Generalized Hebbian Algorithm

The Generalized Hebbian Algorithm (GHA), also known in the literature as Sanger's rule, is a linear feedforward neural network model for unsupervised learning with applications primarily in principal components analysis. First defined in 1989,^[1] it is similar to Oja's rule in its formulation and stability, except it can be applied to networks with multiple outputs. The name originates because of the similarity between the algorithm and a hypothesis made by Donald Hebb^[2] about the way in which synaptic strengths in the brain are modified in response to experience, i.e., that changes are proportional to the correlation between the firing of pre- and post-synaptic neurons.^[3]

Theory

GHA combines Oja's rule with the Gram-Schmidt process to produce a learning rule of the form

\,\Delta w_{ij} ~ = ~ \eta\left(y_i x_j - y_i \sum_{k=1}^{i} w_{kj} y_k \right)

,^[4]

where $w ij$ defines the synaptic weight or connection strength between the $j$ th input and $i$ th output neurons, $x$ and $y$ are the input and output vectors, respectively, and $η$ is the learning rate parameter.

Derivation

In matrix form, Oja's rule can be written

\,\frac{d w(t)}{d t} ~ = ~ w(t) Q - \mathrm{diag} [w(t) Q w(t)^{\mathrm{T}}] w(t)

and the Gram-Schmidt algorithm is

\,\Delta w(t) ~ = ~ -\mathrm{lower} [w(t) w(t)^{\mathrm{T}}] w(t)

where $w (t)$ is any matrix, in this case representing synaptic weights, $Q = η x x T$ is the autocorrelation matrix, simply the outer product of inputs, $diag$ is the function that diagonalizes a matrix, and $lower$ is the function that sets all matrix elements on or above the diagonal equal to 0. We can combine these equations to get our original rule in matrix form,

\,\Delta w(t) ~ = ~ \eta(t) \left(\mathbf{y}(t) \mathbf{x}(t)^{\mathrm{T}} - \mathrm{LT}[\mathbf{y}(t)\mathbf{y}(t)^{\mathrm{T}}] w(t)\right)

where the function $LT$ sets all matrix elements above the diagonal equal to 0, and note that our output $y (t) = w (t) x (t)$ is a linear neuron.^[1]

Stability and PCA

^[5] ^[6]

Applications

GHA is used in applications where a self-organizing map is necessary, or where a feature or principal components analysis can be used. Examples of such cases include artificial intelligence and speech and image processing.

Its importance comes from the fact that learning is a single-layer process—that is, a synaptic weight changes only depending on the response of the inputs and outputs of that layer, thus avoiding the multi-layer dependence associated with the backpropagation algorithm. It also has a simple and predictable trade-off between learning speed and accuracy of convergence as set by the learning rate parameter $η$ .^[5]

References

1 2 Sanger, Terence D. (1989). "Optimal unsupervised learning in a single-layer linear feedforward neural network" (PDF). Neural Networks. 2 (6): 459–473. doi:10.1016/0893-6080(89)90044-0. Retrieved 2007-11-24.
↑ Hebb, D.O. (1949). "The Organization of Behavior". New York: Wiley & Sons.
↑ Hertz, John; Anders Krough; Richard G. Palmer (1991). "Introduction to the Theory of Neural Computation". Redwood City, CA: Addison-Wesley Publishing Company. ISBN 0201515601.
↑ Gorrell, Genevieve (2006), "Generalized Hebbian Algorithm for Incremental Singular Value Decomposition in Natural Language Processing.", EACL
1 2 Haykin, Simon (1998). Neural Networks: A Comprehensive Foundation (2 ed.). Prentice Hall. ISBN 0-13-273350-1.
↑ Oja, Erkki (November 1982). "Simplified neuron model as a principal component analyzer". Journal of Mathematical Biology. 15 (3): 267–273. doi:10.1007/BF00275687. PMID 7153672. BF00275687. Retrieved 2007-11-22.

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