[Paper Review] On the Information-theoretic Limits of Graphical Model Selection for Gaussian Time Series

Gábor Hannák, Alexander Jung+1 more|arXiv (Cornell University)|Mar 4, 2014

Fault Detection and Control Systems4 citations

TL;DR

This paper establishes information-theoretic lower bounds on the sample size required for reliable conditional independence graph (CIG) selection in multivariate stationary Gaussian time series, using Fano's inequality and spectral smoothness constraints. It shows that a simple nonparametric selection scheme achieves these bounds for sparse CIGs, revealing that sample size requirements are independent of spectral smoothness, implying existing sufficient conditions may be suboptimal.

ABSTRACT

We consider the problem of inferring the conditional independence graph (CIG) of a multivariate stationary dicrete-time Gaussian random process based on a finite length observation. Using information-theoretic methods, we derive a lower bound on the error probability of any learning scheme for the underlying process CIG. This bound, in turn, yields a minimum required sample-size which is necessary for any algorithm regardless of its computational complexity, to reliably select the true underlying CIG. Furthermore, by analysis of a simple selection scheme, we show that the information-theoretic limits can be achieved for a subclass of processes having sparse CIG. We do not assume a parametric model for the observed process, but require it to have a sufficiently smooth spectral density matrix (SDM).

Motivation & Objective

• To determine the minimum sample size required for any algorithm to reliably infer the conditional independence graph (CIG) of a multivariate stationary Gaussian time series.
• To establish a fundamental lower bound on error probability in CIG selection using information-theoretic methods, independent of computational complexity.
• To assess whether existing nonparametric graphical model selection schemes achieve these information-theoretic limits.
• To investigate the role of spectral smoothness and sparsity in determining the sample complexity of CIG learning.

Proposed method

• Derives a lower bound on the error probability of any CIG learning scheme using Fano's inequality applied to a set of plausible CIGs.
• Imposes smoothness constraints on the spectral density matrix (SDM) via the moment condition μx = ∑|m|‖Rx[m]‖∞ < ∞ to ensure spectral regularity.
• Analyzes a nonparametric selection scheme based on Fourier-domain inner products of estimated process components to recover neighborhood structures.
• Uses concentration inequalities and Gaussian tail bounds to derive sufficient conditions on sample size for reliable neighborhood recovery.
• Applies union bounds across nodes to control the overall graph selection error probability.
• Compares the derived lower bound with performance of a specific selection scheme to assess tightness of the limit.

Experimental results

Research questions

• RQ1 What is the minimum sample size required for any algorithm to reliably select the true conditional independence graph (CIG) of a multivariate Gaussian time series?
• RQ2 How does the required sample size depend on the spectral smoothness of the process, as quantified by the ACF moment μx?
• RQ3 Can a simple, nonparametric selection scheme achieve the information-theoretic lower bound on sample size for CIG selection?
• RQ4 Is the sample complexity independent of the correlation width (i.e., smoothness of the SDM), as suggested by the analysis?
• RQ5 For which class of processes does the information-theoretic limit become tight, and can it be achieved by a computationally efficient scheme?

Key findings

• The information-theoretic lower bound on sample size for reliable CIG selection is independent of the spectral smoothness (i.e., correlation width) of the process.
• For a subclass of processes with extremely sparse CIGs, the derived lower bound is tight and can be achieved by a simple nonparametric selection scheme.
• The required sample size for reliable CIG learning does not scale with the smoothness of the spectral density matrix, contrary to common assumptions.
• A sufficient condition for reliable selection via the proposed scheme is N ≥ 32B⁴ log(p²/δ)/ρ_min², which matches the information-theoretic limit up to logarithmic factors.
• The analysis reveals that the sufficient conditions in prior work (e.g., [9]) are not optimal, as they likely overestimate the required sample size.
• The error probability of the selection scheme decays exponentially with sample size, with rate depending on the minimum eigenvalue gap ρ_min and spectral norm bound B.