Uncertainty Bounds for Multivariate Machine Learning Predictions on High-Strain Brittle Fracture

Simulation of the crack network evolution on high strain rate impact experiments performed in brittle materials is very compute-intensive. The cost increases even more if multiple simulations are needed to account for the randomness in crack length, location, and orientation, which is inherently found in real-world materials. Constructing a machine learning emulator can make the process faster by orders of magnitude. There has been little work, however, on assessing the error associated with their predictions. Estimating these errors is imperative for meaningful overall uncertainty quantification. In this work, we extend the heteroscedastic uncertainty estimates to bound a multiple output machine learning emulator. We find that the response prediction is robust with a somewhat conservative estimate of uncertainty.

Convolutional Dictionary Learning: A Comparative Review and New Algorithms

Convolutional sparse representations are a form of sparse representation with a dictionary that has a structure that is equivalent to convolution with a set of linear filters. While effective algorithms have recently been developed for the convolutional sparse coding problem, the corresponding dictionary learning problem is substantially more challenging. Furthermore, although a number of different approaches have been proposed, the absence of thorough comparisons between them makes it difficult to determine which of them represents the current state of the art. The present work both addresses this deficiency and proposes some new approaches that outperform existing ones in certain contexts. A thorough set of performance comparisons indicates a very wide range of performance differences among the existing and proposed methods, and clearly identifies those that are the most effective.

Online Convolutional Dictionary Learning

While a number of different algorithms have recently been proposed for convolutional dictionary learning, this remains an expensive problem. The single biggest impediment to learning from large training sets is the memory requirements, which grow at least linearly with the size of the training set since all existing methods are batch algorithms. The work reported here addresses this limitation by extending online dictionary learning ideas to the convolutional context.

Multiclass Data Segmentation using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation compressed sensing and image processing. A multiclass extension is introduced using the Gibbs simplex, with the functional's double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm is a uses a graph adaptation of the classical numerical Merriman-Bence-Osher (MBO) scheme, which alternates between diffusion and thresholding. We demonstrate the performance of both algorithms experimentally on synthetic data, grayscale and color images, and several benchmark data sets such as MNIST, COIL and WebKB. We also make use of fast numerical solvers for finding the eigenvectors and eigenvalues of the graph Laplacian, and take advantage of the sparsity of the matrix. Experiments indicate that the results are competitive with or better than the current state-of-the-art multiclass segmentation algorithms.

Spectral Clustering with Epidemic Diffusion

Spectral clustering is widely used to partition graphs into distinct modules or communities. Existing methods for spectral clustering use the eigenvalues and eigenvectors of the graph Laplacian, an operator that is closely associated with random walks on graphs. We propose a new spectral partitioning method that exploits the properties of epidemic diffusion. An epidemic is a dynamic process that, unlike the random walk, simultaneously transitions to all the neighbors of a given node. We show that the replicator, an operator describing epidemic diffusion, is equivalent to the symmetric normalized Laplacian of a reweighted graph with edges reweighted by the eigenvector centralities of their incident nodes. Thus, more weight is given to edges connecting more central nodes. We describe a method that partitions the nodes based on the componentwise ratio of the replicator's second eigenvector to the first, and compare its performance to traditional spectral clustering techniques on synthetic graphs with known community structure. We demonstrate that the replicator gives preference to dense, clique-like structures, enabling it to more effectively discover communities that may be obscured by dense intercommunity linking.

Multiclass Semi-Supervised Learning on Graphs using Ginzburg-Landau Functional Minimization

We present a graph-based variational algorithm for classification of high-dimensional data, generalizing the binary diffuse interface model to the case of multiple classes. Motivated by total variation techniques, the method involves minimizing an energy functional made up of three terms. The first two terms promote a stepwise continuous classification function with sharp transitions between classes, while preserving symmetry among the class labels. The third term is a data fidelity term, allowing us to incorporate prior information into the model in a semi-supervised framework. The performance of the algorithm on synthetic data, as well as on the COIL and MNIST benchmark datasets, is competitive with state-of-the-art graph-based multiclass segmentation methods.

Multiclass Diffuse Interface Models for Semi-Supervised Learning on Graphs

We present a graph-based variational algorithm for multiclass classification of high-dimensional data, motivated by total variation techniques. The energy functional is based on a diffuse interface model with a periodic potential. We augment the model by introducing an alternative measure of smoothness that preserves symmetry among the class labels. Through this modification of the standard Laplacian, we construct an efficient multiclass method that allows for sharp transitions between classes. The experimental results demonstrate that our approach is competitive with the state of the art among other graph-based algorithms.