Convolutional Neural Networks (CNNs) (LeCun et al., 1989) have become a widely used and powerful tool for computer vision tasks, in large part due to their ability to achieve translation

The development of efficient machine learning models for molecular systems representation is becoming crucial in scientific research.

Neural Information Processing Systems http://nips.cc/

We introduce a novel co-learning paradigm for manifolds naturally admitting an action of a transformation group $\mathcal{G}$, motivated by recent developments on learning a manifold from attached fibre bundle structures. We utilize a representation theoretic mechanism that canonically associates multiple independent vector bundles over a common base manifold, which provides multiple views for the geometry of the underlying manifold. The consistency across these fibre bundles provide a common base for performing unsupervised manifold co-learning through the redundancy created artificially across irreducible representations of the transformation group. We demonstrate the efficacy of our proposed algorithmic paradigm through drastically improved robust nearest neighbor identification in cryo-electron microscopy image analysis and the clustering accuracy in community detection.

A longstanding challenge in neuroscience is to understand neural mechanisms underlying the brain's remarkable ability to learn and detect transformations of objects due to motion. Translations and rotations of images can be viewed as orthogonal transformations in the space of pixel intensity vectors. Every orthogonal transformation can be decomposed into rotations within irreducible two-dimensional subspaces (or representations). For sets of commuting transformations, known as toroidal groups, Cohen and Welling proposed a mathematical framework for learning the irreducible representations. We explore the possibility that the brain also learns irreducible representations using a biologically plausible learning mechanism. The first is based on SVD of the anti-symmetrized outer product of the vectors representing consecutive images and is implemented by a single-layer neural network. The second is based on PCA of the difference between consecutive frames and is implemented in a two-layer network but with greater biological plausibility. Both networks learn image rotations (replicating Cohen and Welling's results) as well as translations. It would be interesting to search for the proposed networks in nascent connectomics and physiology datasets.

We present a convolutional network that is equivariant to rigid body motions.

Many learning machines make use of an internal representation [Bengio et al., 2013] to help inform

H = {( γ,t )|γ [0 , 2π ), t R }. R is the quotient space SE (3)/( SO (2) R ) up to isomorphism. SO (3) is a semidirect product group as stated in [18].