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Keywords
(17)
Convergence To Equilibrium
Energy Efficient
Energy Function
Euclidean Distance
Geometric Structure
hebbian learning
High Dimensionality
hopfield network
Local Recurrence
Multidimensional Scaling
Network Structure
Numerical Method
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Hebbian Learning of Recurrent Connections: A Geometrical Perspective
Hebbian Learning of Recurrent Connections: A Geometrical Perspective,10.1162/NECO_a_00322,Neural Computation,Mathieu N. GaltierOlivier,Olivier D. Faug
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Hebbian Learning of Recurrent Connections: A Geometrical Perspective
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Mathieu N. GaltierOlivier
,
Olivier D. Faugeras
,
Paul C. Bressloff
We show how a
Hopfield network
with modifiable recurrent connections undergoing slow
Hebbian learning
can extract the underlying geometry of an input space. First, we use a slow and fast analysis to derive an averaged system whose dynamics derives from an
energy function
and therefore always converges to equilibrium points. The equilibria reflect the correlation structure of the inputs, a global object extracted through local recurrent interactions only. Second, we use numerical methods to illustrate how learning extracts the hidden geometrical structure of the inputs. Indeed,
multidimensional scaling
methods make it possible to project the final connectivity matrix onto a
Euclidean distance
matrix in a highdimensional space, with the neurons labeled by spatial position within this space. The resulting
network structure
turns out to be roughly convolutional. The residual of the projection defines the nonconvolutional part of the connectivity, which is minimized in the process. Finally, we show how restricting the dimension of the space where the neurons live gives rise to patterns similar to cortical maps. We motivate this using an energy efficiency argument based on wire length minimization. Finally, we show how this approach leads to the emergence of
ocular dominance
or orientation columns in
primary visual cortex
via the selforganization of recurrent rather than feedforward connections. In addition, we establish that the nonconvolutional (or longrange) connectivity is patchy and is coaligned in the case of orientation learning.
Journal:
Neural Computation  NECO
, pp. 138, 2011
DOI:
10.1162/NECO_a_00322
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