Weshow that the eigenvalues decay polynomially, quantify the rate of decay, and derive measures that reflect the composition of hierarchical features in these networks.

PDEs and can be evaluated in linear time.

Substring kernels are classical tools for representing biological sequences or text.

There has been exploding interest in embracing Transformer-based architectures for medical image segmentation. However, the lack of large-scale annotated medical datasets make achieving performances equivalent to those in natural images challenging. Convolutional networks, in contrast, have higher inductive biases and consequently, are easily trainable to high performance. Recently, the ConvNeXt architecture attempted to modernize the standard ConvNet by mirroring Transformer blocks. In this work, we improve upon this to design a modernized and scalable convolutional architecture customized to challenges of data-scarce medical settings. We introduce MedNeXt, a Transformer-inspired large kernel segmentation network which introduces - 1) A fully ConvNeXt 3D Encoder-Decoder Network for medical image segmentation, 2) Residual ConvNeXt up and downsampling blocks to preserve semantic richness across scales, 3) A novel technique to iteratively increase kernel sizes by upsampling small kernel networks, to prevent performance saturation on limited medical data, 4) Compound scaling at multiple levels (depth, width, kernel size) of MedNeXt. This leads to state-of-the-art performance on 4 tasks on CT and MRI modalities and varying dataset sizes, representing a modernized deep architecture for medical image segmentation. Our code is made publicly available at: https://github.com/MIC-DKFZ/MedNeXt.

Motivation: The size of available omics datasets is steadily increasing with technological advancement in recent years. While this increase in sample size can be used to improve the performance of relevant prediction tasks in healthcare, models that are optimized for large datasets usually operate as black boxes. In high stakes scenarios, like healthcare, using a black-box model poses safety and security issues. Without an explanation about molecular factors and phenotypes that affected the prediction, healthcare providers are left with no choice but to blindly trust the models. We propose a new type of artificial neural network, named Convolutional Omics Kernel Network (COmic). By combining convolutional kernel networks with pathway-induced kernels, our method enables robust and interpretable end-to-end learning on omics datasets ranging in size from a few hundred to several hundreds of thousands of samples. Furthermore, COmic can be easily adapted to utilize multi-omics data. Results: We evaluated the performance capabilities of COmic on six different breast cancer cohorts. Additionally, we trained COmic models on multi-omics data using the METABRIC cohort. Our models performed either better or similar to competitors on both tasks. We show how the use of pathway-induced Laplacian kernels opens the black-box nature of neural networks and results in intrinsically interpretable models that eliminate the need for post-hoc explanation models.

Partial differential equations (PDEs) play a dominant role in the mathematical modeling of many complex dynamical processes. Solving these PDEs often requires prohibitively high computational costs, especially when multiple evaluations must be made for different parameters or conditions. After training, neural operators can provide PDEs solutions significantly faster than traditional PDE solvers. In this work, invariance properties and computational complexity of two neural operators are examined for transport PDE of a scalar quantity. Neural operator based on graph kernel network (GKN) operates on graph-structured data to incorporate nonlocal dependencies. Here we propose a modified formulation of GKN to achieve frame invariance. Vector cloud neural network (VCNN) is an alternate neural operator with embedded frame invariance which operates on point cloud data. GKN-based neural operator demonstrates slightly better predictive performance compared to VCNN. However, GKN requires an excessively high computational cost that increases quadratically with the increasing number of discretized objects as compared to a linear increase for VCNN.

The classical development of neural networks has been primarily for mappings between a finite-dimensional Euclidean space and a set of classes, or between two finite-dimensional Euclidean spaces. The purpose of this work is to generalize neural networks so that they can learn mappings between infinite-dimensional spaces (operators). The key innovation in our work is that a single set of network parameters, within a carefully designed network architecture, may be used to describe mappings between infinite-dimensional spaces and between different finite-dimensional approximations of those spaces. We formulate approximation of the infinite-dimensional mapping by composing nonlinear activation functions and a class of integral operators. The kernel integration is computed by message passing on graph networks. This approach has substantial practical consequences which we will illustrate in the context of mappings between input data to partial differential equations (PDEs) and their solutions. In this context, such learned networks can generalize among different approximation methods for the PDE (such as finite difference or finite element methods) and among approximations corresponding to different underlying levels of resolution and discretization. Experiments confirm that the proposed graph kernel network does have the desired properties and show competitive performance compared to the state of the art solvers.

Recently, non-stationary spectral kernels have drawn much attention, owing to its powerful feature representation ability in revealing long-range correlations and input-dependent characteristics. However, non-stationary spectral kernels are still shallow models, thus they are deficient to learn both hierarchical features and local interdependence. In this paper, to obtain hierarchical and local knowledge, we build an interpretable convolutional spectral kernel network (\texttt{CSKN}) based on the inverse Fourier transform, where we introduce deep architectures and convolutional filters into non-stationary spectral kernel representations. Moreover, based on Rademacher complexity, we derive the generalization error bounds and introduce two regularizers to improve the performance. Combining the regularizers and recent advancements on random initialization, we finally complete the learning framework of \texttt{CSKN}. Extensive experiments results on real-world datasets validate the effectiveness of the learning framework and coincide with our theoretical findings.

Substring kernels are classical tools for representing biological sequences or text. However, when large amounts of annotated data is available, models that allow end-to-end training such as neural networks are often prefered. Links between recurrent neural networks (RNNs) and substring kernels have recently been drawn, by formally showing that RNNs with specific activation functions were points in a reproducing kernel Hilbert space (RKHS). In this paper, we revisit this link by generalizing convolutional kernel networks---originally related to a relaxation of the mismatch kernel---to model gaps in sequences. It results in a new type of recurrent neural network which can be trained end-to-end with backpropagation, or without supervision by using kernel approximation techniques. We experimentally show that our approach is well suited to biological sequences, where it outperforms existing methods for protein classification tasks.