Markov random fields are used to model high dimensional distributions in a number of applied areas. Much recent interest has been devoted to the reconstruction of the dependency structure from independent samples from the Markov random fields. We analyze a simple algorithm for reconstructing the underlying graph defining a Markov random field on @math nodes and maximum degree @math given observations. We show that under mild non-degeneracy conditions it reconstructs the generating graph with high probability using @math samples where @math depend on the local interactions. For most local interaction @math are of order @math . Our results are optimal as a function of @math up to a multiplicative constant depending on @math and the strength of the local interactions. Our results seem to be the first results for general models that guarantee that the generating model is reconstructed. Furthermore, we provide explicit @math running time bound. In cases where the measure on the graph has correlation decay, the running time is @math for all fixed @math . We also discuss the effect of observing noisy samples and show that as long as the noise level is low, our algorithm is effective. On the other hand, we construct an example where large noise implies non-identifiability even for generic noise and interactions. Finally, we briefly show that in some simple cases, models with hidden nodes can also be recovered.
Abbeel, @cite_3 considered the problem of reconstructing graphical models based on factor graphs, and proposed a polynomial time and sample complexity algorithm. However, the goal of their algorithm was not to reconstruct the true structure, but rather to produce a model whose distribution is close in Kullback-Leibler divergence to the true distribution. In applications it is often of interest to reconstruct the true structure which give some insights into the underlying structure of the inferred model.
@cite_3
2147071755 2132346211 2034230784 2950618930
We study the computational and sample complexity of parameter and structure learning in graphical models. Our main result shows that the class of factor graphs with bounded degree can be learned in polynomial time and from a polynomial number of training examples, assuming that the data is generated by a network in this class. This result covers both parameter estimation for a known network structure and structure learning. It implies as a corollary that we can learn factor graphs for both Bayesian networks and Markov networks of bounded degree, in polynomial time and sample complexity. Importantly, unlike standard maximum likelihood estimation algorithms, our method does not require inference in the underlying network, and so applies to networks where inference is intractable. We also show that the error of our learned model degrades gracefully when the generating distribution is not a member of the target class of networks. In addition to our main result, we show that the sample complexity of parameter learning in graphical models has an O(1) dependence on the number of variables in the model when using the KL-divergence normalized by the number of variables as the performance criterion. We propose a new graph-based semi-supervised learning (SSL) algorithm and demonstrate its application to document categorization. Each document is represented by a vertex within a weighted undirected graph and our proposed framework minimizes the weighted Kullback-Leibler divergence between distributions that encode the class membership probabilities of each vertex. The proposed objective is convex with guaranteed convergence using an alternating minimization procedure. Further, it generalizes in a straightforward manner to multi-class problems. We present results on two standard tasks, namely Reuters-21578 and WebKB, showing that the proposed algorithm significantly outperforms the state-of-the-art. Summary form only. Inspired by Kolmogorov's structure function for finite sets as models of data in the algorithmic theory of information we adapt the construct to families of probability models to avoid the noncomputability problem. The picture of modeling looks then as follows: The models in the family have a double index, where the first specifies a structure, ranging over a finite or a countable set, and the second consists of parameter values, ranging over a continuum. An optimal structure index can be determined by the MDL (Minimum Description Length) principle in a two-part code, where the sum of the code lengths for the structure and the data is minimized. The latter is obtained from the universal NML (Normalized Maximum Likelihood) model for the subfamily of models having a specified structure. The determination of the optimal model in the optimized structure is more difficult. It requires a partition of the parameter space into equivalence classes, each associated with a model, in such a way that the Kullback-Leibler distance between any two adjacent models is equal and that the models are optimally distinguishable from the given amount of data. This notion of distinguishability is a modification of a related idea of Balasubramanian. The particular model, specified by the observed data, is the simplest one that incorporates all the properties in the data that can be extracted with the model class considered. In this paper, we address the problem of learning the structure of a pairwise graphical model from samples in a high-dimensional setting. Our first main result studies the sparsistency, or consistency in sparsity pattern recovery, properties of a forward-backward greedy algorithm as applied to general statistical models. As a special case, we then apply this algorithm to learn the structure of a discrete graphical model via neighborhood estimation. As a corollary of our general result, we derive sufficient conditions on the number of samples n, the maximum node-degree d and the problem size p, as well as other conditions on the model parameters, so that the algorithm recovers all the edges with high probability. Our result guarantees graph selection for samples scaling as n = Omega(d^2 log(p)), in contrast to existing convex-optimization based algorithms that require a sample complexity of (d^3 log(p)). Further, the greedy algorithm only requires a restricted strong convexity condition which is typically milder than irrepresentability assumptions. We corroborate these results using numerical simulations at the end.