758 research outputs found
A U-statistic estimator for the variance of resampling-based error estimators
We revisit resampling procedures for error estimation in binary classification in terms of U-statistics. In particular, we exploit the fact that the error rate estimator involving all learning-testing splits is a U-statistic. Therefore, several standard theorems on properties of U-statistics apply.
In particular, it has minimal variance among all unbiased estimators and is asymptotically normally distributed. Moreover, there is an unbiased estimator for this minimal variance if the total sample size is at least the double learning set size plus two. In this case, we exhibit such an estimator which is another U-statistic. It enjoys, again, various optimality properties and yields an asymptotically exact hypothesis test of the equality of error rates when two learning algorithms are compared. Our statements apply to any deterministic learning algorithms under weak non-degeneracy assumptions.
In an application to tuning parameter choice in lasso regression on a gene expression data set, the test does not reject the null hypothesis of equal rates between two different parameters
Self-Supervised Learning with Lie Symmetries for Partial Differential Equations
Machine learning for differential equations paves the way for computationally
efficient alternatives to numerical solvers, with potentially broad impacts in
science and engineering. Though current algorithms typically require simulated
training data tailored to a given setting, one may instead wish to learn useful
information from heterogeneous sources, or from real dynamical systems
observations that are messy or incomplete. In this work, we learn
general-purpose representations of PDEs from heterogeneous data by implementing
joint embedding methods for self-supervised learning (SSL), a framework for
unsupervised representation learning that has had notable success in computer
vision. Our representation outperforms baseline approaches to invariant tasks,
such as regressing the coefficients of a PDE, while also improving the
time-stepping performance of neural solvers. We hope that our proposed
methodology will prove useful in the eventual development of general-purpose
foundation models for PDEs
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