A Sufficient-Statistic Reduction of the Information Bottleneck to a Low-Dimensional Problem

Abstract: We show that if the conditional distribution p(C | T) factors through a sufficient statistic φ(T), then the Information Bottleneck (IB) problem for (T, C) is exactly equivalent to the IB problem for (φ(T), C). The reduction is loss-free: it preserves the full IB curve, the Lagrangian optimum at every trade-off parameter \b{eta}, and the optimal representations up to pullback through φ. As a result, the computational complexity of solving the IB problem is governed by the dimension of the sufficient statistic rather than the ambient dimension of the source. This identifies an exact structural condition under which the generic IB problem becomes tractable, and gives a formal bridge between the discrete and linear-Gaussian regimes. We then show that the classical Gaussian IB solution of Chechik, Globerson, Tishby and Weiss is an immediate corollary of this reduction, and we state a nonlinear-Gaussian generalisation. A small numerical example illustrates the practical consequence: when a low-dimensional sufficient statistic is available, the exact IB curve can be computed on the reduced problem at a cost determined by the statistic rather than by the ambient source dimension.

• The paper establishes that IB optimization is exactly equivalent when using a sufficient statistic, preserving the IB curve, Lagrangian formulation, and minimizer correspondence.
• It leverages data-processing inequalities and mutual information analysis to reduce the computational complexity from high-dimensional sources to a lower-dimensional sufficient statistic.
• The reduction method efficiently recovers known results in both Gaussian and non-Gaussian settings, offering practical benefits for solving otherwise intractable IB problems.

Sufficient-Statistic Reduction of the Information Bottleneck to a Low-Dimensional Problem

Overview

This paper presents a formal reduction principle for the Information Bottleneck (IB) problem based on the existence of sufficient statistics. The primary result demonstrates that if the conditional distribution $p(C|T)$ factors through a sufficient statistic $\phi(T)$, then the IB optimization problem for $(T, C)$ is exactly equivalent to that for $(\phi(T), C)$. This equivalence holds at the level of the entire IB curve, the Lagrangian formulation for all $\beta$, and, crucially, the minimizing representations themselves (modulo “pullback” via $\phi$). The computational complexity of IB is thereby governed by the dimension of the sufficient statistic rather than the ambient dimension of the source, making many otherwise intractable IB problems computationally manageable.

Structural Reduction of IB via Sufficiency

The paper establishes a rigorous equivalence between the IB problems for $(T, C)$ and $((T), C)$ when $\phi(T)$ is sufficient for $C$ given $\phi(T)$0. The main theorem (Theorem 1) is multifaceted:

• Curve Preservation: The IB curve, which describes the maximal achievable relevance $\phi(T)$1 for each allowed rate $\phi(T)$2, is exactly preserved under reduction to the sufficient statistic.
• Lagrangian Equivalence: For any tradeoff parameter $\phi(T)$3, the minimal IB Lagrangian over all encoders $\phi(T)$4 is equal to the minimal Lagrangian over encoders $\phi(T)$5, where $\phi(T)$6.
• Minimizer Correspondence: There is a canonical mapping (pullback) between optimizers of the two forms via $\phi(T)$7. Specifically, any optimizer in the reduced problem lifts to an optimizer in the original, and vice versa.
• Critical $\phi(T)$8 Preservation: The phase transition structure—the set of $\phi(T)$9 values where the optimal representation undergoes bifurcation (“kinks” in the IB curve)—is preserved.

These claims hold exactly, not as approximations, and do not depend on the specifics of the cardinality or distributional family beyond existence of the requisite Markov structure.

The paper provides complete proofs, leveraging the data-processing inequality, conditional independence, and careful analysis of mutual information in the presence of sufficient statistics. This structure leads to a lossless and algorithmically meaningful collapse of the original IB optimization to a potentially much lower-dimensional version.

Implications: Computational and Theoretical

A direct consequence of this reduction is that the computational cost of solving the IB problem is determined by the dimension of the sufficient statistic and by the structure of $(T, C)$0, not by the (possibly much larger) dimension of $(T, C)$1. In the discrete case, if the sufficient statistic $(T, C)$2 takes $(T, C)$3 values and $(T, C)$4 takes $(T, C)$5 values, Blahut–Arimoto (BA) iteration can efficiently compute the IB solution over the $(T, C)$6 joint. This principle applies even if $(T, C)$7 is very high-dimensional or continuous.

This reduction provides an explicit bridge between the tractable (discrete, low-cardinality) IB and the closed-form linear-Gaussian IB, and further extends to settings with nonlinear, non-Gaussian sufficient statistics.

Gaussian and Beyond: Specializations

The reduction theorem immediately recovers the classical Gaussian IB solution of Chechik et al., where the conditional distribution $(T, C)$8 is Gaussian with mean linear in $(T, C)$9. In this case, the sufficient statistic is the conditional mean, a linear function of $(\phi(T), C)$0; thus, the dimensionality of the effective IB problem is at most $(\phi(T), C)$1. All IB optimizers and bifurcation points in the Gaussian regime are characterized by this lower-dimensional projection, providing a conceptual explanation for prior empirical results and justifying the observed rank bounds.

The result extends to nonlinear-Gaussian cases, including additive-noise models and mixture-of-Gaussians where the sufficient statistic may be a nonlinear or categorical function of $(\phi(T), C)$2. Formally, whenever $(\phi(T), C)$3, the IB problem reduces in the same manner, regardless of the original data dimension.

Importantly, this generalizes prior work on exponential-family IB formulations and clarifies the structural role sufficient statistics play in governing IB complexity and tractability.

Numerical Illustration and Variational IB

A numerical example demonstrates the practical implications. For a synthetic setting where $(\phi(T), C)$4 exhibits a nonlinear, low-dimensional sufficient statistic, the reduced IB problem is solved efficiently via BA, with wall-clock times orders of magnitude lower than variational IB methods. The deterministic BA method completely avoids the posterior collapse pathologies observed in VIB (where the learnt representation can degenerate to triviality for large $(\phi(T), C)$5), highlighting both the computational—and statistical—advantages of reduction via sufficiency.

Discussion and Theoretical Context

The sufficiency-based reduction clarifies the computational and statistical structure of IB. In practical settings—including classification, regression with additive noise, and many generative models—the conditional distribution $(\phi(T), C)$6 typically factors through a low-dimensional sufficient statistic. The IB reduction enables efficient, population-level computation of relevance–compression tradeoffs and supports the exact solution of problems that would otherwise be prohibitive due to data dimensionality. When the sufficient statistic is not known, the result serves as a design principle: estimate a sufficient statistic first, then solve IB in the reduced space.

The result generalizes classical data-processing results for sufficient statistics within the full IB optimization framework. The theoretical development produces an exact preservation of not only the value functions but also the entire optimizer structure and bifurcation analysis, in contrast to previous surrogate or variational approaches.

Future Directions

Further work could extend the reduction to finite-sample regimes, conditional and multi-view IB variants, or its integration as a preprocessing or regularization layer within neural IB solvers.

Conclusion

This paper rigorously establishes that the Information Bottleneck problem collapses without loss to a lower-dimensional instance whenever $(\phi(T), C)$7 factors through a sufficient statistic. This result preserves all information-theoretic and variational structure and yields exact, practical, and computationally efficient procedures for high-dimensional IB problems. The theorem provides a unifying lens to understand the tractability of various IB regimes and has direct implications for efficient algorithm design, theoretical analysis, and the future application of IB in complex machine learning systems.

Reference: "A Sufficient-Statistic Reduction of the Information Bottleneck to a Low-Dimensional Problem" (2604.26744)