BACKGROUND: Integrating genomic datasets from homogenous or disparate sources to identify genes that are commonly or uniquely expressed remains a largely underexplored area. Such integrative analysis can reveal biologically relevant genes that are common or exclusive across datasets or within specific conditions or cohorts. Identifying these gene expression profiles and employing them to classify disease status can aid in the development of vaccines, diagnostics and targeted therapeutics with efficacy against difficult-to-treat medically important pathogens and cancer. RESULTS: This work develops new methodologies to integrate transcriptomic patterns from the lungs and spleen tissues infected by Francisella tularensis – Schu4 and Live Vaccine Strain (LVS). Our objective is to (i) identify biologically relevant gene features indicative of respiratory infection, disease severity, and bacterial dissemination to the spleen, and (ii) develop a Weighted [Formula: see text]-norm Non-Parallel Support Vector Machines ([Formula: see text]-WNPSVM) that will utilize the selected genes to predict disease status. The [Formula: see text]-WNPSVM is trained on the lungs data and validated on the spleen data, introducing a form of transfer learning, with uninfected controls and Schu4 or LVS samples as classes. Currently, a direct application of existing NPSVM-type methods to analyze gene expression datasets, where the number of genes significantly exceeds the number of samples, is computationally impractical due to their large memory requirements. This work addresses these challenges and also generalizes to models of similar formulations by incorporating dimensionality reduction and gene selection into the NPSVM-type frameworks. The [Formula: see text]-WNPSVM method outperforms traditional machine learning techniques such as ANN, XGBoost, AdaBoost, GradBoost, KNN, SVM, Naive Bayes, Random Forest, Logistic Regression, and Decision Tree, achieving a [Formula: see text] balanced accuracy on imbalanced data. CONCLUSIONS: We discovered sets of 235 genes exclusively expressed in the lungs and spleen tissues and utilized them to classify bacterial strains and controls, enabling prediction of disease status. Gene ontology is performed to reveal underlying metabolic pathways. Our analysis shows that signal transduction and disease (cancer) pathways are the most significant pathways activated in the lungs while gene expression (transcription), immune system, and disease (cancer) pathways are activated in the spleen. Collectively, these pathways indicate a significant host response to infection, including how the bacteria interact with host tissues during dissemination.

We present a novel feature selection technique, Sparse Linear Centroid-Encoder (SLCE). The algorithm uses a linear transformation to reconstruct a point as its class centroid and, at the same time, uses the ℓ1-norm penalty to filter out unnecessary features from the input data. The original formulation of the optimization problem is nonconvex, but we propose a two-step approach, where each step is convex. In the first step, we solve the linear Centroid-Encoder, a convex optimization problem over a matrix A. In the second step, we only search for a sparse solution over a diagonal matrix B while keeping A fixed. Unlike other linear methods, e.g., Sparse Support Vector Machines and Lasso, Sparse Linear Centroid-Encoder uses a single model for multi-class data. We present an in-depth empirical analysis of the proposed model and show that it promotes sparsity on various data sets, including high-dimensional biological data. Our experimental results show that SLCE has a performance advantage over some state-of-the-art neural network-based feature selection techniques.

We introduce a nonlinear model, Masked Sparse Bottleneck Centroid-Encoder (MSBCE), for determining the features that discriminate between two or more classes. The algorithm aims to extract discriminatory features in groups while reconstructing the class centroids in the ambient space and simultaneously use additional penalty terms in the bottleneck layer to decrease within-class scatter and increase the separation of different class centroids. The model has a sparsity-promoting layer (SPL) with a one-to-one connection to the input layer. Along with the primary objective, we minimize the $l_{2,1}$-norm of the sparse layer, which filters out unnecessary features from input data. During training, we update class centroids by taking the Hadamard product of the centroids and weights of the sparse layer, thus masking the irrelevant features from the target. Therefore the proposed method learns to reconstruct the critical features of the class centroids. The algorithm is applied to various real-world data sets, including high-dimensional biological, image, speech, and accelerometer sensor data. We compared our method to different state-of-the-art feature selection techniques, including supervised Concrete Autoencoders (SCAE), Feature Selection Networks (FsNet), Stochastic Gates (STG), and LassoNet. We empirically showed that MSBCE features often produced better classification accuracy than other methods on the sequester test sets, setting new state-of-the-art results. Apart from achieving state-of-the-art results, this is the first time feature selection is done on two high-energy particle physics data sets, SUSY and HIGGS, with 4.5 million and 10 million samples, respectively. The selected features, which are only 25-30% of the total number of variables, attained almost similar prediction rates compared to all features.

