Probabilistic partition of unity networks for high-dimensional regression problems

• URL: http://arxiv.org/abs/2210.02694v2
• Date: Sun, 11 Jun 2023 06:16:34 GMT
• Title: Probabilistic partition of unity networks for high-dimensional regression problems
• Authors: Tiffany Fan, Nathaniel Trask, Marta D'Elia, Eric Darve
• Abstract summary: We explore the partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems. We propose a general framework focusing on adaptive dimensionality reduction. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments.
• Score: 1.0227479910430863
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: We explore the probabilistic partition of unity network (PPOU-Net) model in the context of high-dimensional regression problems and propose a general framework focusing on adaptive dimensionality reduction. With the proposed framework, the target function is approximated by a mixture of experts model on a low-dimensional manifold, where each cluster is associated with a local fixed-degree polynomial. We present a training strategy that leverages the expectation maximization (EM) algorithm. During the training, we alternate between (i) applying gradient descent to update the DNN coefficients; and (ii) using closed-form formulae derived from the EM algorithm to update the mixture of experts model parameters. Under the probabilistic formulation, step (ii) admits the form of embarrassingly parallelizable weighted least-squares solves. The PPOU-Nets consistently outperform the baseline fully-connected neural networks of comparable sizes in numerical experiments of various data dimensions. We also explore the proposed model in applications of quantum computing, where the PPOU-Nets act as surrogate models for cost landscapes associated with variational quantum circuits.

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