It is known that the membership in a given reproducing kernel Hilbert space (RKHS) of the samples of a Gaussian process $X$ is controlled by a certain nuclear dominance condition. However, it is less clear how to identify a "small" set of functions (not necessarily a vector space) that contains the samples. This article presents a general approach for identifying such sets. We use scaled RKHSs, which can be viewed as a generalisation of Hilbert scales, to define the sample support set as the largest set which is contained in every element of full measure under the law of $X$ in the $\sigma$-algebra induced by the collection of scaled RKHS. This potentially non-measurable set is then shown to consist of those functions that can be expanded in terms of an orthonormal basis of the RKHS of the covariance kernel of $X$ and have their squared basis coefficients bounded away from zero and infinity, a result suggested by the Karhunen-Lo\`{e}ve theorem.

One of the requirements of the population-based approach to Structural Health Monitoring (SHM) proposed in the earlier papers in this sequence, is that structures be represented by points in an abstract space. Furthermore, these spaces should be metric spaces in a loose sense; i.e. there should be some measure of distance applicable to pairs of points; similar structures should then be close in the metric. However, this geometrical construction is not enough for the framing of problems in data-based SHM, as it leaves undefined the notion of feature spaces. Interpreting the feature values on a structure-by-structure basis as a type of field over the space of structures, it seems sensible to borrow an idea from modern theoretical physics, and define feature assignments as sections in a vector bundle over the structure space. With this idea in place, one can interpret the effect of environmental and operational variations as gauge degrees of freedom, as in modern gauge field theories. This paper will discuss the various geometrical structures required for an abstract theory of feature spaces in SHM, and will draw analogies with how these structures have shown their power in modern physics. In the second part of the paper, the problem of determining the normal condition cross section of a feature bundle is addressed. The solution is provided by the application of Graph Neural Networks (GNN), a versatile non-Euclidean machine learning algorithm which is not restricted to inputs and outputs from vector spaces. In particular, the algorithm is well suited to operating directly on the sort of graph structures which are an important part of the proposed framework for PBSHM. The solution of the normal section problem is demonstrated for a heterogeneous population of truss structures for which the feature of interest is the first natural frequency.

Another is the family of optimal transport central to many machine learning algorithms, with distances (Villani, 2008), including the Wasserstein-1 metric. Lipschitz constrained variants recently gaining In general, variational representations are supremums attention. Inspired by this, we generalize the of integral formulas taken over sets of functions, such as the so-called tight variational representation of f-Donsker-Varadhan formula (Donsker & Varadhan, 1976) divergences in the case of probability measures for the Kullback-Leibler divergence or the Kantorovich-on compact metric spaces to be taken over the Rubinstein formula (Villani, 2008) for the Wasserstein-1 space of Lipschitz functions vanishing at an arbitrary metric. Informally speaking, one can implement (Nowozin base point, characterize functions achieving et al., 2016; Arjovsky et al., 2017) such a formula by constructing the supremum in the variational representation, a real-valued neural network taking samples from propose a practical algorithm to calculate the the two probability measures as inputs, which is then trained tight convex conjugate of f-divergences compatible to maximize the integral formula in order to approximate with automatic differentiation frameworks, the supremum, resulting in a learned proxy to the actual define the Moreau-Yosida approximation of f-divergence of said probability measures. Implementing the divergences with respect to the Wasserstein-1 metric, Kantorovich-Rubinstein formula in such a way involves and derive the corresponding variational formulas, restricting the Lipschitz constant of the neural network (Gulrajani providing a generalization of a number et al., 2017; Petzka et al., 2018; Miyato et al., 2018), of recent results, novel special cases of interest which effectively stabilizes the approximation procedure.

Embedding methods for mixed-curvature spaces are powerful techniques for low-distortion and low-dimensional representation of complex data structures. Nevertheless, little is known regarding downstream learning and optimization in the embedding space. Here, we address for the first time the problem of linear classification in a product space form -- a mix of Euclidean, spherical, and hyperbolic spaces with different dimensions. First, we revisit the definition of a linear classifier on a Riemannian manifold by using geodesics and Riemannian metrics which generalize the notions of straight lines and inner products in vector spaces, respectively. Second, we prove that linear classifiers in $d$-dimensional constant curvature spaces can shatter exactly $d+1$ points: Hence, Euclidean, hyperbolic and spherical classifiers have the same expressive power. Third, we formalize linear classifiers in product space forms, describe a novel perceptron classification algorithm, and establish rigorous convergence results. We support our theoretical findings with simulation results on several datasets, including synthetic data, MNIST and Omniglot. Our results reveal that learning methods applied to small-dimensional embeddings in product space forms significantly outperform their algorithmic counterparts in Euclidean spaces.

In many applications of machine learning, such as recommender systems, where an input characterizing a user should be matched with a target representing an ordering of a large number of items, accessing fully supervised data (,) is not an option. Instead, one should expect weak information on the target, which could be a list of previously taken (if items are online courses), watched (if items are plays), etc., items by a user characterized by the feature vector. This motivates weakly supervised learning, aiming at learning a mapping from inputs to targets in such a setting where tools from supervised learning can not be applied off-the-shelves. Recent applications of weakly supervised learning showcase impressive results in solving complex tasks such as action retrieval on instructional videos (Miech et al., 2019), image semantic segmentation (Papandreou et al., 2015), salient object detection (Wang et al., 2017), 3D pose estimation (Dabral et al., 2018), text-to-speech synthesis (Jia et al., 2018), to name a few. However, those applications of weakly supervised learning are usually based on clever heuristics, and theoretical foundations of learning from weakly supervised data are scarce, especially when compared to statistical learning literature on supervised learning (Vapnik, 1995; Boucheron et al., 2005; Steinwart and Christmann, 2008). We aim to provide a step in this direction. In this paper, we focus on partial labelling, a popular instance of weak supervision, approached with a structured prediction point of view Ciliberto et al. (2020). We detail this setup in Section 2. Our contributions are organized as follows.

