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The information bottleneck method
Naftali Tishby, Fernando C. Pereira, William Bialek
TL;DR
Choosing relevant features is difficult when a suitable distortion function is unavailable. The information bottleneck compresses a signal while preserving information about another variable, and its self-consistent equations can be solved by a convergent iterative method.
Problem
Relevant compression requires identifying which features matter, but rate-distortion theory requires a distortion function that may be difficult to specify.
Source: §1 · pp. 2–3Method
The information bottleneck compresses X into ˜X while preserving information about Y, with KL divergence between p(y|x) and p(y|˜x) emerging as the effective distortion.
Source: §3.1 · p. 9, §3.2 · p. 12Results
The self-consistent equations admit a convergent iterative re-estimation algorithm.
Source: §1 · p. 4, §3.3 · p. 14Takeaways & Limitations
The framework treats relevance as task-dependent information and encompasses prediction, filtering, and learning as special cases.
Source: §1 · p. 2Obtaining a sufficiently good sample of the joint distribution p(x,y) is a learning-theory issue left beyond this paper’s scope.
Source: §3 · p. 8
Abstract
from arXiv · showhide
We define the relevant information in a signal $x\in X$ as being the information that this signal provides about another signal $y\in \Y$. Examples include the information that face images provide about the names of the people portrayed, or the information that speech sounds provide about the words spoken. Understanding the signal $x$ requires more than just predicting $y$, it also requires specifying which features of $\X$ play a role in the prediction. We formalize this problem as that of finding a short code for $\X$ that preserves the maximum information about $\Y$. That is, we squeeze the information that $\X$ provides about $\Y$ through a `bottleneck' formed by a limited set of codewords $\tX$. This constrained optimization problem can be seen as a generalization of rate distortion theory in which the distortion measure $d(x,\x)$ emerges from the joint statistics of $\X$ and $\Y$. This approach yields an exact set of self consistent equations for the coding rules $X \to \tX$ and $\tX \to \Y$. Solutions to these equations can be found by a convergent re-estimation method that generalizes the Blahut-Arimoto algorithm. Our variational principle provides a surprisingly rich framework for discussing a variety of problems in signal processing and learning, as will be described in detail elsewhere.
1 Introduction
The paper frames relevant information as information in a signal that helps predict another variable, addressing the difficulty of choosing which signal features matter. It proposes an information-theoretic compression principle and derives self-consistent equations and a convergent algorithm for finding relevant representations.
1 Introduction
- The paper asks how to quantify meaningful information, arguing that lossy compression offers a natural approach to representing information relevant to another signal.The motivating examples include speech sounds and the words they convey.Source: §1 · p. 2
- Rate-distortion theory requires a distortion function that implicitly selects relevant features, but those features may be unknown, making the choice difficult in applications such as speech recognition.The paper identifies this as a feature-selection problem that rate-distortion theory alone does not resolve.Source: §1 · pp. 2–3
- An additional variable, such as a speech transcript or speaker identity, can specify which signal features are relevant to the prediction task.The choice of that variable determines the relevant components in each domain.Source: §1 · p. 3
- The paper derives self-consistent equations and an iterative algorithm for representations that capture relevant structure, and proves the algorithm converges.Source: §1 · pp. 3–4
2 Relevant quantization
The section develops quantization as a compressed representation, reviews rate-distortion theory’s rate–distortion tradeoff, and describes the Blahut–Arimoto algorithm and its scope.
2 Relevant quantization
- Quantization maps signal values to codebook representatives, while the rate is measured by mutual information I(X; ˜X), not generally by codebook entropy H(˜X).Reducing rate alone can discard signal details, so an additional constraint is needed.Source: §2 · pp. 4–5
2.1 Relevance through distortion: Rate distortion theory
- Rate-distortion theory minimizes rate subject to an expected-distortion constraint, with higher rate permitting smaller achievable distortion.The distortion function specifies which aspects of the signal count as relevant.Source: §2.1 · p. 5
- The rate-distortion variational formulation yields an exponential-form solution and a converging iterative procedure for determining p(˜x|x) and p(˜x).Source: §2.1 · pp. 5–6, §2.2 · p. 7
2.2 The Blahut–Arimoto algorithm
- Alternating Blahut–Arimoto updates converge to a unique minimum of the functional, but optimize the partition for fixed representatives rather than selecting optimal representatives.For finite data, selecting representatives is a separate joint-optimization problem similar to EM and need not have a unique solution.Source: §2.2 · p. 8
3 Relevance through another variable: The Information Bottleneck
The Information Bottleneck seeks compressed representations of X that retain information about a dependent relevance variable Y, assuming access to their joint distribution. It derives self-consistent assignments and an alternating algorithm, with solution structure varying by β and codebook size.
3 Relevance through another variable: The Information Bottleneck
- The method assumes access to a joint distribution p(x, y) with positive mutual information; obtaining a sufficiently good sample is beyond the paper’s scope.Source: §3 · p. 8
3.1 A new variational principle
- The bottleneck minimizes compression cost while preserving information about Y, with β controlling the tradeoff between meaningful information and representation detail.Source: §3.1 · p. 9
3.2 Self-consistent equations
- The equations derive KL divergence as effective distortion and jointly determine assignments and representatives, but the formal solution is implicit because p(y|X~) depends on the assignments.Source: §3.2 · pp. 10, 12
3.3 The information bottleneck iterative algorithm
- Alternating updates over p(X~|X), p(X~), and p(Y|X~) converge, but separate convexity does not establish a unique solution.Source: §3.3 · pp. 13–14
3.4 The structure of the solutions
- For each codebook cardinality, increasing β traces convex information-plane curves that bifurcate at critical β, yielding a hierarchy of quantizations.Source: §3.4 · pp. 14–15
Further work
- The principle is presented as a unified framework for prediction, filtering, and learning, with applications including semantic clustering and document classification.Source: Further work · p. 15