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

Variations in the states of patterns can be exploited for their discriminatory information and should not be discarded as noise. A pattern recognition system compares a data set of unlabeled patterns having variations of state in a set-by-set comparison with labeled arrays of individual data sets of multiple patterns also having variations of state. The individual data sets are each mapped to a point on a parameter space, and the points of each labeled array define a subset of the parameter space. If the point associated with the data set of unlabeled patterns satisfies a similarity criterion on the parameter space subset of a labeled array, the data set of unlabeled patterns is assigned to the class attributed to that labeled array.

Nonlinear set to set pattern recognition

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.

A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

We propose an algorithm for detecting anomalies in video sequences. In order to build an appropriate model, video of nominal activity is utilized to construct an anomaly free representation of the data. The resulting model produces alarm notifications when anomalous activity is observed. The approach involves characterizing segments of video as subspaces and invoking the geometric framework of Grassmann manifolds, i.e., the space of k-dimensional subspaces of n-dimensional space, Gr(k, n). With subspaces treated as points on Gr(k, n) together with a suitably chosen Grassmannian metric, one can exploit novel aspects of the geometry of the data for the purpose of anomaly detection. This mathematical framework is used to extend the Multivariate State Estimation Technique to the context of Grassmann manifolds. We present an application to the ETHZ Living Room Data Set for detecting anomalous activities.

http://ieeexplore.ieee.org/iel7/7222825/7237120/07237167.pdf

Identity maps and their extensions on parameter spaces: Applications to anomaly detection in video

Maximal Cohen–Macaulay Modules and Gorenstein Algebras

We extend the self-organizing mapping algorithm to the problem of visualizing data on Grassmann manifolds. In this setting, a collection of k points in n-dimensions is represented by a k-dimensional subspace, e.g., via the singular value or QR-decompositions. Data assembled in this way is challenging to visualize given abstract points on the Grassmannian do not reside in Euclidean space. The extension of the SOM algorithm to this geometric setting only requires that distances between two points can be measured and that any given point can be moved towards a presented pattern. The similarity between two points on the Grassmannian is measured in terms of the principal angles between subspaces, e.g., the chordal distance. Further, we employ a formula for moving one subspace towards another along the shortest path, i.e., the geodesic between two points on the Grassmannian. This enables a faithful implementation of the SOM approach for visualizing data consisting of k-dimensional subspaces of n-dimensional Euclidean space. We illustrate the resulting algorithm on a hyperspectral imaging application.

Chris Peterson

Papers

Nonlinear set to set pattern recognition

A GPU-Oriented Algorithm Design for Secant-Based Dimensionality Reduction

Identity maps and their extensions on parameter spaces: Applications to anomaly detection in video

Maximal Cohen–Macaulay Modules and Gorenstein Algebras

Visualizing data sets on the Grassmannian using self-organizing mappings