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
If we suppose that the latent positions are i.i.d. from some distribution, then the latent space approach is closely related to the theory of exchangeable random graphs . For exchangeable graphs, we have a (measurable) link function @math and each vertex is associated with a latent i.i.d. uniform @math random variable denoted @math . Conditioned on the @math , the adjacency of vertices @math and @math is determined by a Bernoulli trial with parameter @math . For a treatment of exchangeable graphs and estimation using the method of moments, see @cite_6 .
@cite_6
2020272293 2127475907 2737338020 2963577681
Let W1,…,Wn be independent random subsets of [m]= 1,…,m . Assuming that each Wi is uniformly distributed in the class of d-subsets of [m] we study the uniform random intersection graph Gs(n,m,d) on the vertex set W1,…Wn , defined by the adjacency relation: Wi∼Wj whenever |Wi∩Wj|≧s. For even n we show that as n,m→∞ the edge density threshold for the property that Gs(n,m,d) contains a perfect matching is asymptotically the same as that for Gs(n,m,d) being connected. We extend spectral methods to random graphs with skewed degree distributions through a degree based normalization closely connected to the normalized Laplacian. The normalization is based on intuition drawn from perturbation theory of random matrices, and has the effect of boosting the expectation of the random adjacency matrix without increasing the variances of its entries, leading to better perturbation bounds. The primary implication of this result lies in the realm of spectral analysis of random graphs with skewed degree distributions, such as the ubiquitous "power law graphs". Mihail and Papadimitriou (2002) argued that for randomly generated graphs satisfying a power law degree distribution, spectral analysis of the adjacency matrix simply produces the neighborhoods of the high degree nodes as its eigenvectors, and thus miss any embedded structure. We present a generalization of their model, incorporating latent structure, and prove that after applying our transformation, spectral analysis succeeds in recovering the latent structure with high probability. We present a randomised distributed maximal independent set (MIS) algorithm for arbitrary graphs of size @math that halts in time @math with probability @math , each message containing @math bit: thus its bit complexity per channel is @math (the bit complexity is the number of bits we need to solve a distributed task, it measures the communication complexity). We assume that the graph is anonymous: unique identities are not available to distinguish the processes; we only assume that each vertex distinguishes between its neighbours by locally known channel names. Furthermore we do not assume that the size (or an upper bound on the size) of the graph is known. This algorithm is optimal (modulo a multiplicative constant) for the bit complexity and improves the best previous randomised distributed MIS algorithms (deduced from the randomised PRAM algorithm due to Luby) for general graphs which is @math per channel (it halts in time @math and the size of each message is @math ). This result is based on a powerful and general technique for converting unrealistic exchanges of messages containing real numbers drawn at random on each vertex of a network into exchanges of bits. Then we consider a natural question: what is the impact of a vertex inclusion in the MIS on distant vertices? We prove that this impact vanishes rapidly as the distance grows for bounded-degree vertices. We provide a counter-example that shows this result does not hold in general. We prove also that these results remain valid for Luby's algorithm presented by Lynch and by Wattenhofer. This question remains open for the variant given by Peleg. Despite the recent advance of Generative Adversarial Networks (GANs) in high-fidelity image synthesis, there lacks enough understandings on how GANs are able to map the latent code sampled from a random distribution to a photo-realistic image. Previous work assumes the latent space learned by GAN follows a distributed representation but observes the vector arithmetic phenomenon of the output's semantics in latent space. In this work, we interpret the semantics hidden in the latent space of well-trained GANs. We find that the latent code for well-trained generative models, such as ProgressiveGAN and StyleGAN, actually learns a disentangled representation after some linear transformations. We make a rigorous analysis on the encoding of various semantics in the latent space as well as their properties, and then study how these semantics are correlated to each other. Based on our analysis, we propose a simple and general technique, called InterFaceGAN, for semantic face editing in latent space. Given a synthesized face, we are able to faithfully edit its various attributes such as pose, expression, age, presence of eyeglasses, without retraining the GAN model. Furthermore, we show that even the artifacts occurred in output images are able to be fixed using same approach. Extensive results suggest that learning to synthesize faces spontaneously brings a disentangled and controllable facial attribute representation