Balanced Realizations, Model Order Reduction, and the Hankel Operator

Scherpen, Jacquelien M.A.

doi:10.1201/b10384-6

Cited by 3 publications

(5 citation statements)

References 52 publications

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“…These parameters and the noise variances were chosen so that the system could not be well approximated by a lower-order one—i.e., so that the uncontrolled and controlled systems were “truly” second- and third-order (respectively). This was accomplished by ensuring that the Hankel singular values [ 14 ] for the system, with output matrix C = [1 0] and input matrix set by the noise variances, were within one order of magnitude of each other; that is, ensuring that the transfer function from noise to joint angle had roughly equal power in all modes. For the uncontrolled system, this was achieved with Σ θ = diag([5 e -7, 5 e -5]); for the controlled system, Σ θ = diag([5 e -5, 1 e -6]) and .…”

Section: Methodsmentioning

confidence: 99%

Learning to Estimate Dynamical State with Probabilistic Population Codes

2015

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Tracking moving objects, including one’s own body, is a fundamental ability of higher organisms, playing a central role in many perceptual and motor tasks. While it is unknown how the brain learns to follow and predict the dynamics of objects, it is known that this process of state estimation can be learned purely from the statistics of noisy observations. When the dynamics are simply linear with additive Gaussian noise, the optimal solution is the well known Kalman filter (KF), the parameters of which can be learned via latent-variable density estimation (the EM algorithm). The brain does not, however, directly manipulate matrices and vectors, but instead appears to represent probability distributions with the firing rates of population of neurons, “probabilistic population codes.” We show that a recurrent neural network—a modified form of an exponential family harmonium (EFH)—that takes a linear probabilistic population code as input can learn, without supervision, to estimate the state of a linear dynamical system. After observing a series of population responses (spike counts) to the position of a moving object, the network learns to represent the velocity of the object and forms nearly optimal predictions about the position at the next time-step. This result builds on our previous work showing that a similar network can learn to perform multisensory integration and coordinate transformations for static stimuli. The receptive fields of the trained network also make qualitative predictions about the developing and learning brain: tuning gradually emerges for higher-order dynamical states not explicitly present in the inputs, appearing as delayed tuning for the lower-order states.

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Section: Methodsmentioning

confidence: 99%

Learning to Estimate Dynamical State with Probabilistic Population Codes

2015

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“…This makes the system stable and oscillatory (which first-order models cannot be). (2) Σ x was chosen so that the Hankel singular values (see Scherpen, 2015) for the system, with output matrix C = 1 0 and input matrix set by Σ x , were within one order of magnitude of each other; that is, ensuring that the transfer function from noise to position had roughly equal power in all modes. This was achieved with Σ x = diag( 5e-7, 5e-5 ).…”

Section: Appendix a Information Retention And Identifiabilitymentioning

confidence: 99%

Recurrent Exponential-Family Harmoniums without Backprop-Through-Time

Makin,

Dichter,

Sabes

2016

Preprint

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Exponential-family harmoniums (EFHs), which extend restricted Boltzmann machines (RBMs) from Bernoulli random variables to other exponential families (Welling et al., 2005), are generative models that can be trained with unsupervised-learning techniques, like contrastive divergence (Hinton et al., 2006;Hinton, 2002), as density estimators for static data. Methods for extending RBMs-and likewise EFHs-to data with temporal dependencies have been proposed previously (Sutskever and Hinton, 2007;Sutskever et al., 2009), the learning procedure being validated by qualitative assessment of the generative model. Here we propose and justify, from a very different perspective, an alternative training procedure, proving sufficient conditions for optimal inference under that procedure. The resulting algorithm can be learned with only forward passes through the data-backpropthrough-time is not required, as in previous approaches. The proof exploits a recent result about information retention in density estimators , and applies it to a "recurrent EFH" (rEFH) by induction. Finally, we demonstrate optimality by simulation, testing the rEFH: (1) as a filter on training data generated with a linear dynamical system, the position of which is noisily reported by a population of "neurons" with Poissondistributed spike counts; and (2) with the qualitative experiments proposed by Sutskever et al. (2009).

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“…In the following sections we review balancing of linear and nonlinear systems as introduced in [29] and [36]. See also [37] for a good survey on balancing for linear and nonlinear systems.…”

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confidence: 99%

“…where σ 2 i is an eigenvalue of H * H with the corresponding eigenvector Ψ i , ordered as σ 1 ≥ • • • ≥ σ n > 0 and σ i≥n+1 = 0 are the Hankel singular values of the input-output system Σ. For square linear systems, the nonzero eigenvalues of the Hankel operator associated to the system are the nonzero eigenvalues of the cross Gramian defined as the solution, W x , of AW x + W x A + BC = 0 [37].…”

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Kernel Methods for the Approximation of Nonlinear Systems

Bouvrie¹,

Hamzi²

2017

SIAM J. Control Optim.

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We introduce a data-driven model approximation method for nonlinear control systems, drawing on recent progress in machine learning and statistical dimensionality reduction. The method is based on embedding the nonlinear system in a high (or infinite) dimensional reproducing kernel Hilbert space (RKHS) where linear balanced truncation may be carried out implicitly. This leads to a nonlinear reduction map which can be combined with a representation of the system belonging to a RKHS to give a closed, reduced order dynamical system which captures the essential input-output characteristics of the original model. Working in RKHS provides a convenient, general functional-analytical framework for theoretical understanding. Empirical simulations illustrating the approach are also provided. 1 2 U Z −1 . We also note that the problem of finding the coordinate change Q can be seen as an optimization problem [1] of the form min Q trace[QW c Q * + Q − * W o Q −1 ]. 5

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Balanced Realizations, Model Order Reduction, and the Hankel Operator

Cited by 3 publications

References 52 publications

Learning to Estimate Dynamical State with Probabilistic Population Codes

Learning to Estimate Dynamical State with Probabilistic Population Codes

Recurrent Exponential-Family Harmoniums without Backprop-Through-Time

Kernel Methods for the Approximation of Nonlinear Systems

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