English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Open

Linear principal component analysis (PCA) learns (semi-)orthogonaltransformations by orienting the axes to maximize variance. Consequently, itcan only identify orthogonal axes whose variances are clearly distinct, but itcannot identify the subsets of axes whose variances are roughly equal. Itcannot eliminate the subspace rotational indeterminacy: it fails to disentanglecomponents with equal variances (eigenvalues), resulting, in each eigensubspace, in randomly rotated axes. In this paper, we propose $\sigma$-PCA, amethod that (1) formulates a unified model for linear and nonlinear PCA, thelatter being a special case of linear independent component analysis (ICA), and(2) introduces a missing piece into nonlinear PCA that allows it to eliminate,from the canonical linear PCA solution, the subspace rotational indeterminacy-- without whitening the inputs. Whitening, a preprocessing step which convertsthe inputs into unit-variance inputs, has generally been a prerequisite stepfor linear ICA methods, which meant that conventional nonlinear PCA could notnecessarily preserve the orthogonality of the overall transformation, could notdirectly reduce dimensionality, and could not intrinsically order by variances.We offer insights on the relationship between linear PCA, nonlinear PCA, andlinear ICA -- three methods with autoencoder formulations for learning speciallinear transformations from data, transformations that are (semi-)orthogonalfor PCA, and arbitrary unit-variance for ICA. As part of our formulation,nonlinear PCA can be seen as a method that maximizes both variance andstatistical independence, lying in the middle between linear PCA and linearICA, serving as a building block for learning linear transformations that areidentifiable.

$\sigma$-PCA: a building block for neural learning of identifiable linear transformations