Learning Generative Graph Prototypes using Simplified Von Neumann Entropy
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1 Learning Generative Graph Prototypes using Simplified Von Neumann Entropy Lin Han, Edwin R. Hancock and Richard C. Wilson Department of Computer Science The University of York YO10 5DD, UK
2 Graph representations Advantages: can capture object or scene structure in a manner that is invariant to changes in viewpoint. Abstract scene contents in an efficient way. Disadvantages: can be fragile (sensitive to noise and segmentation error). Available pattern recognition/machine learning methodology limited.
3 Graph Representations from images
4 Learning with graphs Cluster similar objects, and represent them using a class prototype (e.g. median) using (dis) similarities. Extract features and perform central clustering. Construct a generative model to capture distribution of structural variations using probability distributions.
5 .. is difficult because Graphs are not vectors: There is no natural ordering of nodes and edges. Correspondences must be used to establish order. Structural variations: Numbers of nodes and edges are not fixed. They can vary due to segmentation error. Not easily summarized: Since they do not reside in a vector space, mean and covariance hard to characterise.
6 Generative Models Structural domain: define probability distribution over prototype structure. Prototype together with parameters of distribution minimise description length (Torsello and Hancock, PAMI 2007). Spectral domain: embed nodes of graphs into vectorspace using spectral decomposition. Construct point distribution model over embedded positions of nodes (Bai, Wilson and Hancock, CVIU 2009).
7 Aim Combine spectral and structural methods. Use description length criterion. Apply to graphs rather than trees.
8 Prior work IJCV 2007 (Torsello, Robles-Kelly, Hancock) shape classes from edit distance using pairwise clustering. PAMI 06 and Pattern Recognition 05 (Wilson, Luo and Hancock) graph clustering using spectral features and polynomials. PAMI 07 (Torsello and Hancock) generative model for variations in tree structure using description length. CVIU09 (Xiao, Wilson and Hancock) generative model from heat-kernel embedding of graphs.
9 Structural learning Using description length
10 Description length Wallace+Freeman: minimum message length. Rissanen: minimum description length. Use log-posterior probability to locate model that is optimal with respect to code-length.
11 Similarities/differences MDL: selection of model is aim; model parameters are simply a means to this end. Parameters usually maximum likelihood. Prior on parameters is flat. MML: Recovery of model parameters is central. Parameter prior may be more complex.
12 Coding scheme Usually assumed to follow an exponential distribution. Alternatives are universal codes and predictive codes. MML has two part codes (model+parameters). In MDL the codes may be one or two-part.
13 Method Model is supergraph (i.e. Graph prototypes) formed by graph union. Sample data observation model: Bernoulli distribution over nodes and edges. Mode: complexity: Von-Neumann entropy of supergraphs. Fitting criterion: MDL-like-make ML estimates of the Bernoulli parameters MML-like: two-part code for data-model fit + supergraph complexity.
14 Learn supergraph using MDL Follow Torsello and Hancock and pose the problem of learning generative model for graphs as that of learning a supergraph representation. Required probability distributions is an extension of model developed by Luo and Hancock. Use von Neumann entropy to control supergraph s complexity. Develop an EM algorithm in which the node correspondences and the supergraph edge probability matrix are treated as missing data.
15 Probabilistic Framework Here the structure of the sample graphs and the supergraph are represented by their Adjacency matrices V1 V1 V2 V3 V4 V2 V V1 A= V2 V3 V4 V4
16 Observation model Given a sample graph and a supergraph along with their assignment matrix, the a posteriori probabilities of the sample graphs given the structure of the supergraph and the node correspondences is defined as
17 Data code-length For the sample graph-set and the supergraph, the set of assignment is. Under the assumption that the graphs in are independent samples from the distribution, the likelihood of the sample graphs can be written Code length of observed data
18 Model complexity -Von Neumann entropy Measured by the von Neumann entropy associated with the Laplacian eigenspectrum of graphs (Passerini, Severini, 2008) Comes from quantum mechanics and is entropy associated with density matrix.
19 Quadratic Approximination Approximation by quadratic entropy In terms of matrix traces
20 Simplified Entropy Normalized Laplacian where V is the number of nodes. Normalized Laplacian squared Collect terms together, von Neumann entropy simplifies to
21 Overall code-length According to Rissanen and Grunwald s minimum description length criterion, we encode and transmit the sample graphs and the supergraph structure. This leads to a two-part message whose total length is given We consider both the node correspondence information between graphs S and the structure of the supergraph M as missing data and locate M by minimizing the overall code-length using EM algorithm.
22 Expectation + Maximization M-step : Recover correspondence matrices: Take partial derivative of the weighted loglikelihood function and soft assign. Modify supergraph structure : E-step: Compute the a posteriori probability of the nodes in the sample graphs being matching to those of the supergraph.
23 Experiments Delaunay graphs from images of different objects. COIL dataset Toys dataset
24 Experiment---validation COIL dataset: model complexity increase, graph data log-likelihood increase, overall code length decrease during iterations. Toys dataset: model complexity decrease, graph data log-likelihood increase, overall code length decrease during iterations.
25 Experiment---classification task We compare the performance of our learned supergraph on classification task with two alternative constructions, the median graph and the supergraph learned without using MDL. The table below shows the average classification rates from 10-fold cross validation, which are followed by their standard errors.
26 Experiment---generate new samples
27 Conclusion We have shown how a supergraph or generative model of graph structure can be learned under minimum description length. We propose a variant of EM algorithm to locate the structure of the supergraph. In our experiments, we demonstrate that our supergraph learning method is valid and the supergraph learned is effective for classification.
28 References 1. Friedman, N., Koller, D.:Being Bayesian about network structure.a Bayesian approach to structure discovery in Bayesian networks. Machine Learning,95 125(2003) 2. Christmas, W. J., Kittler, J., Petrou, M.: Modeling compatibility coefficient distribution. Image and Vsion Computing 14, (1996) 3. Bagdanov, A.D., Worring, M.: First order Gaussian graphs for efficient structure classification. Pattern Recognition 36, (2003) 4. Luo, B., Hancock, E.R.: A spectral approach to learning structural variations in graphs. Pattern Recognition 39, (2006) 5. Torsello, A., Hancock, E.R.: Learning shape-classes using a mixture of tree-unions. IEEE PAMI 28, (2006) 6. Luo, B., Hancock, E.R.: Structural graph matching using the EM alogrithm and singular value decomposition. IEEE PAMI 23, (2001) 7. Rissanen, J.: Modelling by Shortest Data Description. Automatica, (1978) 8. Passerini, F., Severini, S.: The von-neumann entropy of networks. arxiv: (2008) 9. Grunwald, P.: Minimum Description Length Tutorial. Advances in Minimum Description Length: Theory and Applications (2005) 10. Gold, S., Rangarajan, A.: A Graduated Assignment Algorithm for Graph Matching. IEEE PAMI 18, (1996) 11. Wilson, R.C., Zhu, P.: A study of graph spectra for comparing graphs and trees. Pattern Recognition 41, (2008) 12. Wilson, R.C., Hancock, E.R.: Structural matching by discrete relaxation. IEEE PAMI 19, (1997) 13. Han, L., Wilson, R.C., Hancock, E.H.: A Supergraph-based Generative Model. ICPR, pp (2010) 14. Figueiredo, M.A.T., Jain, A.K.: Unsupervised learning of finite mixture models. IEEE PAMI 24, (2002)
29 Thanks! And Questions?
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