Abstract The contribution of our work is two-fold. First, we propose a novel feature selection technique, sparsity-promoted centroid-encoder (SCE). The model uses the nonlinear mapping of artificial neural networks to reconstruct a sample as its class centroid and, at the same time, apply aℓ1-penalty to the weights of a sparsity promoting layer, placed between the input and first hidden layer, to select discriminative features from input data. Using the proposed method, we designed a feature selection framework that first ranks each feature and then, compiles the optimal set using validation samples. The second part of our study investigates the role of stochastic optimization, such as Adam, in minimizingℓ1-norm. The empirical analysis shows that the hyper-parameters of Adam (mini-batch size, learning rate, etc.) play a crucial role in promoting feature sparsity by SCE. We apply our technique to numerous real-world data sets and find that it significantly outperforms other state-of-the-art methods, including LassoNet, stochastic gates (STG), feature selection networks (FsNet), supervised concrete autoencoder (CAE), deep feature selection (DFS), and random forest (RF).

Previous work has shown that a neural network with the rectified linear unit (ReLU) activation function leads to a convex polyhedral decomposition of the input space. These decompositions can be represented by a dual graph with vertices corresponding to polyhedra and edges corresponding to polyhedra sharing a facet, which is a subgraph of a Hamming graph. This paper illustrates how one can utilize the dual graph to detect and analyze adversarial attacks in the context of digital images. When an image passes through a network containing ReLU nodes, the firing or non-firing at a node can be encoded as a bit (1 for ReLU activation, 0 for ReLU non-activation). The sequence of all bit activations identifies the image with a bit vector, which identifies it with a polyhedron in the decomposition and, in turn, identifies it with a vertex in the dual graph. We identify ReLU bits that are discriminators between non-adversarial and adversarial images and examine how well collections of these discriminators can ensemble vote to build an adversarial image detector. Specifically, we examine the similarities and differences of ReLU bit vectors for adversarial images, and their non-adversarial counterparts, using a pre-trained ResNet-50 architecture. While this paper focuses on adversarial digital images, ResNet-50 architecture, and the ReLU activation function, our methods extend to other network architectures, activation functions, and types of datasets.

Finding prototypes (e.g., mean and median) for a dataset is central to a number of common machine learning algorithms. Subspaces have been shown to provide useful, robust representations for datasets of images, videos and more. Since subspaces correspond to points on a Grassmann manifold, one is led to consider the idea of a subspace prototype for a Grassmann-valued dataset. While a number of different subspace prototypes have been described, the calculation of some of these prototypes has proven to be computationally expensive while other prototypes are affected by outliers and produce highly imperfect clustering on noisy data. This work proposes a new subspace prototype, the flag median, and introduces the FlagIRLS algorithm for its calculation. We provide evidence that the flag median is robust to outliers and can be used effectively in algorithms like Linde-Buzo-Grey (LBG) to produce improved clusterings on Grassmannians. Numerical experiments include a synthetic dataset, the MNIST handwritten digits dataset, the Mind's Eye video dataset and the UCF YouTube action dataset. The flag median is compared the other leading algorithms for computing prototypes on the Grassmannian, namely, the l2-median and to the flag mean. We find that using FlagIRLS to compute the flag median converges in 4 iterations on a synthetic dataset. We also see that Grassmannian LBG with a codebook size of 20 and using the flag median produces at least a 10% improvement in cluster purity over Grassmannian LBG using the flag mean or l2-median on the Mind's Eye dataset.

