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In graph analysis, a classic task consists in computing similarity measures between (groups of) nodes. In latent space random graphs, nodes are associated to unknown latent variables. One may then seek to compute distances directly in the latent space, using only the graph structure. In this paper, we show that it is possible to consistently estimate entropic-regularized Optimal Transport (OT) distances between groups of nodes in the latent space. We provide a general stability result for entropic OT with respect to perturbations of the cost matrix. We then apply it to several examples of random graphs, such as graphons or $\epsilon$-graphs on manifolds. Along the way, we prove new concentration results for the so-called Universal Singular Value Thresholding estimator, and for the estimation of geodesic distances on a manifold.

We present two algorithms designed to learn a pattern of correspondence between two data sets in situations where it is desirable to match elements that exhibit a relationship belonging to a known parametric model. In the motivating case study, the challenge is to better understand micro-RNA (miRNA) regulation in the striatum of Huntington's disease (HD) model mice. The two data sets contain miRNA and messenger-RNA (mRNA) data, respectively, each data point consisting in a multi-dimensional profile. The biological hypothesis is that if a miRNA induces the degradation of a target mRNA or blocks its translation into proteins, or both, then the profile of the former should be similar to minus the profile of the latter (a particular form of affine relationship).The algorithms unfold in two stages. During the first stage, an optimal transport plan P and an optimal affine transformation are learned, using the Sinkhorn-Knopp algorithm and a mini-batch gradient descent. During the second stage, P is exploited to derive either several co-clusters or several sets of matched elements.We share codes that implement our algorithms. A simulation study illustrates how they work and perform. A brief summary of the real data application in the motivating case-study further illustrates the applicability and interest of the algorithms.

Graph-SLAM is a well-established algorithm for constructing a topological map of the environment while simultaneously attempting the localisation of the robot. It relies on scan matching algorithms to align noisy observations along robot's movements to compute an estimate of the current robot's location. We propose a fundamentally different approach to scan matching tasks to improve the estimation of roto-translation displacements and therefore the performance of the full SLAM algorithm. A Monte-Carlo approach is used to generate weighted hypotheses of the geometrical displacement between two scans, and then we cluster these hypotheses to compute the displacement that results in the best alignment. To cope with clusterization on roto-translations, we propose a novel clustering approach that robustly extends Gaussian Mean-Shift to orientations by factorizing the kernel density over the roto-translation components. We demonstrate the effectiveness of our method in an extensive set of experiments using both synthetic data and the Intel Research Lab's benchmarking datasets. The results confirms that our approach has superior performance in terms of matching accuracy and runtime computation than the state-of-the-art iterative point-based scan matching algorithms.

Successful quantitative investment usually relies on precise predictions of the future movement of the stock price. Recently, machine learning based solutions have shown their capacity to give more accurate stock prediction and become indispensable components in modern quantitative investment systems. However, the i.i.d. assumption behind existing methods is inconsistent with the existence of diverse trading patterns in the stock market, which inevitably limits their ability to achieve better stock prediction performance. In this paper, we propose a novel architecture, Temporal Routing Adaptor (TRA), to empower existing stock prediction models with the ability to model multiple stock trading patterns. Essentially, TRA is a lightweight module that consists of a set of independent predictors for learning multiple patterns as well as a router to dispatch samples to different predictors. Nevertheless, the lack of explicit pattern identifiers makes it quite challenging to train an effective TRA-based model. To tackle this challenge, we further design a learning algorithm based on Optimal Transport (OT) to obtain the optimal sample to predictor assignment and effectively optimize the router with such assignment through an auxiliary loss term. Experiments on the real-world stock ranking task show that compared to the state-of-the-art baselines, e.g., Attention LSTM and Transformer, the proposed method can improve information coefficient (IC) from 0.053 to 0.059 and 0.051 to 0.056 respectively. Our dataset and code used in this work are publicly available: //github.com/microsoft/qlib/tree/main/examples/benchmarks/TRA.

We propose a scalable Gromov-Wasserstein learning (S-GWL) method and establish a novel and theoretically-supported paradigm for large-scale graph analysis. The proposed method is based on the fact that Gromov-Wasserstein discrepancy is a pseudometric on graphs. Given two graphs, the optimal transport associated with their Gromov-Wasserstein discrepancy provides the correspondence between their nodes and achieves graph matching. When one of the graphs has isolated but self-connected nodes ($i.e.$, a disconnected graph), the optimal transport indicates the clustering structure of the other graph and achieves graph partitioning. Using this concept, we extend our method to multi-graph partitioning and matching by learning a Gromov-Wasserstein barycenter graph for multiple observed graphs; the barycenter graph plays the role of the disconnected graph, and since it is learned, so is the clustering. Our method combines a recursive $K$-partition mechanism with a regularized proximal gradient algorithm, whose time complexity is $\mathcal{O}(K(E+V)\log_K V)$ for graphs with $V$ nodes and $E$ edges. To our knowledge, our method is the first attempt to make Gromov-Wasserstein discrepancy applicable to large-scale graph analysis and unify graph partitioning and matching into the same framework. It outperforms state-of-the-art graph partitioning and matching methods, achieving a trade-off between accuracy and efficiency.

