Figure 1: Our method can generate infinite images of diverse and complex scenes that transition naturally from one into another. It does so without any conditioning and trains without any supervision from a dataset of unrelated square images.
We develop a framework for comparing data manifolds, aimed, in particular, towards the evaluation of deep generative models. We describe a novel tool, Cross-Barcode(P,Q), that, given a pair of distributions in a high-dimensional space, tracks multiscale topology spacial discrepancies between manifolds on which the distributions are concentrated. Based on the Cross-Barcode, we introduce the Manifold Topology Divergence score (MTop-Divergence) and apply it to assess the performance of deep generative models in various domains: images, 3D-shapes, time-series, and on different datasets: MNIST, Fashion MNIST, SVHN, CIFAR10, FFHQ, chest X-ray images, market stock data, ShapeNet. We demonstrate that the MTop-Divergence accurately detects various degrees of mode-dropping, intramode collapse, mode invention, and image disturbance. Our algorithm scales well (essentially linearly) with the increase of the dimension of the ambient highdimensional space. It is one of the first TDA-based practical methodologies that can be applied universally to datasets of different sizes and dimensions, including the ones on which the most recent GANs in the visual domain are trained. The proposed method is domain agnostic and does not rely on pre-trained networks.
We apply methods of topological data analysis to loss functions to gain insights on learning of deep neural networks and their generalization properties We study global properties of the loss function's gradient flow. We use topological data analysis of the loss function and its Morse complex to relate local behaviour along gradient trajectories with global properties of the loss surface. We define neural network's Topological Obstructions' score («TO-score») with help of robust topological invariants (barcodes of loss function) that quantify the "badness" of local minima for gradient-based optimization. We have made several experiments for computing these invariants, for small neural networks, and for fully connected, convolutional and ResNetlike neural networks on different datasets: MNIST, Fashion MNIST, CIFAR10, SVHN. Our two principal observations are 1) the neural network's barcode and TO-score decrease with the increase of the neural network's depth and width 2) there is an intriguing connection between the length of minima's segments in the barcode and the minima's generalization error.
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