2,414 research outputs found

    The Army of One (Sample): the Characteristics of Sampling-based Probabilistic Neural Representations

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    There is growing evidence that humans and animals represent the uncertainty associated with sensory stimuli and utilize this uncertainty during planning and decision making in a statistically optimal way. Recently, a nonparametric framework for representing probabilistic information has been proposed whereby neural activity encodes samples from the distribution over external variables. Although such sample-based probabilistic representations have strong empirical and theoretical support, two major issues need to be clarified before they can be considered as viable candidate theories of cortical computation. First, in a fluctuating natural environment, can neural dynamics provide sufficient samples to accurately estimate a stimulus? Second, can such a code support accurate learning over biologically plausible time-scales? Although it is well known that sampling is statistically optimal if the number of samples is unlimited, biological constraints mean that estimation and learning in the cortex must be supported by a relatively small number of possibly dependent samples. We explored these issues in a cue combination task by comparing a neural circuit that employed a sampling-based representation to an optimal estimator. For static stimuli, we found that a single sample is sufficient to obtain an estimator with less than twice the optimal variance, and that performance improves with the inverse square root of the number of samples. For dynamic stimuli, with linear-Gaussian evolution, we found that the efficiency of the estimation improves significantly as temporal information stabilizes the estimate, and because sampling does not require a burn-in phase. Finally, we found that using a single sample, the dynamic model can accurately learn the parameters of the input neural populations up to a general scaling factor, which disappears for modest sample size. These results suggest that sample-based representations can support estimation and learning using a relatively small number of samples and are therefore highly feasible alternatives for performing probabilistic cortical computations.
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    Stochastic Gradient Hamiltonian Monte Carlo

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    Hamiltonian Monte Carlo (HMC) sampling methods provide a mechanism for defining distant proposals with high acceptance probabilities in a Metropolis-Hastings framework, enabling more efficient exploration of the state space than standard random-walk proposals. The popularity of such methods has grown significantly in recent years. However, a limitation of HMC methods is the required gradient computation for simulation of the Hamiltonian dynamical system-such computation is infeasible in problems involving a large sample size or streaming data. Instead, we must rely on a noisy gradient estimate computed from a subset of the data. In this paper, we explore the properties of such a stochastic gradient HMC approach. Surprisingly, the natural implementation of the stochastic approximation can be arbitrarily bad. To address this problem we introduce a variant that uses second-order Langevin dynamics with a friction term that counteracts the effects of the noisy gradient, maintaining the desired target distribution as the invariant distribution. Results on simulated data validate our theory. We also provide an application of our methods to a classification task using neural networks and to online Bayesian matrix factorization.Comment: ICML 2014 versio

    Fine Tuning Classical and Quantum Molecular Dynamics using a Generalized Langevin Equation

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    Generalized Langevin Equation (GLE) thermostats have been used very effectively as a tool to manipulate and optimize the sampling of thermodynamic ensembles and the associated static properties. Here we show that a similar, exquisite level of control can be achieved for the dynamical properties computed from thermostatted trajectories. By developing quantitative measures of the disturbance induced by the GLE to the Hamiltonian dynamics of a harmonic oscillator, we show that these analytical results accurately predict the behavior of strongly anharmonic systems. We also show that it is possible to correct, to a significant extent, the effects of the GLE term onto the corresponding microcanonical dynamics, which puts on more solid grounds the use of non-equilibrium Langevin dynamics to approximate quantum nuclear effects and could help improve the prediction of dynamical quantities from techniques that use a Langevin term to stabilize dynamics. Finally we address the use of thermostats in the context of approximate path-integral-based models of quantum nuclear dynamics. We demonstrate that a custom-tailored GLE can alleviate some of the artifacts associated with these techniques, improving the quality of results for the modelling of vibrational dynamics of molecules, liquids and solids
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