An overview of the self-organizing map algorithm, on which the papers in this issue are based, is presented in this article.

The Self-Organizing Map

Dependence of biodiesel fuel properties on the structure of fatty acid alkyl esters

IEEE Transactions on Pattern Analysis and Machine Intelligence

Production of liquid biofuels from renewable resources

Progress and recent trends in biodiesel fuels

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.

Estimating chemical concentrations from compressed hyperspectral images

A numerical semigroup is a submonoid of $\mathbb N$ with finite complement in $\mathbb N$. A generalized numerical semigroup is a submonoid of $\mathbb{N}^{d}$ with finite complement in $\mathbb{N}^{d}$. In the context of numerical semigroups, Wilf's conjecture is a long standing open problem whose study has led to new mathematics and new ways of thinking about monoids. A natural extension of Wilf's conjecture, to the class of $\mathcal C$-semigroups, was proposed by Garcia-Garcia, Marin-Aragon, and Vigneron-Tenorio. In this paper, we propose a different generalization of Wilf's conjecture, to the setting of generalized numerical semigroups, and prove the conjecture for several large families including the irreducible, symmetric, and monomial case. We also discuss the relationship of our conjecture to the extension proposed by Garcia-Garcia, Marin-Aragon, and Vigneron-Tenorio.

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

Apolarity is a important tool in commutative algebra and algebraic geometry which studies a form, f, by the action of polynomial differential operators on f. The quotient of all polynomial differential operators by those which annihilate f is called the apolar algebra of f. In general, the apolar algebra of a form is useful for determining its Waring decomposition, which consists of writing the form as a sum of powers of linear forms with as few summands as possible. In this article we study the apolar algebra of a product of linear forms, which generalizes the case of monomials and connects to the geometry of hyperplane arrangements. In the first part of the article we provide a bound on the Waring rank of a product of linear forms under certain genericity assumptions; for this we use the defining equations of so-called star configurations due to Geramita, Harbourne, and Migliore. In the second part of the article we use the computer algebra system Bertini, which operates by homotopy continuation methods, to solve certain rank equations for catalecticant matrices. Our computations suggest that, up to a change of variables, there are exactly six homogeneous polynomials of degree six in three variables which factor completely as a product of linear forms defining an irreducible multi-arrangement and whose apolar algebras have dimension six in degree three. As a consequence of these calculations, we find six cases of such forms with cactus rank six, five of which also have Waring rank six. Among these are products defining subarrangements of the braid and Hessian arrangements.

https://dl.acm.org/doi/pdf/10.1145/3373207.3404014

On the apolar algebra of a product of linear forms

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.

Dual Graphs of Polyhedral Decompositions for the Detection of Adversarial Attacks

Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheres by a free linear action of a finite cyclic group, allows a deeper analysis of their structure. In this paper, we consider the problem of moments for the distance function between randomly selected pairs of points on homogeneous three-dimensional lens spaces. We give a derivation of a recursion relation for the moments, a formula for the $k$th moment, and a formula for the moment generating function, as well as an explicit formula for the volume of balls of all radii in these lens spaces.

Chris Peterson

Papers

Estimating chemical concentrations from compressed hyperspectral images

A generalization of Wilf's conjecture for Generalized Numerical Semigroups

On the apolar algebra of a product of linear forms

Dual Graphs of Polyhedral Decompositions for the Detection of Adversarial Attacks

Distributions of Distances and Volumes of Balls in Homogeneous Lens Spaces