  The proliferation of optical, electron, and scanning probe microscopies gives
rise to large volumes of imaging data of objects as diversified as cells,
bacteria, pollen, to nanoparticles and atoms and molecules. In most cases, the
experimental data streams contain images having arbitrary rotations and
translations within the image. At the same time, for many cases, small amounts
of labeled data are available in the form of prior published results, image
collections, and catalogs, or even theoretical models. Here we develop an
approach that allows generalizing from a small subset of labeled data with a
weak orientational disorder to a large unlabeled dataset with a much stronger
orientational (and positional) disorder, i.e., it performs a classification of
image data given a small number of examples even in the presence of a
distribution shift between the labeled and unlabeled parts. This approach is
based on the semi-supervised rotationally invariant variational autoencoder
(ss-rVAE) model consisting of the encoder-decoder "block" that learns a
rotationally (and translationally) invariant continuous latent representation
of data and a classifier that encodes data into a finite number of discrete
classes. The classifier part of the trained ss-rVAE inherits the rotational
(and translational) invariances and can be deployed independently of the other
parts of the model. The performance of the ss-rVAE is illustrated using the
synthetic data sets with known factors of variation. We further demonstrate its
application for experimental data sets of nanoparticles, creating nanoparticle
libraries and disentangling the representations defining the physical factors
of variation in the data. The code reproducing the results is available at
https://github.com/ziatdinovmax/Semi-Supervised-VAE-nanoparticles.
