69,263 research outputs found
A geometrical analysis of global stability in trained feedback networks
Recurrent neural networks have been extensively studied in the context of
neuroscience and machine learning due to their ability to implement complex
computations. While substantial progress in designing effective learning
algorithms has been achieved in the last years, a full understanding of trained
recurrent networks is still lacking. Specifically, the mechanisms that allow
computations to emerge from the underlying recurrent dynamics are largely
unknown. Here we focus on a simple, yet underexplored computational setup: a
feedback architecture trained to associate a stationary output to a stationary
input. As a starting point, we derive an approximate analytical description of
global dynamics in trained networks which assumes uncorrelated connectivity
weights in the feedback and in the random bulk. The resulting mean-field theory
suggests that the task admits several classes of solutions, which imply
different stability properties. Different classes are characterized in terms of
the geometrical arrangement of the readout with respect to the input vectors,
defined in the high-dimensional space spanned by the network population. We
find that such approximate theoretical approach can be used to understand how
standard training techniques implement the input-output task in finite-size
feedback networks. In particular, our simplified description captures the local
and the global stability properties of the target solution, and thus predicts
training performance
On the linear quadratic data-driven control
The classical approach for solving control problems is model based: first a model representation is derived from given data of the plant and then a control law is synthesized using the model and the control specifications. We present an alternative approach that circumvents the explicit identification of a model representation. The considered control problem is finite horizon linear quadratic tracking. The results are derived assuming exact data and the optimal trajectory is constructed off-line
Stochastic filtering via L2 projection on mixture manifolds with computer algorithms and numerical examples
We examine some differential geometric approaches to finding approximate
solutions to the continuous time nonlinear filtering problem. Our primary focus
is a new projection method for the optimal filter infinite dimensional
Stochastic Partial Differential Equation (SPDE), based on the direct L2 metric
and on a family of normal mixtures. We compare this method to earlier
projection methods based on the Hellinger distance/Fisher metric and
exponential families, and we compare the L2 mixture projection filter with a
particle method with the same number of parameters, using the Levy metric. We
prove that for a simple choice of the mixture manifold the L2 mixture
projection filter coincides with a Galerkin method, whereas for more general
mixture manifolds the equivalence does not hold and the L2 mixture filter is
more general. We study particular systems that may illustrate the advantages of
this new filter over other algorithms when comparing outputs with the optimal
filter. We finally consider a specific software design that is suited for a
numerically efficient implementation of this filter and provide numerical
examples.Comment: Updated and expanded version published in the Journal reference
below. Preprint updates: January 2016 (v3) added projection of Zakai Equation
and difference with projection of Kushner-Stratonovich (section 4.1). August
2014 (v2) added Galerkin equivalence proof (Section 5) to the March 2013 (v1)
versio
Order reduction methods for solving large-scale differential matrix Riccati equations
We consider the numerical solution of large-scale symmetric differential
matrix Riccati equations. Under certain hypotheses on the data, reduced order
methods have recently arisen as a promising class of solution strategies, by
forming low-rank approximations to the sought after solution at selected
timesteps. We show that great computational and memory savings are obtained by
a reduction process onto rational Krylov subspaces, as opposed to current
approaches. By specifically addressing the solution of the reduced differential
equation and reliable stopping criteria, we are able to obtain accurate final
approximations at low memory and computational requirements. This is obtained
by employing a two-phase strategy that separately enhances the accuracy of the
algebraic approximation and the time integration. The new method allows us to
numerically solve much larger problems than in the current literature.
Numerical experiments on benchmark problems illustrate the effectiveness of the
procedure with respect to existing solvers
Towards an exact reconstruction of a time-invariant model from time series data
Dynamic processes in biological systems may be profiled by measuring system properties over time. One way of representing such time series data is through weighted interaction networks, where the nodes in the network represent the measurables and the weighted edges represent interactions between any pair of nodes. Construction of these network models from time series data may involve seeking a robust data-consistent and time-invariant model to approximate and describe system dynamics. Many problems in mathematics, systems biology and physics can be recast into this form and may require finding the most consistent solution to a set of first order differential equations. This is especially challenging in cases where the number of data points is less than or equal to the number of measurables. We present a novel computational method for network reconstruction with limited time series data. To test our method, we use artificial time series data generated from known network models. We then attempt to reconstruct the original network from the time series data alone. We find good agreement between the original and predicted networks
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