Abstract In this work we show that, using the eigen-decomposition of the adjacencymatrix, we can consistently estimate latent positions for random dot productgraphs provided the latent positions are i.i.d. from some distribution. If classlabels are observed for a number of vertices tending to inﬁnity, then we showthat the remaining vertices can be classiﬁed with error converging to Bayes opti-mal using the k -nearest-neighbors classiﬁcation rule. We evaluate the proposedmethods on simulated data and a graph derived from Wikipedia. 1 Introduction The classical statistical pattern recognition setting involves( X , Y ),( X 1 , Y 1 ),...,( X n , Y n ) i . i˘ . d . F X , Y ,where the X i : 7!R d are observed feature vectors and the Y i : 7!f0,1gare ob-served class labels for some probability space . We deﬁne D=f( X i , Y i )gas the train-ing set. The goal is to learn a classiﬁer h (;D): R d !f0,1gsuch that the probability oferror P[ h ( X ;D)6= Y jD]approaches Bayes optimal as n !1for all distributions
Many latent space approaches seek to generalize the stochastic blockmodel to allow for variation within blocks. For example, the mixed membership model of @cite_10 posits that a vertex could have partial membership in multiple blocks. In @cite_13 , latent vectors are presumed to be drawn from a mixture of multivariate normal distributions with the link function depending on the distance between the latent vectors. They use Bayesian techniques to estimate the latent vectors.
@cite_13 @cite_10
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In this paper we present a fully Bayesian latent variable model which exploits conditional nonlinear(in)-dependence structures to learn an efficient latent representation. The latent space is factorized to represent shared and private information from multiple views of the data. In contrast to previous approaches, we introduce a relaxation to the discrete segmentation and allow for a "softly" shared latent space. Further, Bayesian techniques allow us to automatically estimate the dimensionality of the latent spaces. The model is capable of capturing structure underlying extremely high dimensional spaces. This is illustrated by modelling unprocessed images with tenths of thousands of pixels. This also allows us to directly generate novel images from the trained model by sampling from the discovered latent spaces. We also demonstrate the model by prediction of human pose in an ambiguous setting. Our Bayesian framework allows us to perform disambiguation in a principled manner by including latent space priors which incorporate the dynamic nature of the data. We propose a temporal latent space model for link prediction in dynamic social networks, where the goal is to predict links over time based on a sequence of previous graph snapshots. The model assumes that each user lies in an unobserved latent space, and interactions are more likely to occur between similar users in the latent space representation. In addition, the model allows each user to gradually move its position in the latent space as the network structure evolves over time. We present a global optimization algorithm to effectively infer the temporal latent space. Two alternative optimization algorithms with local and incremental updates are also proposed, allowing the model to scale to larger networks without compromising prediction accuracy. Empirically, we demonstrate that our model, when evaluated on a number of real-world dynamic networks, significantly outperforms existing approaches for temporal link prediction in terms of both scalability and predictive power. Given data, deep generative models, such as variational autoencoders (VAE) and generative adversarial networks (GAN), train a lower dimensional latent representation of the data space. The linear Euclidean geometry of data space pulls back to a nonlinear Riemannian geometry on the latent space. The latent space thus provides a low-dimensional nonlinear representation of data and classical linear statistical techniques are no longer applicable. In this paper we show how statistics of data in their latent space representation can be performed using techniques from the field of nonlinear manifold statistics. Nonlinear manifold statistics provide generalizations of Euclidean statistical notions including means, principal component analysis, and maximum likelihood fits of parametric probability distributions. We develop new techniques for maximum likelihood inference in latent space, and adress the computational complexity of using geometric algorithms with high-dimensional data by training a separate neural network to approximate the Riemannian metric and cometric tensor capturing the shape of the learned data manifold. We develop an information-theoretic view of the stochastic block model, a popular statistical model for the large-scale structure of complex networks. A graph G from such a model is generated by first assigning vertex labels at random from a finite alphabet, and then connecting vertices with edge probabilities depending on the labels of the endpoints. In the case of the symmetric two-group model, we establish an explicit ‘single-letter’ characterization of the per-vertex mutual information between the vertex labels and the graph, when the graph average degree diverges. The explicit expression of the mutual information is intimately related to estimation-theoretic quantities, and -in particular- reveals a phase transition at the critical point for community detection. Below the critical point the per-vertex mutual information is asymptotically the same as if edges were independent of the vertex labels. Correspondingly, no algorithm can estimate the partition better than random guessing. Conversely, above the threshold, the per-vertex mutual information is strictly smaller than the independent-edges upper bound. In this regime there exists a procedure that estimates the vertex labels better than random guessing.