16,782 research outputs found

    Approaching Mathematics Utopia?

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    When Does a Mixture of Products Contain a Product of Mixtures?

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    We derive relations between theoretical properties of restricted Boltzmann machines (RBMs), popular machine learning models which form the building blocks of deep learning models, and several natural notions from discrete mathematics and convex geometry. We give implications and equivalences relating RBM-representable probability distributions, perfectly reconstructible inputs, Hamming modes, zonotopes and zonosets, point configurations in hyperplane arrangements, linear threshold codes, and multi-covering numbers of hypercubes. As a motivating application, we prove results on the relative representational power of mixtures of product distributions and products of mixtures of pairs of product distributions (RBMs) that formally justify widely held intuitions about distributed representations. In particular, we show that a mixture of products requiring an exponentially larger number of parameters is needed to represent the probability distributions which can be obtained as products of mixtures.Comment: 32 pages, 6 figures, 2 table

    Using deep learning to understand and mitigate the qubit noise environment

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    Understanding the spectrum of noise acting on a qubit can yield valuable information about its environment, and crucially underpins the optimization of dynamical decoupling protocols that can mitigate such noise. However, extracting accurate noise spectra from typical time-dynamics measurements on qubits is intractable using standard methods. Here, we propose to address this challenge using deep learning algorithms, leveraging the remarkable progress made in the field of image recognition, natural language processing, and more recently, structured data. We demonstrate a neural network based methodology that allows for extraction of the noise spectrum associated with any qubit surrounded by an arbitrary bath, with significantly greater accuracy than the current methods of choice. The technique requires only a two-pulse echo decay curve as input data and can further be extended either for constructing customized optimal dynamical decoupling protocols or for obtaining critical qubit attributes such as its proximity to the sample surface. Our results can be applied to a wide range of qubit platforms, and provide a framework for improving qubit performance with applications not only in quantum computing and nanoscale sensing but also in material characterization techniques such as magnetic resonance.Comment: Accepted for publication, 15 pages, 10 figure

    A constitutive model for simple shear of dense frictional suspensions

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    Discrete particle simulations are used to study the shear rheology of dense, stabilized, frictional particulate suspensions in a viscous liquid, toward development of a constitutive model for steady shear flows at arbitrary stress. These suspensions undergo increasingly strong continuous shear thickening (CST) as solid volume fraction ϕ\phi increases above a critical volume fraction, and discontinuous shear thickening (DST) is observed for a range of ϕ\phi. When studied at controlled stress, the DST behavior is associated with non-monotonic flow curves of the steady-state stress as a function of shear rate. Recent studies have related shear thickening to a transition between mostly lubricated to predominantly frictional contacts with the increase in stress. In this study, the behavior is simulated over a wide range of the dimensionless parameters (ϕ,σ~(\phi,\tilde{\sigma}, and μ)\mu), with σ~=σ/σ0\tilde{\sigma} = \sigma/\sigma_0 the dimensionless shear stress and μ\mu the coefficient of interparticle friction: the dimensional stress is σ\sigma, and σ0F0/a2\sigma_0 \propto F_0/ a^2, where F0F_0 is the magnitude of repulsive force at contact and aa is the particle radius. The data have been used to populate the model of the lubricated-to-frictional rheology of Wyart and Cates [Phys. Rev. Lett.{\bf 112}, 098302 (2014)], which is based on the concept of two viscosity divergences or \textquotedblleft jamming\textquotedblright\ points at volume fraction ϕJ0=ϕrcp\phi_{\rm J}^0 = \phi_{\rm rcp} (random close packing) for the low-stress lubricated state, and at ϕJ(μ)<ϕJ0\phi_{\rm J} (\mu) < \phi_{\rm J}^0 for any nonzero μ\mu in the frictional state; a generalization provides the normal stress response as well as the shear stress. A flow state map of this material is developed based on the simulation results.Comment: 12 pages, 10 figure
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