1,317 research outputs found
Criticality in Formal Languages and Statistical Physics
We show that the mutual information between two symbols, as a function of the
number of symbols between the two, decays exponentially in any probabilistic
regular grammar, but can decay like a power law for a context-free grammar.
This result about formal languages is closely related to a well-known result in
classical statistical mechanics that there are no phase transitions in
dimensions fewer than two. It is also related to the emergence of power-law
correlations in turbulence and cosmological inflation through recursive
generative processes. We elucidate these physics connections and comment on
potential applications of our results to machine learning tasks like training
artificial recurrent neural networks. Along the way, we introduce a useful
quantity which we dub the rational mutual information and discuss
generalizations of our claims involving more complicated Bayesian networks.Comment: Replaced to match final published version. Discussion improved,
references adde
Diffusion of Context and Credit Information in Markovian Models
This paper studies the problem of ergodicity of transition probability
matrices in Markovian models, such as hidden Markov models (HMMs), and how it
makes very difficult the task of learning to represent long-term context for
sequential data. This phenomenon hurts the forward propagation of long-term
context information, as well as learning a hidden state representation to
represent long-term context, which depends on propagating credit information
backwards in time. Using results from Markov chain theory, we show that this
problem of diffusion of context and credit is reduced when the transition
probabilities approach 0 or 1, i.e., the transition probability matrices are
sparse and the model essentially deterministic. The results found in this paper
apply to learning approaches based on continuous optimization, such as gradient
descent and the Baum-Welch algorithm.Comment: See http://www.jair.org/ for any accompanying file
On the Inability of Markov Models to Capture Criticality in Human Mobility
We examine the non-Markovian nature of human mobility by exposing the
inability of Markov models to capture criticality in human mobility. In
particular, the assumed Markovian nature of mobility was used to establish a
theoretical upper bound on the predictability of human mobility (expressed as a
minimum error probability limit), based on temporally correlated entropy. Since
its inception, this bound has been widely used and empirically validated using
Markov chains. We show that recurrent-neural architectures can achieve
significantly higher predictability, surpassing this widely used upper bound.
In order to explain this anomaly, we shed light on several underlying
assumptions in previous research works that has resulted in this bias. By
evaluating the mobility predictability on real-world datasets, we show that
human mobility exhibits scale-invariant long-range correlations, bearing
similarity to a power-law decay. This is in contrast to the initial assumption
that human mobility follows an exponential decay. This assumption of
exponential decay coupled with Lempel-Ziv compression in computing Fano's
inequality has led to an inaccurate estimation of the predictability upper
bound. We show that this approach inflates the entropy, consequently lowering
the upper bound on human mobility predictability. We finally highlight that
this approach tends to overlook long-range correlations in human mobility. This
explains why recurrent-neural architectures that are designed to handle
long-range structural correlations surpass the previously computed upper bound
on mobility predictability
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