An official website of the United States government
Abstract The renormalization group (RG) is a class of theoretical techniques used to explain the collective physics of interacting, many-body systems. It has been suggested that the RG formalism may be useful in finding and interpreting emergent low-dimensional structure in complex systems outside of the traditional physics context, such as in biology or computer science. In such contexts, one common dimensionality-reduction framework already in use is information bottleneck (IB), in which the goal is to compress an ‘input’ signalXwhile maximizing its mutual information with some stochastic ‘relevance’ variableY. IB has been applied in the vertebrate and invertebrate processing systems to characterize optimal encoding of the future motion of the external world. Other recent work has shown that the RG scheme for the dimer model could be ‘discovered’ by a neural network attempting to solve an IB-like problem. This manuscript explores whether IB and any existing formulation of RG are formally equivalent. A class of soft-cutoff non-perturbative RG techniques are defined by families of non-deterministic coarsening maps, and hence can be formally mapped onto IB, and vice versa. For concreteness, this discussion is limited entirely to Gaussian statistics (GIB), for which IB has exact, closed-form solutions. Under this constraint, GIB has a semigroup structure, in which successive transformations remain IB-optimal. Further, the RG cutoff scheme associated with GIB can be identified. Our results suggest that IB can be used toimposea notion of ‘large scale’ structure, such as biological function, on an RG procedure.
more »
« less
Nonlinear Information Bottleneck
Information bottleneck (IB) is a technique for extracting information in one random variable X that is relevant for predicting another random variable Y. IB works by encoding X in a compressed “bottleneck” random variable M from which Y can be accurately decoded. However, finding the optimal bottleneck variable involves a difficult optimization problem, which until recently has been considered for only two limited cases: discrete X and Y with small state spaces, and continuous X and Y with a Gaussian joint distribution (in which case optimal encoding and decoding maps are linear). We propose a method for performing IB on arbitrarily-distributed discrete and/or continuous X and Y, while allowing for nonlinear encoding and decoding maps. Our approach relies on a novel non-parametric upper bound for mutual information. We describe how to implement our method using neural networks. We then show that it achieves better performance than the recently-proposed “variational IB” method on several real-world datasets.
more »
« less
- Award ID(s):
- 1648973
- PAR ID:
- 10142780
- Date Published:
- Journal Name:
- Entropy
- Volume:
- 21
- Issue:
- 12
- ISSN:
- 1099-4300
- Page Range / eLocation ID:
- 1181
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
The new 5G communications standard increases data rates and supports low-latency communication that places constraints on the computational complexity of channel decoders. 5G low-density parity-check (LDPC) codes have the so-called protograph-based raptor-like (PBRL) structure which offers inherent rate-compatibility and excellent performance. Practical LDPC decoder implementations use message-passing decoding with finite precision, which becomes coarse as complexity is more severely constrained. Performance degrades as the precision becomes more coarse. Recently, the information bottleneck (IB) method was used to design mutual-information-maximizing lookup tables that replace conventional finite-precision node computations. The IB approach exchanges messages represented by integers with very small bit width. This paper extends the IB principle to the flexible class of PBRL LDPC codes as standardized in 5G. The extensions include puncturing and rate-compatible IB decoder design. As an example of the new approach, a 4-bit information bottleneck decoder is evaluated for PBRL LDPC codes over a typical range of rates. Frame error rate simulations show that the proposed scheme outperforms offset min-sum decoding algorithms and operates very close to double-precision sum-product belief propagation decoding.more » « less
-
Estimation of mutual information from observed samples is a basic primitive in machine learning, useful in several learning tasks including correlation mining, information bottleneck, Chow-Liu tree, and conditional independence testing in (causal) graphical models. While mutual information is a quantity well-defined for general probability spaces, estimators have been developed only in the special case of discrete or continuous pairs of random variables. Most of these estimators operate using the 3H -principle, i.e., by calculating the three (differential) entropies of X, Y and the pair (X,Y). However, in general mixture spaces, such individual entropies are not well defined, even though mutual information is. In this paper, we develop a novel estimator for estimating mutual information in discrete-continuous mixtures. We prove the consistency of this estimator theoretically as well as demonstrate its excellent empirical performance. This problem is relevant in a wide-array of applications, where some variables are discrete, some continuous, and others are a mixture between continuous and discrete components.more » « less
-
Information bottleneck (IB) and privacy funnel (PF) are two closely related optimization problems which have found applications in machine learning, design of privacy algorithms, capacity problems (e.g., Mrs. Gerber’s Lemma), and strong data processing inequalities, among others. In this work, we first investigate the functional properties of IB and PF through a unified theoretical framework. We then connect them to three information-theoretic coding problems, namely hypothesis testing against independence, noisy source coding, and dependence dilution. Leveraging these connections, we prove a new cardinality bound on the auxiliary variable in IB, making its computation more tractable for discrete random variables. In the second part, we introduce a general family of optimization problems, termed “bottleneck problems”, by replacing mutual information in IB and PF with other notions of mutual information, namely f-information and Arimoto’s mutual information. We then argue that, unlike IB and PF, these problems lead to easily interpretable guarantees in a variety of inference tasks with statistical constraints on accuracy and privacy. While the underlying optimization problems are non-convex, we develop a technique to evaluate bottleneck problems in closed form by equivalently expressing them in terms of lower convex or upper concave envelope of certain functions. By applying this technique to a binary case, we derive closed form expressions for several bottleneck problems.more » « less
-
We generalize the information bottleneck (IB) and privacy funnel (PF) problems by introducing the notion of a sensitive attribute, which arises in a growing number of applications. In this generalization, we seek to construct representations of observations that are maximally (or minimally) informative about a target variable, while also satisfying constraints with respect to a variable corresponding to the sensitive attribute. In the Gaussian and discrete settings, we show that by suitably approximating the Kullback-Liebler (KL) divergence defining traditional Shannon mutual information, the generalized IB and PF problems can be formulated as semi-definite programs (SDPs), and thus efficiently solved, which is important in applications of high-dimensional inference. We validate our algorithms on synthetic data and demonstrate their use in imposing fairness in machine learning on real data as an illustrative application.more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.3390/e21121181
