[Paper Review] Gaussian Lower Bound for the Information Bottleneck Limit

Amichai Painsky, Naftali Tishby|arXiv (Cornell University)|Nov 7, 2017

Distributed Sensor Networks and Detection Algorithms21 references5 citations

TL;DR

This paper introduces a Gaussian lower bound for the Information Bottleneck (IB) curve by maximizing the jointly Gaussian component of arbitrary data via nonlinear transformations, enabling efficient, analytically tractable IB-like representation learning. The method outperforms naive Gaussianization and is bounded above by nonlinear CCA, revealing fundamental limits on linearizing non-Gaussian data through second-order statistics.

ABSTRACT

The Information Bottleneck (IB) is a conceptual method for extracting the most compact, yet informative, representation of a set of variables, with respect to the target. It generalizes the notion of minimal sufficient statistics from classical parametric statistics to a broader information-theoretic sense. The IB curve defines the optimal trade-off between representation complexity and its predictive power. Specifically, it is achieved by minimizing the level of mutual information (MI) between the representation and the original variables, subject to a minimal level of MI between the representation and the target. This problem is shown to be in general NP hard. One important exception is the multivariate Gaussian case, for which the Gaussian IB (GIB) is known to obtain an analytical closed form solution, similar to Canonical Correlation Analysis (CCA). In this work we introduce a Gaussian lower bound to the IB curve; we find an embedding of the data which maximizes its "Gaussian part", on which we apply the GIB. This embedding provides an efficient (and practical) representation of any arbitrary data-set (in the IB sense), which in addition holds the favorable properties of a Gaussian distribution. Importantly, we show that the optimal Gaussian embedding is bounded from above by non-linear CCA. This allows a fundamental limit for our ability to Gaussianize arbitrary data-sets and solve complex problems by linear methods.

Motivation & Objective

• To address the challenge of approximating the IB curve for arbitrary continuous, non-Gaussian data, which is generally intractable.
• To develop a practical and theoretically grounded method for representing complex data using linear methods by maximizing its 'Gaussian part'.
• To establish a fundamental limit on how much information can be captured via second-order statistics alone, using the proposed Gaussian lower bound.
• To improve upon existing bounds—such as Cardoso’s information geometry bound—by introducing a tighter, data-adaptive lower bound via the AGCE method.

Proposed method

• The method seeks transformations $\phi(\underline{X})$ and $\psi(\underline{Y})$ that maximize joint Gaussianity while preserving mutual information between $\underline{X}$ and $\underline{Y}$.
• It applies the Gaussian IB (GIB) to the transformed variables $\underline{U} = \phi(\underline{X})$ and $\underline{V} = \psi(\underline{Y})$, leveraging the closed-form solution of GIB via canonical correlation analysis.
• The optimal transformation is derived using the Alternating Conditional Expectations (ACE) algorithm to maximize correlation under Gaussianity constraints.
• The approach is formalized as a lower bound on the IB curve, where the bound is tightest when the data’s nonlinear dependencies are well-captured by second-order statistics.
• The method is validated via reverse annealing and discretization (Gaussian quadratures) for numerical approximation in continuous settings.
• Theoretical analysis shows the bound is upper-bounded by nonlinear CCA, revealing a fundamental trade-off between Gaussianization and information retention.

Experimental results

Research questions

• RQ1 Can we construct a lower bound for the IB curve that is both analytically tractable and informative for arbitrary non-Gaussian data?
• RQ2 What is the maximal amount of information that can be preserved in a Gaussianized representation, and what limits this preservation?
• RQ3 How does the performance of the proposed Gaussian lower bound compare to naive Gaussianization (i.e., applying GIB directly to raw data)?
• RQ4 What is the theoretical relationship between the proposed bound and nonlinear canonical correlation analysis?
• RQ5 To what extent can nonlinear dependencies be captured using only second-order statistics, and what are the fundamental limits?

Key findings

• The proposed Gaussian lower bound consistently outperforms naive Gaussianization (applying GIB directly to raw data), especially in highly non-Gaussian settings such as exponential and Gaussian mixture models.
• The bound is tightest for low-complexity (highly compressed) representations, where degenerate, Gaussian-like structures are easier to recover.
• The method reveals that the maximal achievable information in a Gaussianized representation is fundamentally limited by the non-linear canonical correlation coefficients between $\underline{X}$ and $\underline{Y}$.
• The proposed bound is tighter than Cardoso’s information geometry bound, demonstrating the advantage of the AGCE-based approach for constructing data-adaptive Gaussian approximations.
• The method shows practical utility in continuous IB approximation, offering a meaningful benchmark even when the true IB curve is intractable to compute.
• Theoretical analysis confirms that the bound is upper-bounded by nonlinear CCA, establishing a fundamental limit on linearizing non-linear problems via second-order statistics.