Constituting highly informative network embeddings is an important tool for network analysis. It encodes network topology, along with other useful side information, into low-dimensional node-based feature representations that can be exploited by statistical modeling. This work focuses on learning context-aware network embeddings augmented with text data. We reformulate the network-embedding problem, and present two novel strategies to improve over traditional attention mechanisms: ($i$) a content-aware sparse attention module based on optimal transport, and ($ii$) a high-level attention parsing module. Our approach yields naturally sparse and self-normalized relational inference. It can capture long-term interactions between sequences, thus addressing the challenges faced by existing textual network embedding schemes. Extensive experiments are conducted to demonstrate our model can consistently outperform alternative state-of-the-art methods.

This paper explores the problem of matching entities across different knowledge graphs. Given a query entity in one knowledge graph, we wish to find the corresponding real-world entity in another knowledge graph. We formalize this problem and present two large-scale datasets for this task based on exiting cross-ontology links between DBpedia and Wikidata, focused on several hundred thousand ambiguous entities. Using a classification-based approach, we find that a simple multi-layered perceptron based on representations derived from RDF2Vec graph embeddings of entities in each knowledge graph is sufficient to achieve high accuracy, with only small amounts of training data. The contributions of our work are datasets for examining this problem and strong baselines on which future work can be based.

Proximal Policy Optimization (PPO) is a highly popular model-free reinforcement learning (RL) approach. However, in continuous state and actions spaces and a Gaussian policy -- common in computer animation and robotics -- PPO is prone to getting stuck in local optima. In this paper, we observe a tendency of PPO to prematurely shrink the exploration variance, which naturally leads to slow progress. Motivated by this, we borrow ideas from CMA-ES, a black-box optimization method designed for intelligent adaptive Gaussian exploration, to derive PPO-CMA, a novel proximal policy optimization approach that can expand the exploration variance on objective function slopes and shrink the variance when close to the optimum. This is implemented by using separate neural networks for policy mean and variance and training the mean and variance in separate passes. Our experiments demonstrate a clear improvement over vanilla PPO in many difficult OpenAI Gym MuJoCo tasks.

We show that for the problem of testing if a matrix $A \in F^{n \times n}$ has rank at most $d$, or requires changing an $\epsilon$-fraction of entries to have rank at most $d$, there is a non-adaptive query algorithm making $\widetilde{O}(d^2/\epsilon)$ queries. Our algorithm works for any field $F$. This improves upon the previous $O(d^2/\epsilon^2)$ bound (SODA'03), and bypasses an $\Omega(d^2/\epsilon^2)$ lower bound of (KDD'14) which holds if the algorithm is required to read a submatrix. Our algorithm is the first such algorithm which does not read a submatrix, and instead reads a carefully selected non-adaptive pattern of entries in rows and columns of $A$. We complement our algorithm with a matching query complexity lower bound for non-adaptive testers over any field. We also give tight bounds of $\widetilde{\Theta}(d^2)$ queries in the sensing model for which query access comes in the form of $\langle X_i, A\rangle:=tr(X_i^\top A)$; perhaps surprisingly these bounds do not depend on $\epsilon$. We next develop a novel property testing framework for testing numerical properties of a real-valued matrix $A$ more generally, which includes the stable rank, Schatten-$p$ norms, and SVD entropy. Specifically, we propose a bounded entry model, where $A$ is required to have entries bounded by $1$ in absolute value. We give upper and lower bounds for a wide range of problems in this model, and discuss connections to the sensing model above.

Knowledge graphs contain rich relational structures of the world, and thus complement data-driven machine learning in heterogeneous data. One of the most effective methods in representing knowledge graphs is to embed symbolic relations and entities into continuous spaces, where relations are approximately linear translation between projected images of entities in the relation space. However, state-of-the-art relation projection methods such as TransR, TransD or TransSparse do not model the correlation between relations, and thus are not scalable to complex knowledge graphs with thousands of relations, both in computational demand and in statistical robustness. To this end we introduce TransF, a novel translation-based method which mitigates the burden of relation projection by explicitly modeling the basis subspaces of projection matrices. As a result, TransF is far more light weight than the existing projection methods, and is robust when facing a high number of relations. Experimental results on the canonical link prediction task show that our proposed model outperforms competing rivals by a large margin and achieves state-of-the-art performance. Especially, TransF improves by 9%/5% in the head/tail entity prediction task for N-to-1/1-to-N relations over the best performing translation-based method.