We investigate the problem of exact cluster recovery using oracle queries. Previous results show that clusters in Euclidean spaces that are convex and separated with a margin can be reconstructed exactly using only $O(\log n)$ same-cluster queries, where $n$ is the number of input points. In this work, we study this problem in the more challenging non-convex setting. We introduce a structural characterization of clusters, called $(\beta,\gamma)$-convexity, that can be applied to any finite set of points equipped with a metric (or even a semimetric, as the triangle inequality is not needed). Using $(\beta,\gamma)$-convexity, we can translate natural density properties of clusters (which include, for instance, clusters that are strongly non-convex in $R^d$) into a graph-theoretic notion of convexity. By exploiting this convexity notion, we design a deterministic algorithm that recovers $(\beta,\gamma)$-convex clusters using $O(k^2 \log n + k^2 (\frac{6}{\beta\gamma})^{dens(X)})$ same-cluster queries, where $k$ is the number of clusters and $dens(X)$ is the density dimension of the semimetric. We show that an exponential dependence on the density dimension is necessary, and we also show that, if we are allowed to make $O(k^2 + k \log n)$ additional queries to a "cluster separation" oracle, then we can recover clusters that have different and arbitrary scales, even when the scale of each cluster is unknown.

Network embedding maps the nodes of a given network into a low-dimensional space such that the semantic similarities among the nodes can be effectively inferred. Most existing approaches use inner-product of node embedding to measure the similarity between nodes leading to the fact that they lack the capacity to capture complex relationships among nodes. Besides, they take the path in the network just as structural auxiliary information when inferring node embeddings, while paths in the network are formed with rich user informations which are semantically relevant and cannot be ignored. In this paper, We propose a novel method called Network Embedding on the Metric of Relation, abbreviated as NEMR, which can learn the embeddings of nodes in a relational metric space efficiently. First, our NEMR models the relationships among nodes in a metric space with deep learning methods including variational inference that maps the relationship of nodes to a gaussian distribution so as to capture the uncertainties. Secondly, our NEMR considers not only the equivalence of multiple-paths but also the natural order of a single-path when inferring embeddings of nodes, which makes NEMR can capture the multiple relationships among nodes since multiple paths contain rich user information, e.g., age, hobby and profession. Experimental results on several public datasets show that the NEMR outperforms the state-of-the-art methods on relevant inference tasks including link prediction and node classification.

Finding an expert plays a crucial role in driving successful collaborations and speeding up high-quality research development and innovations. However, the rapid growth of scientific publications and digital expertise data makes identifying the right experts a challenging problem. Existing approaches for finding experts given a topic can be categorised into information retrieval techniques based on vector space models, document language models, and graph-based models. In this paper, we propose $\textit{ExpFinder}$, a new ensemble model for expert finding, that integrates a novel $N$-gram vector space model, denoted as $n$VSM, and a graph-based model, denoted as $\textit{$\mu$CO-HITS}$, that is a proposed variation of the CO-HITS algorithm. The key of $n$VSM is to exploit recent inverse document frequency weighting method for $N$-gram words and $\textit{ExpFinder}$ incorporates $n$VSM into $\textit{$\mu$CO-HITS}$ to achieve expert finding. We comprehensively evaluate $\textit{ExpFinder}$ on four different datasets from the academic domains in comparison with six different expert finding models. The evaluation results show that $\textit{ExpFinder}$ is a highly effective model for expert finding, substantially outperforming all the compared models in 19% to 160.2%.

Fitting neural networks often resorts to stochastic (or similar) gradient descent which is a noise-tolerant (and efficient) resolution of a gradient descent dynamics. It outputs a sequence of networks parameters, which sequence evolves during the training steps. The gradient descent is the limit, when the learning rate is small and the batch size is infinite, of this set of increasingly optimal network parameters obtained during training. In this contribution, we investigate instead the convergence in the Generative Adversarial Networks used in machine learning. We study the limit of small learning rate, and show that, similar to single network training, the GAN learning dynamics tend, for vanishing learning rate to some limit dynamics. This leads us to consider evolution equations in metric spaces (which is the natural framework for evolving probability laws)that we call dual flows. We give formal definitions of solutions and prove the convergence. The theory is then applied to specific instances of GANs and we discuss how this insight helps understand and mitigate the mode collapse.

We address the problem of estimating intrinsic distances in a manifold from a finite sample. We prove that the metric space defined by the sample endowed with a computable metric known as sample Fermat distance converges a.s. in the sense of Gromov-Hausdorff. The limiting object is the manifold itself endowed with the population Fermat distance, an intrinsic metric that accounts for both the geometry of the manifold and the density that produces the sample. This result is applied to obtain sample persistence diagrams that converge towards an intrinsic persistence diagram. We show that this method outperforms more standard approaches based on Euclidean norm with theoretical results and computational experiments.