Mark Kon

Julio E Castrillon-Candas, Mark Kon

Massive vector field datasets are common in multi-spectral optical and radar sensors, among many other emerging areas of application. In this paper we develop a novel stochastic functional (data) analysis approach for detecting anomalies based on the covariance structure of nominal stochastic behavior across a domain. An optimal vector field Karhunen-Loeve expansion is applied to such random field data. A series of multilevel orthogonal functional subspaces is constructed from the geometry of the domain, adapted from the KL expansion. Detection is achieved by examining the projection of the random field on the multilevel basis. In addition, reliable hypothesis tests are formed that do not require prior assumptions on probability distributions of the data. The method is applied to the important problem of deforestation and degradation in the Amazon forest. This is a complex non-monotonic process, as forests can degrade and recover. Using multi-spectral satellite data from Sentinel-2, the multilevel filter is constructed and anomalies are treated as deviations from the initial state of the forest. Forest anomalies are quantified with robust hypothesis tests. Our approach shows the advantage of using multiple bands of data in a vectorized complex, leading to better anomaly detection beyond the capabilities of scalar-based methods.

Julio Enrique Castrillon-Candas, Mark Kon

In this paper we introduce concepts from uncertainty quantification (UQ) and numerical analysis for the efficient evaluation of stochastic high dimensional Newton iterates. In particular, we develop complex analytic regularity theory of the solution with respect to the random variables. This justifies the application of sparse grids for the computation of stochastic moments. Convergence rates are derived and are shown to be subexponential or algebraic with respect to the number of realizations of random perturbations. Due the accuracy of the method, sparse grids are well suited for computing low probability events with high confidence. We apply our method to the power flow problem. Numerical experiments on the 39 bus New England power system model with large stochastic loads are consistent with the theoretical convergence rates.

Zhiguo Zhang, Mark A. Kon

A multiresolution analysis is a nested chain of related approximation spaces.This nesting in turn implies relationships among interpolation bases in the approximation spaces and their derived wavelet spaces. Using these relationships, a necessary and sufficient condition is given for existence of interpolation wavelets, via analysis of the corresponding scaling functions. It is also shown that any interpolation function for an approximation space plays the role of a special type of scaling function (an interpolation scaling function) when the corresponding family of approximation spaces forms a multiresolution analysis. Based on these interpolation scaling functions, a new algorithm is proposed for constructing corresponding interpolation wavelets (when they exist in a multiresolution analysis). In simulations, our theorems are tested for several typical wavelet spaces, demonstrating our theorems for existence of interpolation wavelets and for constructing them in a general multiresolution analysis.

Trajan Murphy, Akshunna S. Dogra, Hanfeng Gu et al.

''Noisy'' datasets (regimes with low signal to noise ratios, small sample sizes, faulty data collection, etc) remain a key research frontier for classification methods with both theoretical and practical implications. We introduce FINDER, a rigorous framework for analyzing generic classification problems, with tailored algorithms for noisy datasets. FINDER incorporates fundamental stochastic analysis ideas into the feature learning and inference stages to optimally account for the randomness inherent to all empirical datasets. We construct ''stochastic features'' by first viewing empirical datasets as realizations from an underlying random field (without assumptions on its exact distribution) and then mapping them to appropriate Hilbert spaces. The Kosambi-Karhunen-Loéve expansion (KLE) breaks these stochastic features into computable irreducible components, which allow classification over noisy datasets via an eigen-decomposition: data from different classes resides in distinct regions, identified by analyzing the spectrum of the associated operators. We validate FINDER on several challenging, data-deficient scientific domains, producing state of the art breakthroughs in: (i) Alzheimer's Disease stage classification, (ii) Remote sensing detection of deforestation. We end with a discussion on when FINDER is expected to outperform existing methods, its failure modes, and other limitations.