We propose a new tool for visualizing complex, and potentially large and high-dimensional, data sets called Centroid-Encoder (CE). The architecture of the Centroid-Encoder is similar to the autoencoder neural network but it has a modified target, i.e., the class centroid in the ambient space. As such, CE incorporates label information and performs a supervised data visualization. The training of CE is done in the usual way with a training set whose parameters are tuned using a validation set. The evaluation of the resulting CE visualization is performed on a sequestered test set where the generalization of the model is assessed both visually and quantitatively. We present a detailed comparative analysis of the method using a wide variety of data sets and techniques, both supervised and unsupervised, including NCA, non-linear NCA, t-distributed NCA, t-distributed MCML, supervised UMAP, supervised PCA, Colored Maximum Variance Unfolding, supervised Isomap, Parametric Embedding, supervised Neighbor Retrieval Visualizer, and Multiple Relational Embedding. An analysis of variance using PCA demonstrates that a non-linear preprocessing by the CE transformation of the data captures more variance than PCA by dimension.

Abstract A flag is a nested sequence of vector spaces. The type of the flag encodes the sequence of dimensions of the vector spaces making up the flag. A flag manifold is a manifold whose points parameterize all flags of a fixed type in a fixed vector space. This paper provides the mathematical framework necessary for implementing self-organizing mappings on flag manifolds. Flags arise implicitly in many data analysis contexts including wavelet, Fourier, and singular value decompositions. The proposed geometric framework in this paper enables the computation of distances between flags, the computation of geodesics between flags, and the ability to move one flag a prescribed distance in the direction of another flag. Using these operations as building blocks, we implement the SOM algorithm on a flag manifold. The basic algorithm is applied to the problem of parameterizing a set of flags of a fixed type.

New depth sensors, like the Microsoft Kinect, produce streams of human pose data. These discrete pose streams can be viewed as noisy samples of an underlying continuous ideal curve that describes a trajectory through high-dimensional pose space. This paper introduces a technique for generalized curvature analysis (GCA) that determines features along the trajectory which can be used to characterize change and segment motion. Tools are developed for approximating generalized curvatures at mean points along a curve in terms of the singular values of local mean-centered data balls. The features of the GCA algorithm are illustrated on both synthetic and real examples, including data collected from a Kinect II sensor. We also applied GCA to the Carnegie Mellon University Motion Capture (MoCaP) database. Given that GCA scales linearly with the length of the time series we are able to analyze large data sets without down sampling. It is demonstrated that the generalized curvature approximations can be used to segment pose streams into motions and transitions between motions. The GCA algorithm can identify 94.2 percent of the transitions between motions without knowing the set of possible motions in advance, even though the subjects do not stop or pause between motions.

Dimensionality-reduction methods are a fundamental tool in the analysis of large datasets. These algorithms work on the assumption that the "intrinsic dimension" of the data is generally much smaller than the ambient dimension in which it is collected. Alongside their usual purpose of mapping data into a smaller-dimensional space with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the dataset.In this paper, we propose a new statistic that we call the kappa-profile for analysis of large datasets. The kappa-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection that optimally preserves the secants between points in the dataset. From this optimal projection we extract kappa, the norm of the shortest projected secant from among the set of all normalized secants. This kappa can be computed for any dimension k; thus the tuple of kappa values (indexed by dimension) becomes a kappa-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm provide a computationally feasible means of estimating the kappa-profile for large datasets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the kappa-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data.

Dimensionality-reduction techniques are a fundamental tool for extracting useful information from high-dimensional data sets. Because secant sets encode manifold geometry, they are a useful tool for designing meaningful data-reduction algorithms. In one such approach, the goal is to construct a projection that maximally avoids secant directions and hence ensures that distinct data points are not mapped too close together in the reduced space. This type of algorithm is based on a mathematical framework inspired by the constructive proof of Whitney's embedding theorem from differential topology. Computing all (unit) secants for a set of points is by nature computationally expensive, thus opening the door for exploitation of GPU architecture for achieving fast versions of these algorithms. We present a polynomial-time data-reduction algorithm that produces a meaningful low-dimensional representation of a data set by iteratively constructing improved projections within the framework described above. Key to our algorithm design and implementation is the use of GPUs which, among other things, minimizes the computational time required for the calculation of all secant lines. One goal of this report is to share ideas with GPU experts and to discuss a class of mathematical algorithms that may be of interest to the broader GPU community.

