480,621 research outputs found
Tropical Limits of Probability Spaces, Part I: The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable
cardinality of large independent products of the probability
space. If two probability spaces and have the same entropy, there is an
almost measure-preserving bijection between large parts of and
. In this way, and are asymptotically equivalent.
It turns out to be challenging to generalize this notion of asymptotic
equivalence to configurations of probability spaces, which are collections of
probability spaces with measure-preserving maps between some of them.
In this article we introduce the intrinsic Kolmogorov-Sinai distance on the
space of configurations of probability spaces. Concentrating on the large-scale
geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an
asymptotic equivalence relation on sequences of configurations of probability
spaces. We will call the equivalence classes \emph{tropical probability
spaces}.
In this context we prove an Asymptotic Equipartition Property for
configurations. It states that tropical configurations can always be
approximated by homogeneous configurations. In addition, we show that the
solutions to certain Information-Optimization problems are
Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai
distance. It follows from these two statements that in order to solve an
Information-Optimization problem, it suffices to consider homogeneous
configurations.
Finally, we show that spaces of trajectories of length of certain
stochastic processes, in particular stationary Markov chains, have a tropical
limit.Comment: Comment to version 2: Fixed typos, a calculation mistake in Lemma 5.1
and its consequences in Proposition 5.2 and Theorem 6.
Empowerment for Continuous Agent-Environment Systems
This paper develops generalizations of empowerment to continuous states.
Empowerment is a recently introduced information-theoretic quantity motivated
by hypotheses about the efficiency of the sensorimotor loop in biological
organisms, but also from considerations stemming from curiosity-driven
learning. Empowemerment measures, for agent-environment systems with stochastic
transitions, how much influence an agent has on its environment, but only that
influence that can be sensed by the agent sensors. It is an
information-theoretic generalization of joint controllability (influence on
environment) and observability (measurement by sensors) of the environment by
the agent, both controllability and observability being usually defined in
control theory as the dimensionality of the control/observation spaces. Earlier
work has shown that empowerment has various interesting and relevant
properties, e.g., it allows us to identify salient states using only the
dynamics, and it can act as intrinsic reward without requiring an external
reward. However, in this previous work empowerment was limited to the case of
small-scale and discrete domains and furthermore state transition probabilities
were assumed to be known. The goal of this paper is to extend empowerment to
the significantly more important and relevant case of continuous vector-valued
state spaces and initially unknown state transition probabilities. The
continuous state space is addressed by Monte-Carlo approximation; the unknown
transitions are addressed by model learning and prediction for which we apply
Gaussian processes regression with iterated forecasting. In a number of
well-known continuous control tasks we examine the dynamics induced by
empowerment and include an application to exploration and online model
learning
Scale-Space Splatting: Reforming Spacetime for the Cross-Scale Exploration of Integral Measures in Molecular Dynamics
Understanding large amounts of spatiotemporal data from particle-based
simulations, such as molecular dynamics, often relies on the computation and
analysis of aggregate measures. These, however, by virtue of aggregation, hide
structural information about the space/time localization of the studied
phenomena. This leads to degenerate cases where the measures fail to capture
distinct behaviour. In order to drill into these aggregate values, we propose a
multi-scale visual exploration technique. Our novel representation, based on
partial domain aggregation, enables the construction of a continuous
scale-space for discrete datasets and the simultaneous exploration of scales in
both space and time. We link these two scale-spaces in a scale-space space-time
cube and model linked views as orthogonal slices through this cube, thus
enabling the rapid identification of spatio-temporal patterns at multiple
scales. To demonstrate the effectiveness of our approach, we showcase an
advanced exploration of a protein-ligand simulation.Comment: 11 pages, 9 figures, IEEE SciVis 201
Air Pollution and Urban Morphology: A Complex Relation or How to Optimize the Pedestrian Movement in Town
International audienceUrban air pollution is traditionally estimated by using techniques based on geostatistical methods, such as interpolation, applied to a set of data stemming from measures of stations of pollution. Now very often, these stations are in insufficient number or do not measure the same pollutants to allow mapping finely dispersion of air pollution through urban spaces. Numerous studies work then from land registries of broadcasts. Although interesting in a regional scale, these studies bring only not enough information in the understanding of the phenomena to a scale as fine as the intra-urban. So, it is necessary to resort to the fine three-dimensional modelling to dread this intra-urban scale and it is what we describe now
Tropical Limits of Probability Spaces, Part I:The Intrinsic Kolmogorov-Sinai Distance and the Asymptotic Equipartition Property for Configurations
The entropy of a finite probability space measures the observable cardinality of large independent products of the probability space. If two probability spaces and have the same entropy, there is an almost measure-preserving bijection between large parts of and . In this way, and are asymptotically equivalent. It turns out to be challenging to generalize this notion of asymptotic equivalence to configurations of probability spaces, which are collections of probability spaces with measure-preserving maps between some of them. In this article we introduce the intrinsic Kolmogorov-Sinai distance on the space of configurations of probability spaces. Concentrating on the large-scale geometry we pass to the asymptotic Kolmogorov-Sinai distance. It induces an asymptotic equivalence relation on sequences of configurations of probability spaces. We will call the equivalence classes \emph{tropical probability spaces}. In this context we prove an Asymptotic Equipartition Property for configurations. It states that tropical configurations can always be approximated by homogeneous configurations. In addition, we show that the solutions to certain Information-Optimization problems are Lipschitz-con\-tinuous with respect to the asymptotic Kolmogorov-Sinai distance. It follows from these two statements that in order to solve an Information-Optimization problem, it suffices to consider homogeneous configurations. Finally, we show that spaces of trajectories of length of certain stochastic processes, in particular stationary Markov chains, have a tropical limit
Measuring Social Relations in New Classroom Spaces: Development and Validation of the Social Context and Learning Environments (SCALE) Survey
This study addresses the need for reliable and valid information about how the innovative classrooms that are becoming more common on college and university campuses affect teaching and learning. The Social Context and Learning Environments (SCALE) survey was developed though a three-stage process involving almost 1300 college students. Exploratory and confirmatory factor analyses supported a four-factor solution that measures formal and informal aspects of student-student and student-instructor classroom relations. The resulting 26-item instrument can be used by instructors and researchers to measure classroom social context in different types of learning spaces and to guide efforts to improve student outcomes
On the Memory Requirement of Hop-by-hop Routing: Tight Bounds and Optimal Address Spaces
Routing in large-scale computer networks today is built on hop-by-hop routing: packet headers specify the destination address and routers use internal forwarding tables to map addresses to next-hop ports. In this paper we take a new
look at the scalability of this paradigm. We define a new model that reduces forwarding tables to sequential strings, which then lend themselves readily to an information-theoretical analysis. Contrary to previous work, our analysis is not of worst-case nature, but gives verifiable and realizable memory requirement characterizations even when subjected to concrete topologies and routing policies. We formulate the optimal address space design problem as the task to set node addresses in order to minimize certain network-wide entropy-related measures. We derive tight space bounds for many well-known graph families and we propose a simple heuristic to find optimal address spaces for general graphs. Our evaluations suggest that in structured graphs, including most practically important network topologies,
significant memory savings can be attained by forwarding table compression over our optimized address spaces. According to our knowledge, our work is the first to bridge the gap between computer network scalability and information-theory
- …