Oja's rule

Consider a simplified model of a neuron ${\displaystyle y}$ that returns a linear combination of its inputs x using presynaptic weights w:

${\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}}$

Oja's rule defines the change in presynaptic weights w given the output response ${\displaystyle y}$ of a neuron to its inputs x to be

${\displaystyle \,\Delta \mathbf {w} ~=~\mathbf {w} _{n+1}-\mathbf {w} _{n}~=~\eta \,y_{n}(\mathbf {x} _{n}-y_{n}\mathbf {w} _{n}),}$

where η is the learning rate which can also change with time. Note that the bold symbols are vectors and n defines a discrete time iteration. The rule can also be made for continuous iterations as

${\displaystyle \,{\frac {d\mathbf {w} }{dt}}~=~\eta \,y(t)(\mathbf {x} (t)-y(t)\mathbf {w} (t)).}$

The simplest learning rule known is Hebb's rule, which states in conceptual terms that neurons that fire together, wire together. In component form as a difference equation, it is written

${\displaystyle \,\Delta \mathbf {w} ~=~\eta \,y(\mathbf {x} _{n})\mathbf {x} _{n}}$,

or in scalar form with implicit n-dependence,

${\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}$,

where y(x_n) is again the output, this time explicitly dependent on its input vector x.

Hebb's rule has synaptic weights approaching infinity with a positive learning rate. We can stop this by normalizing the weights so that each weight's magnitude is restricted between 0, corresponding to no weight, and 1, corresponding to being the only input neuron with any weight. We do this by normalizing the weight vector to be of length one:

${\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)+\eta \,y(\mathbf {x} )x_{i}}{\left(\sum _{j=1}^{m}[w_{j}(n)+\eta \,y(\mathbf {x} )x_{j}]^{p}\right)^{1/p}}}}$.

Note that in Oja's original paper,[1] p=2, corresponding to quadrature (root sum of squares), which is the familiar Cartesian normalization rule. However, any type of normalization, even linear, will give the same result without loss of generality.

For a small learning rate ${\displaystyle |\eta |\ll 1}$ the equation can be expanded as a Power series in ${\displaystyle \eta }$.[1]

${\displaystyle \,w_{i}(n+1)~=~{\frac {w_{i}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}~+~\eta \left({\frac {yx_{i}}{\left(\sum _{j}w_{j}^{p}(n)\right)^{1/p}}}-{\frac {w_{i}(n)\sum _{j}yx_{j}w_{j}^{p-1}(n)}{\left(\sum _{j}w_{j}^{p}(n)\right)^{(1+1/p)}}}\right)~+~O(\eta ^{2})}$.

For small η, our higher-order terms O(η²) go to zero. We again make the specification of a linear neuron, that is, the output of the neuron is equal to the sum of the product of each input and its synaptic weight to the power of p-1, which in the case of p=2 is synaptic weight itself, or

${\displaystyle \,y(\mathbf {x} )~=~\sum _{j=1}^{m}x_{j}w_{j}^{p-1}}$.

We also specify that our weights normalize to 1, which will be a necessary condition for stability, so

${\displaystyle \,|\mathbf {w} |~=~\left(\sum _{j=1}^{m}w_{j}^{p}\right)^{1/p}~=~1}$,

which, when substituted into our expansion, gives Oja's rule, or

${\displaystyle \,w_{i}(n+1)~=~w_{i}(n)+\eta \,y(x_{i}-w_{i}(n)y)}$.

In analyzing the convergence of a single neuron evolving by Oja's rule, one extracts the first principal component, or feature, of a data set. Furthermore, with extensions using the Generalized Hebbian Algorithm, one can create a multi-Oja neural network that can extract as many features as desired, allowing for principal components analysis.

A principal component a_j is extracted from a dataset x through some associated vector q_j, or a_j = q_j⋅x, and we can restore our original dataset by taking

${\displaystyle \mathbf {x} ~=~\sum _{j}a_{j}\mathbf {q} _{j}}$.

In the case of a single neuron trained by Oja's rule, we find the weight vector converges to q₁, or the first principal component, as time or number of iterations approaches infinity. We can also define, given a set of input vectors X_i, that its correlation matrix R_ij = X_iX_j has an associated eigenvector given by q_j with eigenvalue λ_j. The variance of outputs of our Oja neuron σ²(n) = ⟨y²(n)⟩ then converges with time iterations to the principal eigenvalue, or

${\displaystyle \lim _{n\rightarrow \infty }\sigma ^{2}(n)~=~\lambda _{1}}$.

These results are derived using Lyapunov function analysis, and they show that Oja's neuron necessarily converges on strictly the first principal component if certain conditions are met in our original learning rule. Most importantly, our learning rate η is allowed to vary with time, but only such that its sum is divergent but its power sum is convergent, that is

${\displaystyle \sum _{n=1}^{\infty }\eta (n)=\infty ,~~~\sum _{n=1}^{\infty }\eta (n)^{p}<\infty ,~~~p>1}$.

Our output activation function y(x(n)) is also allowed to be nonlinear and nonstatic, but it must be continuously differentiable in both x and w and have derivatives bounded in time.[2]

There is clear evidence for both long-term potentiation and long-term depression in biological neural networks, along with a normalization effect in both input weights and neuron outputs. However, while there is no direct experimental evidence yet of Oja's rule active in a biological neural network, a biophysical derivation of a generalization of the rule is possible. Such a derivation requires retrograde signalling from the postsynaptic neuron, which is biologically plausible (see neural backpropagation), and takes the form of

${\displaystyle \Delta w_{ij}~\propto ~\langle x_{i}y_{j}\rangle -\epsilon \left\langle \left(c_{\mathrm {pre} }*\sum _{k}w_{ik}y_{k}\right)\cdot \left(c_{\mathrm {post} }*y_{j}\right)\right\rangle ,}$

where as before w_ij is the synaptic weight between the ith input and jth output neurons, x is the input, y is the postsynaptic output, and we define ε to be a constant analogous the learning rate, and c_pre and c_post are presynaptic and postsynaptic functions that model the weakening of signals over time. Note that the angle brackets denote the average and the ∗ operator is a convolution. By taking the pre- and post-synaptic functions into frequency space and combining integration terms with the convolution, we find that this gives an arbitrary-dimensional generalization of Oja's rule known as Oja's Subspace,[4] namely

${\displaystyle \Delta w~=~Cx\cdot w-w\cdot Cy.}$[5]