Xinying Mu, Mark Kon

A machine learning (ML) feature network is a graph that connects ML features in learning tasks based on their similarity. This network representation allows us to view feature vectors as functions on the network. By leveraging function operations from Fourier analysis and from functional analysis, one can easily generate new and novel features, making use of the graph structure imposed on the feature vectors. Such network structures have previously been studied implicitly in image processing and computational biology. We thus describe feature networks as graph structures imposed on feature vectors, and provide applications in machine learning. One application involves graph-based generalizations of convolutional neural networks, involving structured deep learning with hierarchical representations of features that have varying depth or complexity. This extends also to learning algorithms that are able to generate useful new multilevel features. Additionally, we discuss the use of feature networks to engineer new features, which can enhance the expressiveness of the model. We give a specific example of a deep tree-structured feature network, where hierarchical connections are formed through feature clustering and feed-forward learning. This results in low learning complexity and computational efficiency. Unlike "standard" neural features which are limited to modulated (thresholded) linear combinations of adjacent ones, feature networks offer more general feedforward dependencies among features. For example, radial basis functions or graph structure-based dependencies between features can be utilized.

Wenrui Li, Xiaoyu Wang, Yuetian Sun et al.

It has long been a recognized problem that many datasets contain significant levels of missing numerical data. A potentially critical predicate for application of machine learning methods to datasets involves addressing this problem. However, this is a challenging task. In this paper, we apply a recently developed multi-level stochastic optimization approach to the problem of imputation in massive medical records. The approach is based on computational applied mathematics techniques and is highly accurate. In particular, for the Best Linear Unbiased Predictor (BLUP) this multi-level formulation is exact, and is significantly faster and more numerically stable. This permits practical application of Kriging methods to data imputation problems for massive datasets. We test this approach on data from the National Inpatient Sample (NIS) data records, Healthcare Cost and Utilization Project (HCUP), Agency for Healthcare Research and Quality. Numerical results show that the multi-level method significantly outperforms current approaches and is numerically robust. It has superior accuracy as compared with methods recommended in the recent report from HCUP. Benchmark tests show up to 75% reductions in error. Furthermore, the results are also superior to recent state of the art methods such as discriminative deep learning.

Julio Enrique Castrillon-Candas, Dingning Liu, Sicheng Yang et al.

In this paper we develop a Multilevel Orthogonal Subspace (MOS) Karhunen-Loeve feature theory based on stochastic tensor spaces, for the construction of robust machine learning features. Training data is treated as instances of a random field within a relevant Bochner space. Our key observation is that separate machine learning classes can reside predominantly in mostly distinct subspaces. Using the Karhunen-Loeve expansion and a hierarchical expansion of the first (nominal) class, a MOS is constructed to detect anomalous signal components, treating the second class as an outlier of the first. The projection coefficients of the input data into these subspaces are then used to train a Machine Learning (ML) classifier. These coefficients become new features from which much clearer separation surfaces can arise for the underlying classes. Tests in the blood plasma dataset (Alzheimer's Disease Neuroimaging Initiative) show dramatic increases in accuracy. This is in contrast to popular ML methods such as Gradient Boosting, RUS Boost, Random Forest and (Convolutional) Neural Networks.

Shichen Cao, Jingjing Li, Kenric P. Nelson et al.

We present a coupled Variational Auto-Encoder (VAE) method that improves the accuracy and robustness of the probabilistic inferences on represented data. The new method models the dependency between input feature vectors (images) and weighs the outliers with a higher penalty by generalizing the original loss function to the coupled entropy function, using the principles of nonlinear statistical coupling. We evaluate the performance of the coupled VAE model using the MNIST dataset. Compared with the traditional VAE algorithm, the output images generated by the coupled VAE method are clearer and less blurry. The visualization of the input images embedded in 2D latent variable space provides a deeper insight into the structure of new model with coupled loss function: the latent variable has a smaller deviation and a more compact latent space generates the output values. We analyze the histogram of the likelihoods of the input images using the generalized mean, which measures the model's accuracy as a function of the relative risk. The neutral accuracy, which is the geometric mean and is consistent with a measure of the Shannon cross-entropy, is improved. The robust accuracy, measured by the -2/3 generalized mean, is also improved.

Mark A. Kon, Nikolay Nikolaev

We introduce a new discriminant analysis method (Empirical Discriminant Analysis or EDA) for binary classification in machine learning. Given a dataset of feature vectors, this method defines an empirical feature map transforming the training and test data into new data with components having Gaussian empirical distributions. This map is an empirical version of the Gaussian copula used in probability and mathematical finance. The purpose is to form a feature mapped dataset as close as possible to Gaussian, after which standard quadratic discriminants can be used for classification. We discuss this method in general, and apply it to some datasets in computational biology.