34,797 research outputs found
Input and Weight Space Smoothing for Semi-supervised Learning
We propose regularizing the empirical loss for semi-supervised learning by
acting on both the input (data) space, and the weight (parameter) space. We
show that the two are not equivalent, and in fact are complementary, one
affecting the minimality of the resulting representation, the other
insensitivity to nuisance variability. We propose a method to perform such
smoothing, which combines known input-space smoothing with a novel weight-space
smoothing, based on a min-max (adversarial) optimization. The resulting
Adversarial Block Coordinate Descent (ABCD) algorithm performs gradient ascent
with a small learning rate for a random subset of the weights, and standard
gradient descent on the remaining weights in the same mini-batch. It achieves
comparable performance to the state-of-the-art without resorting to heavy data
augmentation, using a relatively simple architecture
Model Reduction and Neural Networks for Parametric PDEs
We develop a general framework for data-driven approximation of input-output maps between infinite-dimensional spaces. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation which, in principle, is defined on infinite-dimensional spaces and, in practice, is robust to the dimension of finite-dimensional approximations of these spaces required for computation. For a class of input-output maps, and suitably chosen probability measures on the inputs, we prove convergence of the proposed approximation methodology. Numerically we demonstrate the effectiveness of the method on a class of parametric elliptic PDE problems, showing convergence and robustness of the approximation scheme with respect to the size of the discretization, and compare our method with existing algorithms from the literature
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