Expressions for the Moore–Penrose inverse of block matrices involving the Schur complement

Castro‐González, N.; Martínez-Serrano, M.F.; Robles, J.

doi:10.1016/j.laa.2015.01.003

Cited by 23 publications

(17 citation statements)

References 15 publications

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“…ln k r (x r |x pa(r ) ; ξ r ). (11) We have an expectation value of a function, ∂ ∂ξ (r ;i) ln k r (x r |x pa(r ) ; ξ r ), that is local in two ways: All arguments of this function, the states and the parameters, are local with respect to the node r . However, the distribution p * ξ , used for the evaluation of the expectation value, depends on the full set of parameters ξ .…”

Section: Locality Of the Euclidean Gradientmentioning

confidence: 99%

“…We exemplify the derivative (11) in the context of binary neurons where it leads to a natural learning rule.…”

Section: Locality Of the Euclidean Gradientmentioning

confidence: 99%

“…We now highlight an important alternative to sampling from p * ξ for the computation of the derivative (11). This alternative is based on the idea that we have, in addition to the generative model M of distributions p ξ , a so-called recognition model Q H |V of conditional distributions q(x H |x V ; η) with which we can approximate p(x H | x V ; ξ).…”

Section: The Wake-sleep Algorithmmentioning

confidence: 99%

“…We have introduced the parameters η for sampling and thereby evaluating the derivative (11). However, there is another remarkable feature of the corresponding extended optimisation problem.…”

Section: The Wake-sleep Algorithmmentioning

confidence: 99%

“…One simplification was expressed in terms of a block diagonal structure of the Fisher information matrix. For the representation of the natural gradient, we evaluated the pseudoinverse of that block diagonal matrix based on the following simple observation (see, e.g., [11] for more general results related to the pseudoinverse of a block matrix):…”

Section: Conclusion: a Natural Gradient Perspective Of The Wake-sleementioning

confidence: 99%

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On the locality of the natural gradient for learning in deep Bayesian networks

2020

Info. Geo.

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We study the natural gradient method for learning in deep Bayesian networks, including neural networks. There are two natural geometries associated with such learning systems consisting of visible and hidden units. One geometry is related to the full system, the other one to the visible sub-system. These two geometries imply different natural gradients. In a first step, we demonstrate a great simplification of the natural gradient with respect to the first geometry, due to locality properties of the Fisher information matrix. This simplification does not directly translate to a corresponding simplification with respect to the second geometry. We develop the theory for studying the relation between the two versions of the natural gradient and outline a method for the simplification of the natural gradient with respect to the second geometry based on the first one. This method suggests to incorporate a recognition model as an auxiliary model for the efficient application of the natural gradient method in deep networks.

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Section: Locality Of the Euclidean Gradientmentioning

confidence: 99%

“…We exemplify the derivative (11) in the context of binary neurons where it leads to a natural learning rule.…”

Section: Locality Of the Euclidean Gradientmentioning

confidence: 99%

Section: The Wake-sleep Algorithmmentioning

confidence: 99%

Section: The Wake-sleep Algorithmmentioning

confidence: 99%

Section: Conclusion: a Natural Gradient Perspective Of The Wake-sleementioning

confidence: 99%

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On the locality of the natural gradient for learning in deep Bayesian networks

2020

Info. Geo.

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A Novel Theoretical Model Development and Simulation of Melt‐Electrospinning Using Kane's and Udwadia–Kalaba Methods

Wubneh

Ayranci

Kim

2022

Advcd Theory and Sims

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A novel analytical model development and simulation of melt-electrospinning process is presented. Unconstrained equations of motion for the description of a discretized melt-electrospun fiber are formulated using Kane's method. The motions of the spinneret and the collector plate are also incorporated into the kinematics formulation to simulate direct writing and three-dimensional printing scenarios. Constraints describing viscoelastic joints in the system and the phenomenon of the melt-electrospun fiber adhering to the collector plate are implemented using the Udwadia Kalaba method. Rotational viscoelastic elements are introduced to mimic the dampening of the whipping motion observed in the unstable region. A novel algorithm is devised to continuously run the simulation with no limit on the total duration and phase transitions. System responses such as fiber diameter, collection size, fiber elongations, and jet speeds are monitored for changes in control parameters including applied voltage, collector distance, and flowrate. The results demonstrate close agreement with experimental observations in the literature. The jet starts relatively straight on the onset of its trajectory and develops subtle whipping motion as it advances toward the collector plate. Reduction in the amplitude of the whipping motion is observed once the free end of the fiber adhered to the collector plate.

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Norm Estimations for the Moore-Penrose Inverse of the Weak Perturbation of Hilbert C ∗-Module Operators

Huang

et al. 2022

Trends in Mathematics

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Expressions for the Moore–Penrose inverse of block matrices involving the Schur complement

Cited by 23 publications

References 15 publications

On the locality of the natural gradient for learning in deep Bayesian networks

On the locality of the natural gradient for learning in deep Bayesian networks

A Novel Theoretical Model Development and Simulation of Melt‐Electrospinning Using Kane's and Udwadia–Kalaba Methods

Norm Estimations for the Moore-Penrose Inverse of the Weak Perturbation of Hilbert C ∗-Module Operators

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