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2025-06-20T15:55:33Z

Derivation: this is not a diagonalization!

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{{short description|Linear feedforward neural network model}}
The '''generalized Hebbian algorithm''', also known in the literature as '''Sanger's rule''', is a linear [[feedforward neural network]] for [[unsupervised learning]] with applications primarily in [[principal components analysis]]. First defined in 1989,<ref name="Sanger89">{{cite journal |last=Sanger |first=Terence D. |author-link=Terence Sanger |year=1989 |title= Optimal unsupervised learning in a single-layer linear feedforward neural network |journal=Neural Networks |volume=2 |issue=6 |pages=459–473 |url=http://courses.cs.washington.edu/courses/cse528/09sp/sanger_pca_nn.pdf |access-date= 2007-11-24 |doi= 10.1016/0893-6080(89)90044-0 |citeseerx=10.1.1.128.6893 }}</ref> 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]]<ref name="Hebb 1949">{{Cite book|last=Hebb|first=D.O.|author-link=Donald Olding Hebb|title=The Organization of Behavior|publisher=Wiley & Sons|location=New York|year=1949|isbn=9781135631918|url=https://books.google.com/books?id=uyV5AgAAQBAJ}}</ref> 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]].<ref name="Hertz, Krough, and Palmer, 1991">{{Cite book|last=Hertz|first=John|author2=Anders Krough |author3=Richard G. Palmer |title=Introduction to the Theory of Neural Computation|publisher=Addison-Wesley Publishing Company|location=Redwood City, CA|year=1991|isbn=978-0201515602}}</ref>

==Theory==
Consider a problem of learning a linear code for some data. Each data is a multi-dimensional vector <math>x \in \R^n</math>, and can be (approximately) represented as a linear sum of linear code vectors <math>w_1, \dots, w_m</math>. When <math>m = n</math>, it is possible to exactly represent the data. If <math>m < n</math>, it is possible to approximately represent the data. To minimize the L2 loss of representation, <math>w_1, \dots, w_m</math> should be the highest principal component vectors.

The generalized Hebbian algorithm is an iterative algorithm to find the highest principal component vectors, in an algorithmic form that resembles [[Unsupervised learning|unsupervised]] Hebbian learning in neural networks.

Consider a one-layered neural network with <math>n</math> input neurons and <math>m</math> output neurons <math>y_1, \dots, y_m</math>. The linear code vectors are the connection strengths, that is, <math>w_{ij}</math> is the [[synaptic weight]] or connection strength between the <math>j</math>-th input and <math>i</math>-th output neurons.

The generalized Hebbian algorithm learning rule is of the form

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

where <math>\eta</math> is the ''[[learning rate]]'' parameter.<ref>{{Citation |last=Gorrell |first=Genevieve |title=Generalized Hebbian Algorithm for Incremental Singular Value Decomposition in Natural Language Processing. |journal=EACL |year=2006 |citeseerx=10.1.1.102.2084}}</ref>

===Derivation===
In matrix form, Oja's rule can be written

:<math>\,\frac{\text{d} w(t)}{\text{d} t} ~ = ~ w(t) Q - \mathrm{diag} [w(t) Q w(t)^{\mathrm{T}}] w(t)</math>,

and the Gram-Schmidt algorithm is

:<math>\,\Delta w(t) ~ = ~ -\mathrm{lower} [w(t) w(t)^{\mathrm{T}}] w(t)</math>,

where {{math|''w''(''t'')}} is any matrix, in this case representing synaptic weights, {{math|''Q'' {{=}} ''η'' '''x''' '''x'''<sup>T</sup>}} is the autocorrelation matrix, simply the outer product of inputs, {{math|diag}} is the function that sets all matrix elements outside of the diagonal equal to 0, and {{math|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,

:<math>\,\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)</math>,

where the function {{math|LT}} sets all matrix elements above the diagonal equal to 0, and note that our output {{math|'''y'''(''t'') {{=}} ''w''(''t'') '''x'''(''t'')}} is a linear neuron.<ref name="Sanger89"/>

===Stability and Principal Components Analysis===
<ref name="Haykin98">{{cite book |last=Haykin |first=Simon |author-link=Simon Haykin |title=Neural Networks: A Comprehensive Foundation |edition=2 |year=1998 |publisher=Prentice Hall |isbn=978-0-13-273350-2 }}</ref>

[[Oja's rule]] is the special case where <math>m = 1</math>.<ref name="Oja82">{{cite journal |last=Oja |first=Erkki |author-link=Erkki Oja |date=November 1982 |title=Simplified neuron model as a principal component analyzer |journal=Journal of Mathematical Biology |volume=15 |issue=3 |pages=267–273 |doi=10.1007/BF00275687 |pmid=7153672 |s2cid=16577977 |id=BF00275687}}</ref> One can think of the generalized Hebbian algorithm as iterating Oja's rule.

With Oja's rule, <math>w_1</math> is learned, and it has the same direction as the largest principal component vector is learned, with length determined by <math>E[x_j] = E[w_{1j} y_1]</math> for all <math>j </math>, where the expectation is taken over all input-output pairs. In other words, the length of the vector <math>w_1</math> is such that we have an [[autoencoder]], with the latent code <math>y_1 = \sum_i w_{1i} x_i </math>, such that <math>E[\| x - y_1 w_1 \|^2] </math> is minimized.

When <math>m = 2 </math>, the first neuron in the hidden layer of the autoencoder still learns as described, since it is unaffected by the second neuron. So, after the first neuron and its vector <math>w_1</math> has converged, the second neuron is effectively running another Oja's rule on the modified input vectors, defined by <math>x' = x - y_1w_1 </math>, which we know is the input vector with the first principal component removed. Therefore, the second neuron learns to code for the second principal component.

By induction, this results in finding the top-<math>m </math> principal components for arbitrary <math>m </math>.

==Applications==
The generalized Hebbian algorithm 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 {{math|η}}.<ref name="Haykin98"/>

[[File:Generalized Hebbian algorithm on 8-by-8 patches of Caltech101.png|thumb|Features learned by generalized Hebbian algorithm running on 8-by-8 patches of [[Caltech 101]].]]
[[File:Principal component analysis of Caltech101.png|thumb|Features found by [[Principal Component Analysis]] on the same Caltech 101 dataset.]]

As an example, (Olshausen and Field, 1996)<ref>{{Cite journal |last1=Olshausen |first1=Bruno A. |last2=Field |first2=David J. |date=June 1996 |title=Emergence of simple-cell receptive field properties by learning a sparse code for natural images |url=https://www.nature.com/articles/381607a0 |journal=Nature |language=en |volume=381 |issue=6583 |pages=607–609 |doi=10.1038/381607a0 |pmid=8637596 |issn=1476-4687|url-access=subscription }}</ref> performed the generalized Hebbian algorithm on 8-by-8 patches of photos of natural scenes, and found that it results in Fourier-like features. The features are the same as the principal components found by principal components analysis, as expected, and that, the features are determined by the <math>64\times 64</math> variance matrix of the samples of 8-by-8 patches. In other words, it is determined by the second-order statistics of the pixels in images. They criticized this as insufficient to capture higher-order statistics which are necessary to explain the Gabor-like features of simple cells in the [[primary visual cortex]].

==See also==
*[[Hebbian learning]]
*[[Factor analysis]]
*[[Contrastive Hebbian learning]]
*[[Oja's rule]]

==References==
{{reflist}}

{{Hebbian learning}}

[[Category:Hebbian theory]]
[[Category:Artificial neural networks]]

Generalized Hebbian algorithm - Revision history

imported>GBainier: /* Derivation */ this is not a diagonalization!