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Train Like a (Var)Pro: Efficient Training of Neural Networks with Variable Projection

SIAM Journal on Mathematics of Data Science

Newman, Elizabeth N.; Ruthotto, Lars R.; Hart, Joseph L.; van Bloemen Waanders, Bart G.

Deep neural networks (DNNs) have achieved state-of-the-art performance across a variety of traditional machine learning tasks, e.g., speech recognition, image classification, and segmentation. The ability of DNNs to efficiently approximate high-dimensional functions has also motivated their use in scientific applications, e.g., to solve partial differential equations and to generate surrogate models. In this paper, we consider the supervised training of DNNs, which arises in many of the above applications. We focus on the central problem of optimizing the weights of the given DNN such that it accurately approximates the relation between observed input and target data. Devising effective solvers for this optimization problem is notoriously challenging due to the large number of weights, nonconvexity, data sparsity, and nontrivial choice of hyperparameters. To solve the optimization problem more efficiently, we propose the use of variable projection (VarPro), a method originally designed for separable nonlinear least-squares problems. Our main contribution is the Gauss--Newton VarPro method (GNvpro) that extends the reach of the VarPro idea to nonquadratic objective functions, most notably cross-entropy loss functions arising in classification. These extensions make GNvpro applicable to all training problems that involve a DNN whose last layer is an affine mapping, which is common in many state-of-the-art architectures. In our four numerical experiments from surrogate modeling, segmentation, and classification, GNvpro solves the optimization problem more efficiently than commonly used stochastic gradient descent (SGD) schemes. Finally, GNvpro finds solutions that generalize well, and in all but one example better than well-tuned SGD methods, to unseen data points.

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TYPE Journal Article YEAR 2021
OSTIDOI

Randomized algorithms for generalized singular value decomposition with application to sensitivity analysis

Numerical Linear Algebra with Applications

Saibaba, Arvind K.; Hart, Joseph L.; van Bloemen Waanders, Bart G.

The generalized singular value decomposition (GSVD) is a valuable tool that has many applications in computational science. However, computing the GSVD for large-scale problems is challenging. Motivated by applications in hyper-differential sensitivity analysis (HDSA), we propose new randomized algorithms for computing the GSVD which use randomized subspace iteration and weighted QR factorization. Detailed error analysis is given which provides insight into the accuracy of the algorithms and the choice of the algorithmic parameters. We demonstrate the performance of our algorithms on test matrices and a large-scale model problem where HDSA is used to study subsurface flow.

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TYPE Journal Article YEAR 2021
ScopusOSTIDOI

Classification of orthostatic intolerance through data analytics

Medical and Biological Engineering and Computing

Hart, Joseph L.; Gilmore, Steven G.; Gremaud, Pierre G.; Olsen, Christian O.; Mehlsen, Jesper M.; Olufsen, Mette O.

Imbalance in the autonomic nervous system can lead to orthostatic intolerance manifested by dizziness, lightheadedness, and a sudden loss of consciousness (syncope); these are common conditions, but they are challenging to diagnose correctly. Uncertainties about the triggering mechanisms and the underlying pathophysiology have led to variations in their classification. This study uses machine learning to categorize patients with orthostatic intolerance. Here we use random forest classification trees to identify a small number of markers in blood pressure, and heart rate time-series data measured during head-up tilt to (a) distinguish patients with a single pathology and (b) examine data from patients with a mixed pathophysiology. Next, we use Kmeans to cluster the markers representing the time-series data. We apply the proposed method analyzing clinical data from 186 subjects identified as control or suffering from one of four conditions: postural orthostatic tachycardia (POTS), cardioinhibition, vasodepression, and mixed cardioinhibition and vasodepression. Classification results confirm the use of supervised machine learning. We were able to categorize more than 95% of patients with a single condition and were able to subgroup all patients with mixed cardioinhibitory and vasodepressor syncope. Clustering results confirm the disease groups and identify two distinct subgroups within the control and mixed groups. The proposed study demonstrates how to use machine learning to discover structure in blood pressure and heart rate time-series data. The methodology is used in classification of patients with orthostatic intolerance. Diagnosing orthostatic intolerance is challenging, and full characterization of the pathophysiological mechanisms remains a topic of ongoing research. This study provides a step toward leveraging machine learning to assist clinicians and researchers in addressing these challenges.

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TYPE Journal Article YEAR 2021
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