Conference Paper

Indian Buffet Processes with Power-law Behavior.

Conference: Advances in Neural Information Processing Systems 22: 23rd Annual Conference on Neural Information Processing Systems 2009. Proceedings of a meeting held 7-10 December 2009, Vancouver, British Columbia, Canada.
Source: DBLP


The Indian buffet process (IBP) is an exchangeable distribution over binary ma- trices used in Bayesian nonparametric featural models. In this paper we propose a three-parameter generalization of the IBP exhibiting power-law behavior. We achieve this by generalizing the beta process (the de Finetti measure of the IBP) to the stable-beta process and deriving the IBP corresponding to it. We find interest- ing relationships between the stable-beta process and the Pitman-Yor process (an- other stochastic process used in Bayesian nonparametric models with interesting power-law properties). We derive a stick-breaking construction for the stable-beta process, and find that our power-law IBP is a good model for word occurrences in document corpora.

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    • "They also appear, up to a transformation s → − log(1 − s), in [12]. Teh and Görur [38] "
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    ABSTRACT: We study random families of subsets of $\mathbb{N}$ that are similar to exchangeable random partitions, but do not require constituent sets to be disjoint: Each element of ${\mathbb{N}}$ may be contained in multiple subsets. One class of such objects, known as Indian buffet processes, has become a popular tool in machine learning. Based on an equivalence between Indian buffet and scale-invariant Poisson processes, we identify a random scaling variable whose role is similar to that played in exchangeable partition models by the total mass of a random measure. Analogous to the construction of exchangeable partitions from normalized subordinators, random families of sets can be constructed from randomly scaled subordinators. Coupling to a heavy-tailed scaling variable induces a power law on the number of sets containing the first $n$ elements. Several examples, with properties desirable in applications, are derived explicitly. A relationship to exchangeable partitions is made precise as a correspondence between scaled subordinators and Poisson-Kingman measures, generalizing a result of Arratia, Barbour and Tavare on scale-invariant processes.
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    • "La collection de mesures binaires (Z 1 , ..., Z n ) définit l'ensemble des relations entre les lecteurs et les livres. Généralisant l'extension du processus du buffet indien (IBP) (Griffiths et Ghahramani, 2005, 2011) appelée IBP stable (Teh et Görür, 2009), Caron (2012) propose le modèle z ij |γ i , w j ∼ Ber (1 − exp(−γ i w j )) où les (w j , θ j ), w j > 0 sont issus d'une mesure complètement aléatoire (CRM, voir section 2) et où chaque lecteur possède son propre paramètre d'intérêt pour la lecture γ i > 0. Ce modèle plus flexible permet une distribution des degrés des lecteurs non Poissonnienne , tout en conservant les propriétés de conjugaison et un processus génératif similaire à l'IBP (stable). Cependant, ce modèle est limité à une approximation de rang 1 de la collection (Z 1 , ..., Z n ) : chaque lecteur n'a qu'un seul paramètre qui régle la quantité de livres qu'il lira et chaque livre ne possède qu'un seul paramètre qui règle sa popularité. "

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    • "We will describe the dynamics using a culinary metaphor (similarly to what some authors do for other models, see Chinese Restaurant [29], Indian Buffet process [15] [16] [33] and their generalizations [4] [5]). We identify the nodes with the customers of a restaurant and the attributes with the dishes, so that the dishes tried by a customer represent the attributes that a node exhibits. "
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    ABSTRACT: The quest for a model that is able to explain, describe, analyze and simulate real-world complex networks is of uttermost practical as well as theoretical interest. In this paper we introduce and study a network model that is based on a latent attribute structure: each node is characterized by a number of features and the probability of the existence of an edge between two nodes depends on the features they share. Features are chosen according to a process of Indian-Buffet type but with an additional random "fitness" parameter attached to each node, that determines its ability to transmit its own features to other nodes. As a consequence, a node's connectivity does not depend on its age alone, so also "young" nodes are able to compete and succeed in acquiring links. One of the advantages of our model for the latent bipartite "node-attribute" network is that it depends on few parameters with a straightforward interpretation. We provide some theoretical, as well experimental, results regarding the power-law behaviour of the model and the estimation of the parameters. By experimental data, we also show how the proposed model for the attribute structure naturally captures most local and global properties (e.g., degree distributions, connectivity and distance distributions) real networks exhibit. keyword: Complex network, social network, attribute matrix, Indian Buffet process
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