Unsupervised learning

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

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.

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 $\tau d\Vert w{\Vert }^2/dt=2(w^{T}Cw)(1-\alpha \Vert w{\Vert }^2)$.