Performing Group Difference Testing on Graph Structured Data From GANs: Analysis and Applications in Neuroimaging.

Generative adversarial networks (GANs) have emerged as a powerful generative model in computer vision. Given their impressive abilities in generating highly realistic images, they are also being used in novel ways in applications in the life sciences. This raises an interesting question when GANs are used in scientific or biomedical studies. Consider the setting where we are restricted to only using the samples from a trained GAN for downstream group difference analysis (and do not have direct access to the real data). Will we obtain similar conclusions? In this work, we explore if "generated" data, i.e., sampled from such GANs can be used for performing statistical group difference tests in cases versus controls studies, common across many scientific disciplines. We provide a detailed analysis describing regimes where this may be feasible. We complement the technical results with an empirical study focused on the analysis of cortical thickness on brain mesh surfaces in an Alzheimer's disease dataset. To exploit the geometric nature of the data, we use simple ideas from spectral graph theory to show how adjustments to existing GANs can yield improvements. We also give a generalization error bound by extending recent results on Neural Network Distance. To our knowledge, our work offers the first analysis assessing whether the Null distribution in "healthy versus diseased subjects" type statistical testing using data generated from the GANs coincides with the one obtained from the same analysis with real data. The code is available at https://github.com/yyxiongzju/GLapGAN.

Nyströmformer: A Nystöm-based Algorithm for Approximating Self-Attention.

Transformers have emerged as a powerful tool for a broad range of natural language processing tasks. A key component that drives the impressive performance of Transformers is the self-attention mechanism that encodes the influence or dependence of other tokens on each specific token. While beneficial, the quadratic complexity of self-attention on the input sequence length has limited its application to longer sequences - a topic being actively studied in the community. To address this limitation, we propose Nyströmformer - a model that exhibits favorable scalability as a function of sequence length. Our idea is based on adapting the Nyström method to approximate standard self-attention with O(n) complexity. The scalability of Nyströmformer enables application to longer sequences with thousands of tokens. We perform evaluations on multiple downstream tasks on the GLUE benchmark and IMDB reviews with standard sequence length, and find that our Nyströmformer performs comparably, or in a few cases, even slightly better, than standard self-attention. On longer sequence tasks in the Long Range Arena (LRA) benchmark, Nyströmformer performs favorably relative to other efficient self-attention methods. Our code is available at https://github.com/mlpen/Nystromformer.

Scaling Recurrent Models via Orthogonal Approximations in Tensor Trains.

Modern deep networks have proven to be very effective for analyzing real world images. However, their application in medical imaging is still in its early stages, primarily due to the large size of three-dimensional images, requiring enormous convolutional or fully connected layers - if we treat an image (and not image patches) as a sample. These issues only compound when the focus moves towards longitudinal analysis of 3D image volumes through recurrent structures, and when a point estimate of model parameters is insufficient in scientific applications where a reliability measure is necessary. Using insights from differential geometry, we adapt the tensor train decomposition to construct networks with significantly fewer parameters, allowing us to train powerful recurrent networks on whole brain image volume sequences. We describe the "orthogonal" tensor train, and demonstrate its ability to express a standard network layer both theoretically and empirically. We show its ability to effectively reconstruct whole brain volumes with faster convergence and stronger confidence intervals compared to the standard tensor train decomposition. We provide code and show experiments on the ADNI dataset using image sequences to regress on a cognition related outcome.

On Training Deep 3D CNN Models with Dependent Samples in Neuroimaging.

There is much interest in developing algorithms based on 3D convolutional neural networks (CNNs) for performing regression and classification with brain imaging data and more generally, with biomedical imaging data. A standard assumption in learning is that the training samples are independently drawn from the underlying distribution. In computer vision, where we have millions of training examples, this assumption is violated but the empirical performance may remain satisfactory. But in many biomedical studies with just a few hundred training examples, one often has multiple samples per participant and/or data may be curated by pooling datasets from a few different institutions. Here, the violation of the independent samples assumption turns out to be more significant, especially in small-to-medium sized datasets. Motivated by this need, we show how 3D CNNs can be modified to deal with dependent samples. We show that even with standard 3D CNNs, there is value in augmenting the network to exploit information regarding dependent samples. We present empirical results for predicting cognitive trajectories (slope and intercept) from morphometric change images derived from multiple time points. With terms which encode dependency between samples in the model, we get consistent improvements over a strong baseline which ignores such knowledge.

Affine arithmetic-based B-spline surface intersection with GPU acceleration.

Because the B-spline surface intersection is a fundamental operation in geometric design software, it is important to make the surface intersection operation robust and efficient. As is well known, affine arithmetic is robust for calculating the surface intersection because it is able to not only find every branch of the intersection, but also deal with some singular cases, such as surface tangency. However, the classical affine arithmetic is defined only for the globally supported polynomials, and its computation is very time consuming, thus hampering its usefulness in practical applications, especially in geometric design. In this paper, we extend affine arithmetic to calculate the range of recursively and locally defined B-spline basis functions, and we accelerate the affine arithmetic-based surface intersection algorithm by using a GPU. Moreover, we develop efficient methods to thin the strip-shaped intersection regions produced by the affine arithmetic-based intersection algorithm, calculate the intersection points, and further improve their accuracy. The many examples presented in this paper demonstrate the robustness and efficiency of this method.