Showing papers by "Chris Peterson published in 2022"

The Flag Median and FlagIRLS

Abstract: 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.

Estimating chemical concentrations from compressed hyperspectral images

Abstract: In this paper we derive two algorithms for estimating concentrations of a known chemical compound from compressed measurements of a hyperspectral image (HSI). It is assumed that each resolved pixel in a scene contains a chemical of known spectral signature, at an unknown concentration. The problem is to estimate the concentration directly from the compressed measurements. Estimated concentrations are either displayed or used as detection scores in a threshold test for presence or absence of chemical. In the first algorithm we use matched filtering and ℓ1 regularization to extract an image of concentrations, directly from compressed data. In the second we model the image of concentrations in a fixed-resolution subspace of the 2D Haar wavelet domain, estimate its parameters in this space, and reconstruct the image of concentrations at a macro-pixel resolution. We evaluate our algorithms by applying them to several long-wave infrared (LWIR) HSI data sets, either synthetically generated or recorded by Physical Sciences Inc. Synthetically-generated data is compressed with a mathematically-defined linear compressor; real HSI data is compressed with PSI’s Digital Micromirror Device (DMD), which is a physical implementation of a mathematically-defined compressor; Fabry-Perot data is raw HSI data recorded by PSI, which is then compressed with a mathematically-defined compressor. We demonstrate for these data sets that estimating concentrations through matched filtering and ℓ1 inversion of compressed measurements yields detection performance that is as good as previously proposed methods that first reconstruct a hyperspectral data cube from compressed data, and then estimate or detect chemical concentrations. The proposed methods save on memory and computation. We demonstrate that detection performance is maintained when resolving concentration maps at a lower resolution, so long as the resolution is not too low.

Dual Graphs of Polyhedral Decompositions for the Detection of Adversarial Attacks

Abstract: 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.

local adaptation and host specificity to copepod intermediate hosts by the Schistocephalus solidus tapeworm

Abstract: We investigated if there was local adaptation and host specify in the tapeworm Schistocephalus solidus to its copepod first intermediate hosts. The tapeworm is locally adapted and host specific to its threespine stickleback second intermediate host. We exposed copepods from five lakes in Vancouver Island (BC, Canada) to local (i.e. same lake) and foreign tapeworms in a reciprocal exposure experiment. Results indicate that the tapeworm is not locally adapted to the copepods, but there was host specificity as a copepod genus was more parasitized than another genus.

Quasi-polynomial growth of numerical and affine semigroups with constrained gaps

Abstract: . A common tool in the theory of numerical semigroups is to interpret a desired class of semigroups as the integer lattice points in a rational polyhedron in order to leverage computational and enumerative techniques from polyhedral geometry. Most arguments of this type make use of a parametrization of numerical semigroups with fixed multiplicity m in terms of their m -Ap´ery sets, giving a representation called Kunz coordinates which obey a collection of inequalities defining the Kunz polyhedron. In this work, we introduce a new class of polyhedra describing numerical semigroups in terms of a truncated addition table of their sporadic elements. Applying a classical theorem of Ehrhart to slices of these polyhedra, we prove that the number of numerical semigroups with n sporadic elements and Frobenius number f is polynomial up to periodicity, or quasi-polynomial, as a function of f for fixed n . We also generalize this approach to higher dimensions to demonstrate quasi-polynomial growth of the number of affine semigroups with a fixed number of elements, and all gaps, contained in an integer dilation of a fixed polytope.