553 research outputs found
Universal Estimation of Directed Information
Four estimators of the directed information rate between a pair of jointly
stationary ergodic finite-alphabet processes are proposed, based on universal
probability assignments. The first one is a Shannon--McMillan--Breiman type
estimator, similar to those used by Verd\'u (2005) and Cai, Kulkarni, and
Verd\'u (2006) for estimation of other information measures. We show the almost
sure and convergence properties of the estimator for any underlying
universal probability assignment. The other three estimators map universal
probability assignments to different functionals, each exhibiting relative
merits such as smoothness, nonnegativity, and boundedness. We establish the
consistency of these estimators in almost sure and senses, and derive
near-optimal rates of convergence in the minimax sense under mild conditions.
These estimators carry over directly to estimating other information measures
of stationary ergodic finite-alphabet processes, such as entropy rate and
mutual information rate, with near-optimal performance and provide alternatives
to classical approaches in the existing literature. Guided by these theoretical
results, the proposed estimators are implemented using the context-tree
weighting algorithm as the universal probability assignment. Experiments on
synthetic and real data are presented, demonstrating the potential of the
proposed schemes in practice and the utility of directed information estimation
in detecting and measuring causal influence and delay.Comment: 23 pages, 10 figures, to appear in IEEE Transactions on Information
Theor
Secrecy Through Synchronization Errors
In this paper, we propose a transmission scheme that achieves information
theoretic security, without making assumptions on the eavesdropper's channel.
This is achieved by a transmitter that deliberately introduces synchronization
errors (insertions and/or deletions) based on a shared source of randomness.
The intended receiver, having access to the same shared source of randomness as
the transmitter, can resynchronize the received sequence. On the other hand,
the eavesdropper's channel remains a synchronization error channel. We prove a
secrecy capacity theorem, provide a lower bound on the secrecy capacity, and
propose numerical methods to evaluate it.Comment: 5 pages, 6 figures, submitted to ISIT 201
Characterization of Information Channels for Asymptotic Mean Stationarity and Stochastic Stability of Non-stationary/Unstable Linear Systems
Stabilization of non-stationary linear systems over noisy communication
channels is considered. Stochastically stable sources, and unstable but
noise-free or bounded-noise systems have been extensively studied in
information theory and control theory literature since 1970s, with a renewed
interest in the past decade. There have also been studies on non-causal and
causal coding of unstable/non-stationary linear Gaussian sources. In this
paper, tight necessary and sufficient conditions for stochastic stabilizability
of unstable (non-stationary) possibly multi-dimensional linear systems driven
by Gaussian noise over discrete channels (possibly with memory and feedback)
are presented. Stochastic stability notions include recurrence, asymptotic mean
stationarity and sample path ergodicity, and the existence of finite second
moments. Our constructive proof uses random-time state-dependent stochastic
drift criteria for stabilization of Markov chains. For asymptotic mean
stationarity (and thus sample path ergodicity), it is sufficient that the
capacity of a channel is (strictly) greater than the sum of the logarithms of
the unstable pole magnitudes for memoryless channels and a class of channels
with memory. This condition is also necessary under a mild technical condition.
Sufficient conditions for the existence of finite average second moments for
such systems driven by unbounded noise are provided.Comment: To appear in IEEE Transactions on Information Theor
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
Universal Densities Exist for Every Finite Reference Measure
As it is known, universal codes, which estimate the entropy rate
consistently, exist for stationary ergodic sources over finite alphabets but
not over countably infinite ones. We generalize universal coding as the problem
of universal densities with respect to a fixed reference measure on a countably
generated measurable space. We show that universal densities, which estimate
the differential entropy rate consistently, exist for finite reference
measures. Thus finite alphabets are not necessary in some sense. To exhibit a
universal density, we adapt the non-parametric differential (NPD) entropy rate
estimator by Feutrill and Roughan. Our modification is analogous to Ryabko's
modification of prediction by partial matching (PPM) by Cleary and Witten.
Whereas Ryabko considered a mixture over Markov orders, we consider a mixture
over quantization levels. Moreover, we demonstrate that any universal density
induces a strongly consistent Ces\`aro mean estimator of conditional density
given an infinite past. This yields a universal predictor with the loss
for a countable alphabet. Finally, we specialize universal densities to
processes over natural numbers and on the real line. We derive sufficient
conditions for consistent estimation of the entropy rate with respect to
infinite reference measures in these domains.Comment: 28 pages, no figure
On the Capacity of Multilevel NAND Flash Memory Channels
In this paper, we initiate a first information-theoretic study on multilevel
NAND flash memory channels with intercell interference. More specifically, for
a multilevel NAND flash memory channel under mild assumptions, we first prove
that such a channel is indecomposable and it features asymptotic equipartition
property; we then further prove that stationary processes achieve its
information capacity, and consequently, as its order tends to infinity, its
Markov capacity converges to its information capacity; eventually, we establish
that its operational capacity is equal to its information capacity. Our results
suggest that it is highly plausible to apply the ideas and techniques in the
computation of the capacity of finite-state channels, which are relatively
better explored, to that of the capacity of multilevel NAND flash memory
channels.Comment: Submitted to IEEE Transactions on Information Theor
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