Learning by Unsupervised Nonlinear Diffusion

This paper proposes and analyzes a novel clustering algorithm that combines graph-based diffusion geometry with density estimation. The proposed method is suitable for data generated from mixtures of distributions with densities that are both multimodal and have nonlinear shapes. A crucial aspect of this algorithm is to introduce time of a data-adapted diffusion process as a scale parameter that is different from the local spatial scale parameter used in many clustering and learning algorithms. We prove estimates for the behavior of diffusion distances with respect to this time parameter under a flexible nonparametric data model, identifying a range of times in which the mesoscopic equilibria of the underlying process are revealed, corresponding to a gap between within-cluster and between-cluster diffusion distances. This analysis is leveraged to prove sufficient conditions guaranteeing the accuracy of the proposed learning by unsupervised nonlinear diffusion (LUND) algorithm. We implement the LUND algorithm numerically and confirm its theoretical properties on illustrative datasets, showing that the proposed method enjoys both theoretical and empirical advantages over current spectral clustering and density-based clustering techniques.

Random Projection Forests

Ensemble methods---particularly those based on decision trees---have recently demonstrated superior performance in a variety of machine learning settings. We introduce a generalization of many existing decision tree methods called "Random Projection Forests" (RPF), which is any decision forest that uses (possibly data dependent and random) linear projections. Using this framework, we introduce a special case, called "Lumberjack", using very sparse random projections, that is, linear combinations of a small subset of features. Lumberjack obtains statistically significantly improved accuracy over Random Forests, Gradient Boosted Trees, and other approaches on a standard benchmark suites for classification with varying dimension, sample size, and number of classes. To illustrate how, why, and when Lumberjack outperforms other methods, we conduct extensive simulated experiments, in vectors, images, and nonlinear manifolds. Lumberjack typically yields improved performance over existing decision trees ensembles, while mitigating computational efficiency and scalability, and maintaining interpretability. Lumberjack can easily be incorporated into other ensemble methods such as boosting to obtain potentially similar gains.

Discovering and Deciphering Relationships Across Disparate Data Modalities

Understanding the relationships between different properties of data, such as whether a connectome or genome has information about disease status, is becoming increasingly important in modern biological datasets. While existing approaches can test whether two properties are related, they often require unfeasibly large sample sizes in real data scenarios, and do not provide any insight into how or why the procedure reached its decision. Our approach, "Multiscale Graph Correlation" (MGC), is a dependence test that juxtaposes previously disparate data science techniques, including k-nearest neighbors, kernel methods (such as support vector machines), and multiscale analysis (such as wavelets). Other methods typically require double or triple the number samples to achieve the same statistical power as MGC in a benchmark suite including high-dimensional and nonlinear relationships - spanning polynomial (linear, quadratic, cubic), trigonometric (sinusoidal, circular, ellipsoidal, spiral), geometric (square, diamond, W-shape), and other functions, with dimensionality ranging from 1 to 1000. Moreover, MGC uniquely provides a simple and elegant characterization of the potentially complex latent geometry underlying the relationship, providing insight while maintaining computational efficiency. In several real data applications, including brain imaging and cancer genetics, MGC is the only method that can both detect the presence of a dependency and provide specific guidance for the next experiment and/or analysis to conduct.

Linear Optimal Low Rank Projection for High-Dimensional Multi-Class Data

Classifying samples into categories becomes intractable when a single sample can have millions to billions of features, such as in genetics or imaging data. Principal Components Analysis (PCA) is widely used to identify a low-dimensional representation of such features for further analysis. However, PCA ignores class labels, such as whether or not a subject has cancer, thereby discarding information that could substantially improve downstream classification performance. We describe an approach, "Linear Optimal Low-rank" projection (LOL), which extends PCA by incorporating the class labels in a fashion that is advantageous over existing supervised dimensionality reduction techniques. We prove, and substantiate with synthetic experiments, that LOL leads to a better representation of the data for subsequent classification than other linear approaches, while adding negligible computational cost. We then demonstrate that LOL substantially outperforms PCA in differentiating cancer patients from healthy controls using genetic data, and in differentiating gender using magnetic resonance imaging data with $>$500 million features and 400 gigabytes of data. LOL therefore allows the solution of previous intractable problems, yet requires only a few minutes to run on a desktop computer.

Path-Based Spectral Clustering: Guarantees, Robustness to Outliers, and Fast Algorithms

We consider the problem of clustering with the longest leg path distance (LLPD) metric, which is informative for elongated and irregularly shaped clusters. We prove finite-sample guarantees on the performance of clustering with respect to this metric when random samples are drawn from multiple intrinsically low-dimensional clusters in high-dimensional space, in the presence of a large number of high-dimensional outliers. By combining these results with spectral clustering with respect to LLPD, we provide conditions under which the eigengap statistic correctly determines the number of clusters for a large class of data sets, and prove guarantees on the number of points mislabeled by the proposed algorithm. Our methods are quite general and provide performance guarantees for spectral clustering with any ultrametric. We also introduce an efficient approximation algorithm, easy to implement, for the LLPD, based on a multiscale analysis of adjacency graphs.

