  To make sense of the world our brains must analyze high-dimensional datasets
streamed by our sensory organs. Because such analysis begins with
dimensionality reduction, modelling early sensory processing requires
biologically plausible online dimensionality reduction algorithms. Recently, we
derived such an algorithm, termed similarity matching, from a Multidimensional
Scaling (MDS) objective function. However, in the existing algorithm, the
number of output dimensions is set a priori by the number of output neurons and
cannot be changed. Because the number of informative dimensions in sensory
inputs is variable there is a need for adaptive dimensionality reduction. Here,
we derive biologically plausible dimensionality reduction algorithms which
adapt the number of output dimensions to the eigenspectrum of the input
covariance matrix. We formulate three objective functions which, in the offline
setting, are optimized by the projections of the input dataset onto its
principal subspace scaled by the eigenvalues of the output covariance matrix.
In turn, the output eigenvalues are computed as i) soft-thresholded, ii)
hard-thresholded, iii) equalized thresholded eigenvalues of the input
covariance matrix. In the online setting, we derive the three corresponding
adaptive algorithms and map them onto the dynamics of neuronal activity in
networks with biologically plausible local learning rules. Remarkably, in the
last two networks, neurons are divided into two classes which we identify with
principal neurons and interneurons in biological circuits.
