858 research outputs found
Improving variational methods via pairwise linear response identities
nference methods are often formulated as variational approximations: these approxima-tions allow easy evaluation of statistics by marginalization or linear response, but theseestimates can be inconsistent. We show that by introducing constraints on covariance, onecan ensure consistency of linear response with the variational parameters, and in so doinginference of marginal probability distributions is improved. For the Bethe approximationand its generalizations, improvements are achieved with simple choices of the constraints.The approximations are presented as variational frameworks; iterative procedures relatedto message passing are provided for finding the minim
Cycle-based Cluster Variational Method for Direct and Inverse Inference
We elaborate on the idea that loop corrections to belief propagation could be
dealt with in a systematic way on pairwise Markov random fields, by using the
elements of a cycle basis to define region in a generalized belief propagation
setting. The region graph is specified in such a way as to avoid dual loops as
much as possible, by discarding redundant Lagrange multipliers, in order to
facilitate the convergence, while avoiding instabilities associated to minimal
factor graph construction. We end up with a two-level algorithm, where a belief
propagation algorithm is run alternatively at the level of each cycle and at
the inter-region level. The inverse problem of finding the couplings of a
Markov random field from empirical covariances can be addressed region wise. It
turns out that this can be done efficiently in particular in the Ising context,
where fixed point equations can be derived along with a one-parameter log
likelihood function to minimize. Numerical experiments confirm the
effectiveness of these considerations both for the direct and inverse MRF
inference.Comment: 47 pages, 16 figure
A Deterministic and Generalized Framework for Unsupervised Learning with Restricted Boltzmann Machines
Restricted Boltzmann machines (RBMs) are energy-based neural-networks which
are commonly used as the building blocks for deep architectures neural
architectures. In this work, we derive a deterministic framework for the
training, evaluation, and use of RBMs based upon the Thouless-Anderson-Palmer
(TAP) mean-field approximation of widely-connected systems with weak
interactions coming from spin-glass theory. While the TAP approach has been
extensively studied for fully-visible binary spin systems, our construction is
generalized to latent-variable models, as well as to arbitrarily distributed
real-valued spin systems with bounded support. In our numerical experiments, we
demonstrate the effective deterministic training of our proposed models and are
able to show interesting features of unsupervised learning which could not be
directly observed with sampling. Additionally, we demonstrate how to utilize
our TAP-based framework for leveraging trained RBMs as joint priors in
denoising problems
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Modeling the Dynamics of Consumer Behavior from Massive Interaction Data
Recent technological innovations (e.g. e-commerce platforms, automated retail stores) have enabled dramatic changes in people's shopping experiences, as well as the accessibility to incredible volumes of consumer-product interaction data. As a result, machine learning (ML) systems can be widely developed to help people navigate relevant information and make decisions. Traditional ML systems have achieved great success on various well-defined problems such as speech recognition and facial recognition. Unlike these tasks where datasets and objectives are clearly benchmarked, modeling consumer behavior can be rather complicated; for example, consumer activities can be affected by real-time shopping contexts, collected interaction data can be noisy and biased, interests from multiple parties (both consumers and producers) can be involved in the predictive objectives.The primary goal of this dissertation is to address the obstacles in modeling consumer activities through computational approaches, but with careful considerations from economic and societal perspectives. Intellectually, such models help us to understand the forces that guide consumer behavior. Methodologically, I build algorithms capable of processing massive interaction datasets by connecting well-developed ML techniques and well-established economic theories. Practically, my work has applications ranging from recommender systems, e-commerce and business intelligence
Viscosity and scale invariance in the unitary Fermi gas
We compute the shear viscosity of the unitary Fermi gas above the superfluid
transition temperature, using a diagrammatic technique that starts from the
exact Kubo formula. The formalism obeys a Ward identity associated with scale
invariance which guarantees that the bulk viscosity vanishes identically. For
the shear viscosity, vertex corrections and the associated Aslamazov-Larkin
contributions are shown to be crucial to reproduce the full Boltzmann equation
result in the high-temperature, low fugacity limit. The frequency dependent
shear viscosity exhibits a Drude-like transport peak and a
power-law tail at large frequencies which is proportional to the Tan contact.
The weight in the transport peak is given by the equilibrium pressure, in
agreement with a sum rule due to Taylor and Randeria. Near the superfluid
transition the peak width is of the order of , thus invalidating a
quasiparticle description. The ratio between the static shear
viscosity and the entropy density exhibits a minimum near the superfluid
transition temperature whose value is larger than the string theory bound
by a factor of about seven.Comment: 34 pages, 9 figures; final form (contains new derivation of sum
rule), accepted for publication in Annals of Physic
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