We propose a sequential algorithm for learning sparse radial basis approximations for streaming data. The initial phase of the algorithm formulates the RBF training as a convex optimization problem with an objective function on the expansion weights while the data fitting problem imposed only as an ℓ∞-norm constraint. Each new data point observed is tested for feasibility, i.e., whether the data fitting constraint is satisfied. If so, that point is discarded and no model update is required. If it is infeasible, a new basic variable is added to the linear program. The result is a primal infeasible-dual feasible solution. The dual simplex algorithm is applied to determine a new optimal solution. A large fraction of the streaming data points does not require updates to the RBF model since they are similar enough to previously observed data and satisfy the data fitting constraints. The structure of the simplex algorithm makes the update to the solution particularly efficient given the inverse of the new basis matrix is easily computed from the old inverse. The second phase of the algorithm involves a non-convex refinement of the convex problem. Given the sparse nature of the LP solution, the computational expense of the non-convex algorithm is greatly reduced. We have also found that a small subset of the training data that includes the novel data identified by the algorithm can be used to train the non-convex optimization problem with substantial computation savings and comparable errors on the test data. We illustrate the method on the Mackey-Glass chaotic time-series, the monthly sunspot data, and a Fort Collins, Colorado weather data set. In each case we compare the results to artificial neural networks (ANN) and standard skew-RBFs.

Many data sets can be viewed as a noisy sampling of an underlying space, and tools from topological data analysis can characterize this structure for the purpose of knowledge discovery. One such tool is persistent homology, which provides a multiscale description of the homological features within a data set. A useful representation of this homological information is a persistence diagram (PD). Efforts have been made to map PDs into spaces with additional structure valuable to machine learning tasks. We convert a PD to a finite- dimensional vector representation which we call a persistence image (PI), and prove the stability of this transformation with respect to small perturbations in the inputs. The discriminatory power of PIs is compared against existing methods, showing significant performance gains. We explore the use of PIs with vector-based machine learning tools, such as linear sparse support vector machines, which identify features containing discriminating topological information. Finally, high accuracy inference of parameter values from the dynamic output of a discrete dynamical system (the linked twist map) and a partial differential equation (the anisotropic Kuramoto-Sivashinsky equation) provide a novel application of the discriminatory power of PIs.

We propose an approach for the representation and classification of hyperspectral data that exploits the geometric framework, the Grassmann manifold, i.e., a parameterization of k-dimensional subspaces of Rn. Multiple pixels from a data class are used to capture the variability of the class information using a subspace representation. We use two metrics defined on the Grassmannian, chordal and geodesic, and several pseudometrics to measure the pairwise distances between the points, i.e., subspaces. Once a distance matrix is generated, classical multidimensional scaling is applied to find a configuration of points with preserved or approximated original distances, thus realizing an embedding of the Grassmannian in Euclidean space. A sparse support vector machine trained in the embedding space simultaneously classifies embedded subspaces and selects a subset of optimal dimensions (features) using a weight ratio criterion. The resulting embedding affords substantial model order reduction for classification and data visualization. In many cases, this framework provides linearly separable representations even when raw data are not linearly separable. We analyze frameworks and compare binary classification results for several distances. Finally, we illustrate the embedding of multiple data classes.

We present an application of persistent homology to the detection of chemical plumes in hyperspectral movies. The pixels of the raw hyperspectral data cubes are mapped to the geometric framework of the real Grassmann manifold G(k, n) (whose points parameterize the k-dimensional subspaces of ℝn) where they are analyzed, contrasting our approach with the more standard framework in Euclidean space. An advantage of this approach is that it allows the time slices in a hyperspectral movie to be collapsed to a sequence of points in such a way that some of the key structure within and between the slices is encoded by the points on the Grassmann manifold. This motivates the search for topological structure, associated with the evolution of the frames of a hyperspectral movie, within the corresponding points on the Grassmann manifold. The proposed framework affords the processing of large data sets, such as the hyperspectral movies explored in this investigation, while retaining valuable discriminative information.