Multiscale Strategies for Computing Optimal Transport

This paper presents a multiscale approach to efficiently compute approximate optimal transport plans between point sets. It is particularly well-suited for point sets that are in high-dimensions, but are close to being intrinsically low-dimensional. The approach is based on an adaptive multiscale decomposition of the point sets. The multiscale decomposition yields a sequence of optimal transport problems, that are solved in a top-to-bottom fashion from the coarsest to the finest scale. We provide numerical evidence that this multiscale approach scales approximately linearly, in time and memory, in the number of nodes, instead of quadratically or worse for a direct solution. Empirically, the multiscale approach results in less than one percent relative error in the objective function. Furthermore, the multiscale plans constructed are of interest by themselves as they may be used to introduce novel features and notions of distances between point sets. An analysis of sets of brain MRI based on optimal transport distances illustrates the effectiveness of the proposed method on a real world data set. The application demonstrates that multiscale optimal transport distances have the potential to improve on state-of-the-art metrics currently used in computational anatomy.

Adaptive Geometric Multiscale Approximations for Intrinsically Low-dimensional Data

We consider the problem of efficiently approximating and encoding high-dimensional data sampled from a probability distribution $\rho$ in $\mathbb{R}^D$, that is nearly supported on a $d$-dimensional set $\mathcal{M}$ - for example supported on a $d$-dimensional Riemannian manifold. Geometric Multi-Resolution Analysis (GMRA) provides a robust and computationally efficient procedure to construct low-dimensional geometric approximations of $\mathcal{M}$ at varying resolutions. We introduce a thresholding algorithm on the geometric wavelet coefficients, leading to what we call adaptive GMRA approximations. We show that these data-driven, empirical approximations perform well, when the threshold is chosen as a suitable universal function of the number of samples $n$, on a wide variety of measures $\rho$, that are allowed to exhibit different regularity at different scales and locations, thereby efficiently encoding data from more complex measures than those supported on manifolds. These approximations yield a data-driven dictionary, together with a fast transform mapping data to coefficients, and an inverse of such a map. The algorithms for both the dictionary construction and the transforms have complexity $C n \log n$ with the constant linear in $D$ and exponential in $d$. Our work therefore establishes adaptive GMRA as a fast dictionary learning algorithm with approximation guarantees. We include several numerical experiments on both synthetic and real data, confirming our theoretical results and demonstrating the effectiveness of adaptive GMRA.

High Dimensional Data Modeling Techniques for Detection of Chemical Plumes and Anomalies in Hyperspectral Images and Movies

We briefly review recent progress in techniques for modeling and analyzing hyperspectral images and movies, in particular for detecting plumes of both known and unknown chemicals. For detecting chemicals of known spectrum, we extend the technique of using a single subspace for modeling the background to a "mixture of subspaces" model to tackle more complicated background. Furthermore, we use partial least squares regression on a resampled training set to boost performance. For the detection of unknown chemicals we view the problem as an anomaly detection problem, and use novel estimators with low-sampled complexity for intrinsically low-dimensional data in high-dimensions that enable us to model the "normal" spectra and detect anomalies. We apply these algorithms to benchmark data sets made available by the Automated Target Detection program co-funded by NSF, DTRA and NGA, and compare, when applicable, to current state-of-the-art algorithms, with favorable results.

Multiscale Markov Decision Problems: Compression, Solution, and Transfer Learning

Many problems in sequential decision making and stochastic control often have natural multiscale structure: sub-tasks are assembled together to accomplish complex goals. Systematically inferring and leveraging hierarchical structure, particularly beyond a single level of abstraction, has remained a longstanding challenge. We describe a fast multiscale procedure for repeatedly compressing, or homogenizing, Markov decision processes (MDPs), wherein a hierarchy of sub-problems at different scales is automatically determined. Coarsened MDPs are themselves independent, deterministic MDPs, and may be solved using existing algorithms. The multiscale representation delivered by this procedure decouples sub-tasks from each other and can lead to substantial improvements in convergence rates both locally within sub-problems and globally across sub-problems, yielding significant computational savings. A second fundamental aspect of this work is that these multiscale decompositions yield new transfer opportunities across different problems, where solutions of sub-tasks at different levels of the hierarchy may be amenable to transfer to new problems. Localized transfer of policies and potential operators at arbitrary scales is emphasized. Finally, we demonstrate compression and transfer in a collection of illustrative domains, including examples involving discrete and continuous statespaces.

Multiscale Geometric Methods for Data Sets II: Geometric Multi-Resolution Analysis

Data sets are often modeled as point clouds in $R^D$, for $D$ large. It is often assumed that the data has some interesting low-dimensional structure, for example that of a $d$-dimensional manifold $M$, with $d$ much smaller than $D$. When $M$ is simply a linear subspace, one may exploit this assumption for encoding efficiently the data by projecting onto a dictionary of $d$ vectors in $R^D$ (for example found by SVD), at a cost $(n+D)d$ for $n$ data points. When $M$ is nonlinear, there are no "explicit" constructions of dictionaries that achieve a similar efficiency: typically one uses either random dictionaries, or dictionaries obtained by black-box optimization. In this paper we construct data-dependent multi-scale dictionaries that aim at efficient encoding and manipulating of the data. Their construction is fast, and so are the algorithms that map data points to dictionary coefficients and vice versa. In addition, data points are guaranteed to have a sparse representation in terms of the dictionary. We think of dictionaries as the analogue of wavelets, but for approximating point clouds rather than functions.