M. Kirby

Nathan Mankovich, Emily King, Chris Peterson et al.

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 $\ell_2$-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 $\ell_2$-median on the Mind's Eye dataset.

Huma Jamil, Yajing Liu, Christina M. Cole et al.

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.

Tomojit Ghosh, Michael Kirby

We introduce a novel nonlinear model, Sparse Adaptive Bottleneck Centroid-Encoder (SABCE), 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 ignoring the irrelevant features from the target. Therefore the proposed method learns to reconstruct the critical components of class centroids rather than the whole 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 SABCE features often produced better classification accuracy than other methods on the sequester test sets, setting new state-of-the-art results.

Tomojit Ghosh, Michael Kirby

We propose a new supervised dimensionality reduction technique called Supervised Linear Centroid-Encoder (SLCE), a linear counterpart of the nonlinear Centroid-Encoder (CE) \citep{ghosh2022supervised}. SLCE works by mapping the samples of a class to its class centroid using a linear transformation. The transformation is a projection that reconstructs a point such that its distance from the corresponding class centroid, i.e., centroid-reconstruction loss, is minimized in the ambient space. We derive a closed-form solution using an eigendecomposition of a symmetric matrix. We did a detailed analysis and presented some crucial mathematical properties of the proposed approach. %We also provide an iterative solution approach based solving the optimization problem using a descent method. We establish a connection between the eigenvalues and the centroid-reconstruction loss. In contrast to Principal Component Analysis (PCA) which reconstructs a sample in the ambient space, the transformation of SLCE uses the instances of a class to rebuild the corresponding class centroid. Therefore the proposed method can be considered a form of supervised PCA. Experimental results show the performance advantage of SLCE over other supervised methods.

Huma Jamil, Yajing Liu, Turgay Caglar et al.

Researchers typically investigate neural network representations by examining activation outputs for one or more layers of a network. Here, we investigate the potential for ReLU activation patterns (encoded as bit vectors) to aid in understanding and interpreting the behavior of neural networks. We utilize Representational Dissimilarity Matrices (RDMs) to investigate the coherence of data within the embedding spaces of a deep neural network. From each layer of a network, we extract and utilize bit vectors to construct similarity scores between images. From these similarity scores, we build a similarity matrix for a collection of images drawn from 2 classes. We then apply Fiedler partitioning to the associated Laplacian matrix to separate the classes. Our results indicate, through bit vector representations, that the network continues to refine class detectability with the last ReLU layer achieving better than 95\% separation accuracy. Additionally, we demonstrate that bit vectors aid in adversarial image detection, again achieving over 95\% accuracy in separating adversarial and non-adversarial images using a simple classifier.

Tomojit Ghosh, Michael Kirby

Autoencoders have been widely used as a nonlinear tool for data dimensionality reduction. While autoencoders don't utilize the label information, Centroid-Encoders (CE)\cite{ghosh2022supervised} use the class label in their learning process. In this study, we propose a sparse optimization using the Centroid-Encoder architecture to determine a minimal set of features that discriminate between two or more classes. The resulting algorithm, Sparse Centroid-Encoder (SCE), extracts discriminatory features in groups using a sparsity inducing $\ell_1$-norm while mapping a point to its class centroid. One key attribute of SCE is that it can extract informative features from a multi-modal data set, i.e., data sets whose classes appear to have multiple clusters. The algorithm is applied to a wide variety of real world data sets, including single-cell data, high dimensional biological data, image data, speech data, and accelerometer sensor data. We compared our method to various state-of-the-art feature selection techniques, including supervised Concrete Autoencoders (SCAE), Feature Selection Network (FsNet), deep feature selection (DFS), Stochastic Gate (STG), and LassoNet. We empirically showed that SCE features often produced better classification accuracy than other methods on sequester test set.

