Novel Lower Bounds on the Entropy Rate of Binary Hidden Markov Processes

Recently, Samorodnitsky proved a strengthened version of Mrs. Gerber's Lemma, where the output entropy of a binary symmetric channel is bounded in terms of the average entropy of the input projected on a random subset of coordinates. Here, this result is applied for deriving novel lower bounds on the entropy rate of binary hidden Markov processes. For symmetric underlying Markov processes, our bound improves upon the best known bound in the very noisy regime. The nonsymmetric case is also considered, and explicit bounds are derived for Markov processes that satisfy the $(1,\infty)$-RLL constraint

Correlation-powered Information Engines and the Thermodynamics of Self-Correction

Information engines can use structured environments as a resource to generate work by randomizing ordered inputs and leveraging the increased Shannon entropy to transfer energy from a thermal reservoir to a work reservoir. We give a broadly applicable expression for the work production of an information engine, generally modeled as a memoryful channel that communicates inputs to outputs as it interacts with an evolving environment. The expression establishes that an information engine must have more than one memory state in order to leverage input environment correlations. To emphasize this functioning, we designed an information engine powered solely by temporal correlations and not by statistical biases, as employed by previous engines. Key to this is the engine's ability to synchronize---the engine automatically returns to a desired dynamical phase when thrown into an unwanted, dissipative phase by corruptions in the input---that is, by unanticipated environmental fluctuations. This self-correcting mechanism is robust up to a critical level of corruption, beyond which the system fails to act as an engine. We give explicit analytical expressions for both work and critical corruption level and summarize engine performance via a thermodynamic-function phase diagram over engine control parameters. The results reveal a new thermodynamic mechanism based on nonergodicity that underlies error correction as it operates to support resilient engineered and biological systems.Comment: 22 pages, 13 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/tos.ht

Deep Unsupervised Learning using Nonequilibrium Thermodynamics

A central problem in machine learning involves modeling complex data-sets using highly flexible families of probability distributions in which learning, sampling, inference, and evaluation are still analytically or computationally tractable. Here, we develop an approach that simultaneously achieves both flexibility and tractability. The essential idea, inspired by non-equilibrium statistical physics, is to systematically and slowly destroy structure in a data distribution through an iterative forward diffusion process. We then learn a reverse diffusion process that restores structure in data, yielding a highly flexible and tractable generative model of the data. This approach allows us to rapidly learn, sample from, and evaluate probabilities in deep generative models with thousands of layers or time steps, as well as to compute conditional and posterior probabilities under the learned model. We additionally release an open source reference implementation of the algorithm

Quantifying Self-Organization with Optimal Wavelets

The optimal wavelet basis is used to develop quantitative, experimentally applicable criteria for self-organization. The choice of the optimal wavelet is based on the model of self-organization in the wavelet tree. The framework of the model is founded on the wavelet-domain hidden Markov model and the optimal wavelet basis criterion for self-organization which assumes inherent increase in statistical complexity, the information content necessary for maximally accurate prediction of the system's dynamics. At the same time the method, presented here for the one-dimensional data of any type, performs superior denoising and may be easily generalized to higher dimensions.Comment: 12 pages, 3 figure

Source Coding When the Side Information May Be Delayed

For memoryless sources, delayed side information at the decoder does not improve the rate-distortion function. However, this is not the case for more general sources with memory, as demonstrated by a number of works focusing on the special case of (delayed) feedforward. In this paper, a setting is studied in which the encoder is potentially uncertain about the delay with which measurements of the side information are acquired at the decoder. Assuming a hidden Markov model for the sources, at first, a single-letter characterization is given for the set-up where the side information delay is arbitrary and known at the encoder, and the reconstruction at the destination is required to be (near) lossless. Then, with delay equal to zero or one source symbol, a single-letter characterization is given of the rate-distortion region for the case where side information may be delayed or not, unbeknownst to the encoder. The characterization is further extended to allow for additional information to be sent when the side information is not delayed. Finally, examples for binary and Gaussian sources are provided.Comment: revised July 201

Occam's Quantum Strop: Synchronizing and Compressing Classical Cryptic Processes via a Quantum Channel

A stochastic process's statistical complexity stands out as a fundamental property: the minimum information required to synchronize one process generator to another. How much information is required, though, when synchronizing over a quantum channel? Recent work demonstrated that representing causal similarity as quantum state-indistinguishability provides a quantum advantage. We generalize this to synchronization and offer a sequence of constructions that exploit extended causal structures, finding substantial increase of the quantum advantage. We demonstrate that maximum compression is determined by the process's cryptic order---a classical, topological property closely allied to Markov order, itself a measure of historical dependence. We introduce an efficient algorithm that computes the quantum advantage and close noting that the advantage comes at a cost---one trades off prediction for generation complexity.Comment: 10 pages, 6 figures; http://csc.ucdavis.edu/~cmg/compmech/pubs/oqs.ht