Many computer vision algorithms employ subspace models to represent data. Many of these approaches benefit from the ability to create an average or prototype for a set of subspaces. The most popular method in these situations is the Karcher mean, also known as the Riemannian center of mass. The prevalence of the Karcher mean may lead some to assume that it provides the best average in all scenarios. However, other subspace averages that appear less frequently in the literature may be more appropriate for certain tasks. The extrinsic manifold mean, the L2-median, and the flag mean are alternative averages that can be substituted directly for the Karcher mean in many applications. This paper evaluates the characteristics and performance of these four averages on synthetic and real-world data. While the Karcher mean generalizes the Euclidean mean to the Grassman manifold, we show that the extrinsic manifold mean, the L2-median, and the flag mean behave more like medians and are therefore more robust to the presence of outliers among the subspaces being averaged. We also show that while the Karcher mean and L2-median are computed using iterative algorithms, the extrinsic manifold mean and flag mean can be found analytically and are thus orders of magnitude faster in practice. Finally, we show that the flag mean is a generalization of the extrinsic manifold mean that permits subspaces with different numbers of dimensions to be averaged. The result is a "cookbook" that maps algorithm constraints and data properties to the most appropriate subspace mean for a given application.

Videos can be naturally represented as multidimensional arrays known as tensors. However, the geometry of the tensor space is often ignored. In this paper, we argue that the underlying geometry of the tensor space is an important property for action classification. We characterize a tensor as a point on a product manifold and perform classification on this space. First, we factorize a tensor relating to each order using a modified High Order Singular Value Decomposition (HOSVD). We recognize each factorized space as a Grassmann manifold. Consequently, a tensor is mapped to a point on a product manifold and the geodesic distance on a product manifold is computed for tensor classification. We assess the proposed method using two public video databases, namely Cambridge-Gesture gesture and KTH human action data sets. Experimental results reveal that the proposed method performs very well on these data sets. In addition, our method is generic in the sense that no prior training is needed.

The theory of illumination subspaces is well developed and has been tested extensively on the Yale Face Database B (YDB) and CMU-PIE (PIE) data sets. This paper shows that if face recognition under varying illumination is cast as a problem of matching sets of images to sets of images, then the minimal principal angle between subspaces is sufficient to perfectly separate matching pairs of image sets from nonmatching pairs of image sets sampled from YDB and PIE. This is true even for subspaces estimated from as few as six images and when one of the subspaces is estimated from as few as three images if the second subspace is estimated from a larger set (10 or more). This suggests that variation under illumination may be thought of as useful discriminating information rather than unwanted noise.

An image-set based face recognition algorithm is proposed that exploits the full geometrical interpretation of Canonical Correlation Analysis (CCA). CCA maximizes the correlation between two linear subspaces associated with image-sets, where an image-set is assumed to contain multiple images of a person's face. When these linear subspaces are viewed as points on a Grassmann manifold, then geodesic distance on the manifold becomes the natural way to compare image-sets. The proposed method is tested on the ORL data set where it achieves a rank one identification rate of 98.75%. The proposed method is also tested on a subset of the Face Recognition Grand Challenge Experiment 4 data. Specifically, 82 probe and 230 gallery subjects with 32 images per probe and gallery image-set. Our algorithm achieves a rank one identification rate of 87% and a verification rate of 81% at a false accept rate of 1/1;000. These results on FRGC are significantly better than the well-known image-set matching algorithm, Mutual Subspace Method (MSM), which does not use geodesic distance. Another important finding is that cohort normalization boosts verification performance by 50% when used in conjunction with image-set matching. These results suggest that excellent levels of face recognition performance are possible when using image-sets, geodesic distance and cohort normalization. Finally, the proposed approach is generic in the sense that no training is required.