Tomojit Ghosh, Michael Kirby

Visualizing high-dimensional data is an essential task in Data Science and Machine Learning. The Centroid-Encoder (CE) method is similar to the autoencoder but incorporates label information to keep objects of a class close together in the reduced visualization space. CE exploits nonlinearity and labels to encode high variance in low dimensions while capturing the global structure of the data. We present a detailed analysis of the method using a wide variety of data sets and compare it with other supervised dimension reduction techniques, including NCA, nonlinear 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. We empirically show that centroid-encoder outperforms most of these techniques. We also show that when the data variance is spread across multiple modalities, centroid-encoder extracts a significant amount of information from the data in low dimensional space. This key feature establishes its value to use it as a tool for data visualization.

Henry Kvinge, Elin Farnell, Julia R. Dupuis et al.

Compressive sensing (CS) is a method of sampling which permits some classes of signals to be reconstructed with high accuracy even when they were under-sampled. In this paper we explore a phenomenon in which bandwise CS sampling of a hyperspectral data cube followed by reconstruction can actually result in amplification of chemical signals contained in the cube. Perhaps most surprisingly, chemical signal amplification generally seems to increase as the level of sampling decreases. In some examples, the chemical signal is significantly stronger in a data cube reconstructed from 10% CS sampling than it is in the raw, 100% sampled data cube. We explore this phenomenon in two real-world datasets including the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset and the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset. Each of these datasets contains the release of a chemical simulant, such as glacial acetic acid, triethyl phospate, and sulfur hexafluoride, and in all cases we use the adaptive coherence estimator (ACE) to detect a target signal in the hyperspectral data cube. We end the paper by suggesting some theoretical justifications for why chemical signals would be amplified in CS sampled and reconstructed hyperspectral data cubes and discuss some practical implications.

Elin Farnell, Henry Kvinge, John P. Dixon et al.

Sampling is a fundamental aspect of any implementation of compressive sensing. Typically, the choice of sampling method is guided by the reconstruction basis. However, this approach can be problematic with respect to certain hardware constraints and is not responsive to domain-specific context. We propose a method for defining an order for a sampling basis that is optimal with respect to capturing variance in data, thus allowing for meaningful sensing at any desired level of compression. We focus on the Walsh-Hadamard sampling basis for its relevance to hardware constraints, but our approach applies to any sampling basis of interest. We illustrate the effectiveness of our method on the Physical Sciences Inc. Fabry-Pérot interferometer sensor multispectral dataset, the Johns Hopkins Applied Physics Lab FTIR-based longwave infrared sensor hyperspectral dataset, and a Colorado State University Swiss Ranger depth image dataset. The spectral datasets consist of simulant experiments, including releases of chemicals such as GAA and SF6. We combine our sampling and reconstruction with the adaptive coherence estimator (ACE) and bulk coherence for chemical detection and we incorporate an algorithmic threshold for ACE values to determine the presence or absence of a chemical. We compare results across sampling methods in this context. We have successful chemical detection at a compression rate of 90%. For all three datasets, we compare our sampling approach to standard orderings of sampling basis such as random, sequency, and an analog of sequency that we term `frequency.' In one instance, the peak signal to noise ratio was improved by over 30% across a test set of depth images.

Henry Kvinge, Elin Farnell, Michael Kirby et al.

Dimensionality-reduction methods are a fundamental tool in the analysis of large data sets. 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 dimension with minimal information loss, dimensionality-reduction techniques implicitly or explicitly provide information about the dimension of the data set. In this paper, we propose a new statistic that we call the $κ$-profile for analysis of large data sets. The $κ$-profile arises from a dimensionality-reduction optimization problem: namely that of finding a projection into $k$-dimensions that optimally preserves the secants between points in the data set. From this optimal projection we extract $κ,$ the norm of the shortest projected secant from among the set of all normalized secants. This $κ$ can be computed for any $k$; thus the tuple of $κ$ values (indexed by dimension) becomes a $κ$-profile. Algorithms such as the Secant-Avoidance Projection algorithm and the Hierarchical Secant-Avoidance Projection algorithm, provide a computationally feasible means of estimating the $κ$-profile for large data sets, and thus a method of understanding and monitoring their behavior. As we demonstrate in this paper, the $κ$-profile serves as a useful statistic in several representative settings: weather data, soundscape data, and dynamical systems data.