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Convolutional Neural Networks (CNN) have recently seen tremendous success in various computer vision tasks. However, their application to problems with high dimensional input and output, such as high-resolution image and video segmentation or 3D medical imaging, has been limited by various factors. Primarily, in the training stage, it is necessary to store network activations for back propagation. In these settings, the memory requirements associated with storing activations can exceed what is feasible with current hardware, especially for problems in 3D. Motivated by the propagation of signals over physical networks, that are governed by the hyperbolic Telegraph equation, in this work we introduce a fully conservative hyperbolic network for problems with high dimensional input and output. We introduce a coarsening operation that allows completely reversible CNNs by using a learnable Discrete Wavelet Transform and its inverse to both coarsen and interpolate the network state and change the number of channels. We show that fully reversible networks are able to achieve results comparable to the state of the art in 4D time-lapse hyper spectral image segmentation and full 3D video segmentation, with a much lower memory footprint that is a constant independent of the network depth. We also extend the use of such networks to Variational Auto Encoders with high resolution input and output., 21 pages, 9 figures, Updated work to include additional numerical experiments, a section about VAEs and learnable wavelets

Recently established equivalences between differential equations and the structure of neural networks enabled some interpretation of training of a neural network as partial-differential-equation (PDE) constrained optimization. We add to the previously established connections, explicit regularization that is particularly beneficial in the case of single large-scale examples with partial annotation. We show that explicit regularization of model parameters in PDE constrained optimization translates to regularization of the network output. Examination of the structure of the corresponding Lagrangian and backpropagation algorithm do not reveal additional computational challenges. A hyperspectral imaging example shows that minimum prior information together with cross-validation for optimal regularization parameters boosts the segmentation accuracy.

Factors that limit the size of the input and output of a neural network include memory requirements for the network states/activations to compute gradients, as well as memory for the convolutional kernels or other weights. The memory restriction is especially limiting for applications where we want to learn how to map volumetric data to the desired output, such as video-to-video. Recently developed fully reversible neural networks enable gradient computations using storage of the network states for a couple of layers only. While this saves a tremendous amount of memory, it is the convolutional kernels that take up most memory if fully reversible networks contain multiple invertible pooling/coarsening layers. Invertible coarsening operators such as the orthogonal wavelet transform cause the number of channels to grow explosively. We address this issue by combining fully reversible networks with layers that contain the convolutional kernels in a compressed form directly. Specifically, we introduce a layer that has a symmetric block-low-rank structure. In spirit, this layer is similar to bottleneck and squeeze-and-expand structures. We contribute symmetry by construction, and a combination of notation and flattening of tensors allows us to interpret these network structures in linear algebraic fashion as a block-low-rank matrix in factorized form and observe various properties. A video segmentation example shows that we can train a network to segment the entire video in one go, which would not be possible, in terms of memory requirements, using non-reversible networks and previously proposed reversible networks.

Height is a classic complex trait with common variants in a growing list of genes known to contribute to the phenotype. Using a genecentric genotyping array targeted toward cardiovascular-related loci, comprising 49,320 SNPs across approximately 2000 loci, we evaluated the association of common and uncommon SNPs with adult height in 114,223 individuals from 47 studies and six ethnicities. Atotal of 64 loci contained a SNP associated with height at array-wide significance (p < 2.4× 10(-6)), with 42 loci surpassing the conventional genome-wide significance threshold (p < 5× 10(-8)). Common variants with minor allele frequencies greater than 5% were observed to be associated with height in 37 previously reported loci. In individuals of European ancestry, uncommon SNPs in IL11 and SMAD3, which would not be genotyped with the use of standard genome-wide genotyping arrays, were strongly associated with height (p < 3× 10(-11)). Conditional analysis within associated regions revealed five additional variants associated with height independent of lead SNPs within the locus, suggesting allelic heterogeneity. Although underpowered to replicate findings from individuals of European ancestry, the direction of effect of associated variants was largely consistent in African American, South Asian, and Hispanic populations. Overall, we show that dense coverage of genes for uncommon SNPs, coupled with large-scale meta-analysis, can successfully identify additional variants associated with a common complex trait.