Information-theoretic limits of selecting binary graphical models in high dimensions (0905.2639v1)

Abstract: The problem of graphical model selection is to correctly estimate the graph structure of a Markov random field given samples from the underlying distribution. We analyze the information-theoretic limitations of the problem of graph selection for binary Markov random fields under high-dimensional scaling, in which the graph size $p$ and the number of edges $k$, and/or the maximal node degree $d$ are allowed to increase to infinity as a function of the sample size $n$. For pairwise binary Markov random fields, we derive both necessary and sufficient conditions for correct graph selection over the class $\mathcal{G}{p,k}$ of graphs on $p$ vertices with at most $k$ edges, and over the class $\mathcal{G}{p,d}$ of graphs on $p$ vertices with maximum degree at most $d$. For the class $\mathcal{G}{p, k}$, we establish the existence of constants $c$ and $c'$ such that if $\numobs < c k \log p$, any method has error probability at least 1/2 uniformly over the family, and we demonstrate a graph decoder that succeeds with high probability uniformly over the family for sample sizes $\numobs > c' k^2 \log p$. Similarly, for the class $\mathcal{G}{p,d}$, we exhibit constants $c$ and $c'$ such that for $n < c d^2 \log p$, any method fails with probability at least 1/2, and we demonstrate a graph decoder that succeeds with high probability for $n > c' d^3 \log p$.

• The paper establishes necessary sample size conditions for accurate graph selection, showing that data below specific thresholds makes successful recovery infeasible.
• Sufficient conditions are derived, indicating the sample sizes needed for a specific strategy to successfully select the graph with high probability.
• The findings provide foundational statistical boundaries for high-dimensional graph selection, guiding data collection practices and theoretical advancements.

Information-theoretic Limits of Selecting Binary Graphical Models in High Dimensions

The paper explores the fundamental problem of graphical model selection, specifically focusing on the information-theoretic boundaries within the scope of high-dimensional binary Markov random fields. The central theme is to accurately estimate the graph structure from sampled data, which is pivotal in numerous domains including image analysis, social network analysis, and computational biology.

The researchers evaluate the conditions under which it is feasible to perform graph selection effectively as the problem scales—that is, as the number of vertices $p$, number of edges $k$, or maximal node degree $d$ tends to infinity with the sample size $n$. The objective is to delineate necessary and sufficient conditions for successful graph selection, essentially providing a rich insight into the complexity and feasibility of the task at hand.

Main Contributions

1. Necessary Conditions for Graph Selection:
   • Graph Class $\mathcal{G}_{p,k}$: For graphs with at most $k$ edges, if the sample size $n$ is less than $c k \log p$, any method will have at least a $1/2$ probability of error, showcasing the infeasibility of exact graph recovery under this sample constraint. These conditions confirm the intuition that as the complexity of the graph increases (either in terms of more edges or higher node degree), so does the information requirement for successful decoding.
   • Graph Class $\mathcal{G}_{p,d}$: In scenarios where the graph has a degree constraint, if the sample size is below $c d^2 \log p$, any method would fail with a probability of at least $1/2$. This constraint illustrates the profound impact of maximum node degree on the sample complexity, highlighting the role of node connectivity in graph identifiability.
2. Sufficient Conditions for Graph Selection:
   • Graph Class $\mathcal{G}_{p,k}$: They demonstrate a graph decoding strategy which requires at least $c' k^2 \log p$ samples for correct graph recovery with high probability. It's a pointer towards an achievable threshold for graph learning when the graph structure possesses a bounded number of edges.
   • Graph Class $\mathcal{G}_{p,d}$: For this bounded-degree scenario, the paper offers a strategy that can successfully select the graph structure using more than $c' d^3 \log p$ samples. This provides a practical boundary that assures data sufficiency in practical high-dimensional scenarios, thus offering a feasible path for scalable graph recovery.

Implications

The exploration of these limits has profound implications for theoretical and practical advancements in areas relying on Markov random fields. Practically, this research informs how large one can expect graphs to grow while still allowing accurate selection, guiding data collection practices in large-scale applications. Theoretically, it suggests potential directions for exploring enhanced algorithms matching these thresholds or possibly revising the assumptions under real-world constraints.

Future Directions

With the rigorous bounds established, future research might explore:

• Algorithmic Developments: Bridging the gap within these bounds by devising algorithms optimizing sample efficiency, potentially leveraging sparsity or distributional properties.
• Extension to Non-binary Variables: Investigating analogous limits in models involving non-binary variables could open avenues in more complex applications across various scientific fields.
• Incorporation of Computational Constraints: Examining trade-offs between computational efficiency and sample complexity which could lead to richer understanding in real-world deployments.

The findings set a foundational understanding of the statistical boundaries inherent in high-dimensional graph selection, paving the way for more informed methodological advancements in dealing with complex, large-scale data.