set of Reg-GXPath expressions as integrity constraints. Specifically, we study the problem of computing preferred repairs based on two different preference criteria, one based on weights and the other based on multisets, showing that in most cases it is possible to retain the same computational complexity as in the case where no preference criterion is available for exploitation.

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.

We consider the problem of explaining the predictions of graph neural networks (GNNs), which otherwise are considered as black boxes. Existing methods invariably focus on explaining the importance of graph nodes or edges but ignore the substructures of graphs, which are more intuitive and human-intelligible. In this work, we propose a novel method, known as SubgraphX, to explain GNNs by identifying important subgraphs. Given a trained GNN model and an input graph, our SubgraphX explains its predictions by efficiently exploring different subgraphs with Monte Carlo tree search. To make the tree search more effective, we propose to use Shapley values as a measure of subgraph importance, which can also capture the interactions among different subgraphs. To expedite computations, we propose efficient approximation schemes to compute Shapley values for graph data. Our work represents the first attempt to explain GNNs via identifying subgraphs explicitly and directly. Experimental results show that our SubgraphX achieves significantly improved explanations, while keeping computations at a reasonable level.

In many real-world network datasets such as co-authorship, co-citation, email communication, etc., relationships are complex and go beyond pairwise. Hypergraphs provide a flexible and natural modeling tool to model such complex relationships. The obvious existence of such complex relationships in many real-world networks naturaly motivates the problem of learning with hypergraphs. A popular learning paradigm is hypergraph-based semi-supervised learning (SSL) where the goal is to assign labels to initially unlabeled vertices in a hypergraph. Motivated by the fact that a graph convolutional network (GCN) has been effective for graph-based SSL, we propose HyperGCN, a novel GCN for SSL on attributed hypergraphs. Additionally, we show how HyperGCN can be used as a learning-based approach for combinatorial optimisation on NP-hard hypergraph problems. We demonstrate HyperGCN's effectiveness through detailed experimentation on real-world hypergraphs.