Arjuna Flenner

Diffuse Interface Models on Graphs for Classification of High Dimensional Data

There are currently several communities working on algorithms for classification of high dimensional data. This work develops a class of variational algorithms that combine recent ideas from spectral methods on graphs with nonlinear edge/region detection methods traditionally used in the PDE-based imaging community. The algorithms are based on the Ginzburg--Landau functional which has classical PDE connections to total variation minimization. Convex-splitting algorithms allow us to quickly find minimizers of the proposed model and take advantage of fast spectral solvers of linear graph-theoretic problems. We present diverse computational examples involving both basic clustering and semisupervised learning for different applications. Case studies include feature identification in images, segmentation in social networks, and segmentation of shapes in high dimensional datasets.

Multiclass Data Segmentation Using Diffuse Interface Methods on Graphs

We present two graph-based algorithms for multiclass segmentation of high-dimensional data on graphs. The algorithms use a diffuse interface model based on the Ginzburg-Landau functional, related to total variation and graph cuts. A multiclass extension is introduced using the Gibbs simplex, with the functionals double-well potential modified to handle the multiclass case. The first algorithm minimizes the functional using a convex splitting numerical scheme. The second algorithm 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, image labeling, 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 in multiclass graph-based segmentation algorithms for high-dimensional data.

Diffuse Interface Methods for Multiclass Segmentation of High-Dimensional Data

Abstract We present two graph-based algorithms for multiclass segmentation of high-dimensional data, motivated by the binary diffuse interface model. One algorithm generalizes Ginzburg–Landau (GL) functional minimization on graphs to the Gibbs simplex. The other algorithm uses a reduction of GL minimization, based on the Merriman–Bence–Osher scheme for motion by mean curvature. These yield accurate and efficient algorithms for semi-supervised learning. Our algorithms outperform existing methods, including supervised learning approaches, on the benchmark datasets that we used. We refer to Garcia-Cardona (2014) for a more detailed illustration of the methods, as well as different experimental examples.

Improving image clustering using sparse text and the wisdom of the crowds

We propose a method to improve image clustering using sparse text and the wisdom of the crowds. In particular, we present a method to fuse two different kinds of document features, image and text features, and use a common dictionary or “wisdom of the crowds” as the connection between the two different kinds of documents. With the proposed fusion matrix, we use topic modeling via non-negative matrix factorization to cluster documents.

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.

Learning representations for improved target identification, scene classification, and information fusion

Object representation is fundamental to Automated Target Recognition (ATR). Many ATR approaches choose a basis, such as a wavelet or Fourier basis, to represent the target. Recently, advancements in Image and Signal processing have shown that object recognition can be improved if, rather than a assuming a basis, a database of training examples is used to learn a representation. We discuss learning representations using Non-parametric Bayesian topic models, and demonstrate how to integrate information from other sources to improve ATR. We apply the method to EO and IR information integration for vehicle target identification and show that the learned representation of the joint EO and IR information improves target identification by 4%. Furthermore, we demonstrate that we can integrate text and imagery data to direct the representation for mission specific tasks and improve performance by 8%. Finally, we illustrate integrating graphical models into representation learning to improve performance by 2%.