Neural Comput - Hebbian learning of recurrent connections: a geometrical perspective.

Tópicos

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Resumo

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 high-dimensional 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 self-organization of recurrent rather than feedforward connections. In addition, we establish that the nonconvolutional (or long-range) connectivity is patchy and is co-aligned in the case of orientation learning.

Resumo Limpo

show hopfield network modifi recurr connect undergo slow hebbian learn can extract under geometri input space first use slow fast analysi deriv averag system whose dynam deriv energi function therefor alway converg equilibrium point equilibria reflect correl structur input global object extract local recurr interact second use numer method illustr learn extract hidden geometr structur input inde multidimension scale method make possibl project final connect matrix onto euclidean distanc matrix highdimension space neuron label spatial posit within space result network structur turn rough convolut residu project defin nonconvolut part connect minim process final show restrict dimens space neuron live give rise pattern similar cortic map motiv use energi effici argument base wire length minim final show approach lead emerg ocular domin orient column primari visual cortex via selforgan recurr rather feedforward connect addit establish nonconvolut longrang connect patchi coalign case orient learn

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