Unsupervised learning

Learning structure from data without supplied target labels. Examples include extracting principal components, clustering and learning representations that efficiently reconstruct inputs.

Learning transformation-tolerant representations by combining input-feature overlap, current-activity Hebbian learning and synaptic competition. A new view partly activates an output already learned for an overlapping view, then strengthens its new features onto that output. Chains of overlapping views can link more distant transformations without an explicit temporal trace; dense enough overlap and suitable competition remain conditions.^[ref]

Units compete to respond to an input, and the winning unit or sparse winning set strengthens its representation of that input. A prototype update is $w_k\leftarrow w_k+\eta (x-w_k)$ for the winner $k$, with $0<\eta <1$. The winner moves toward the sample: $w_k^{\prime }-x=(1-\eta )(w_k-x)$. Competition and constraints prevent all units from learning the same unrestricted average.

Independently varying companions can make an individual object’s features more consistently related than the features of any particular multi-object conjunction. Competitive networks can exploit this difference to learn separate object representations without being shown each object in isolation. The result depends on the training distribution; a fixed co-occurring group can instead be learned as a conjunction.^[ref][ref]

Construct low-dimensional coordinates representing a dissimilarity matrix. Different methods preserve different features: classical multidimensional scaling approximates a centered Gram matrix, while other methods minimize discrepancies between fitted and supplied distances or preserve distance ordering. The method and dimension determine what geometric claims the resulting display supports.

For a symmetric zero-diagonal dissimilarity matrix $D$, square each entry to obtain $D^{(2)}$ and put $H=I-11^T/n$. Take the eigenvectors of $G=-H{D}^{(2)}H/2$ corresponding to its largest positive eigenvalues. Multiplying them by square roots of these eigenvalues gives coordinates. If $D$ contains Euclidean distances, $G$ is a centered Gram matrix, so retaining all positive eigenvalues reconstructs those distances exactly. Retaining fewer approximates the Gram matrix, not generally the sum of squared errors in raw distances. Negative eigenvalues signal failure of an exact Euclidean representation. Coordinates are unchanged in meaning by orthogonal transformations.

For symmetric nonnegative $D$ with zero diagonal, a centered Euclidean realization implies $D_{ij}^2=G_{ii}+G_{jj}-2G_{ij}$ and hence the displayed double-centering identity. Conversely, if the double-centered matrix is positive semidefinite, factor it as $X{X}^T$. Its reconstructed squared distances equal the original squared distances: their difference has zero double-centering, so is of the form $1a^T+a{1}^T$; the zero diagonal forces $a=0$. The minimum exact embedding dimension is $\mathrm{rank}G$.

Represent inputs accurately using few strongly active coefficients, for example by minimizing $\Vert x-Az{\Vert }^2/2+\lambda {\sum }_i|z_i|$. Learning basis functions on natural images can yield localized oriented filters resembling simple-cell receptive fields. This demonstrates one possible functional account of selectivity, not a proof that cortex implements that exact objective or learning algorithm.

Activity-dependent strengthening associated with coactivation of an input and its postsynaptic output. The idealized update $\Delta w_i\propto y{x}_i$ requires normalization, competition or other constraints to prevent indefinite weight growth.

In Hebbian memory weights for a Hopfield network, the local update $\Delta w_{ij}=\eta {\xi }_i{\xi }_j$ records a pairwise association, where $\eta >0$ and ${\xi }_i,{\xi }_j\in \{-1,1\}$. Equal signs strengthen a positive coupling; opposite signs favor a negative coupling. Summing such updates gives a symmetric matrix of pairwise associations. This is a storage prescription: it does not by itself compare the generated probability distribution with the training data, as Boltzmann machine learning does. Bipolar activity also makes joint inactivity contribute positively; this encoding is a mathematical idealization, not a literal statement about inactive biological cells.

A local association rule uses a decaying activity trace to connect representations encountered at nearby times. One implementation has ${\bar{y}}_i(t)=\rho {\bar{y}}_i(t-1)+(1-\rho )y_i(t)$ and $\Delta w_{ij}=\eta {\bar{y}}_i(t-1)x_j(t)$, followed by synaptic normalization. Temporal continuity can link distinct views of one object, but a trace spanning an identity change can create a false association.^[ref]

With $y=w\cdot x$, the squared weight norm $s=|w{|}^2$ satisfies $\tau \dot{s}=2y^2(1-\alpha s^2)$. It moves monotonically toward $s_{\ast }={\alpha }^{-1/2}$ without crossing it. Explicitly $(s-s_{\ast })/(s+s_{\ast })$ equals its initial value times $\exp [-(4\sqrt{\alpha }/\tau ){\int }_0^{t}y(u{)}^{2}du]$. Persistent excitation therefore drives $|w|$ to ${\alpha }^{-1/4}$; without excitation it can remain stationary. This nonlinear normalization avoids the unbounded amplification of unmodified Hebbian learning.

For a linear output $y=w^{T}x$, the second term limits the growth produced by the Hebbian term. On the averaged dynamics with $C=\mathbb{E}[x{x}^T]$, $\tau \dot{w}=Cw-\alpha (w^{T}Cw)w$. For $\alpha >0$, positive-eigenvalue equilibria have norm $1/\sqrt{\alpha }$ and align with eigenvectors of $C$. If inputs are centered, $C$ is their covariance matrix.

At the equilibrium $v_i/\sqrt{\alpha }$, linearization along orthonormal eigenvectors of $C$ gives the displayed rates, with the first equation applying to $j\ne i$. A simple positive largest eigenvalue makes its two normalized eigenvectors locally asymptotically stable; smaller eigenvalues have an unstable direction. Repeated largest eigenvalues give a manifold of equilibria with neutral directions. Generic convergence also needs a nonzero initial component in the largest eigenspace. The squared norm satisfies $\tau d\Vert w{\Vert }^2/dt=2(w^{T}Cw)(1-\alpha \Vert w{\Vert }^2)$.