2018/07/09 by Henry Kvinge, Kvinge, Henry, Elin Farnell +7
Engineering · Mathematics · #Advanced Numerical Analysis Techniques #Computer Vision and Pattern Recognition (cs.CV) #FOS: Computer and information sciences #FOS: Electrical engineering #Image and Video Processing (eess.IV) #Machine Learning (cs.LG) #Signal Processing (eess.SP) #Statistical and numerical algorithms #Tensor decomposition and applications #electronic engineering #information engineering
paper · pdf · doi:10.48550/arxiv.1807.03425
openalex publication_date 2018/07/09 · openalex created_date 2022/08/04 · openalex updated_date 2026/07/28
Dimensionality-reduction techniques are a fundamental tool for extracting\nuseful information from high-dimensional data sets. Because secant sets encode\nmanifold geometry, they are a useful tool for designing meaningful\ndata-reduction algorithms. In one such approach, the goal is to construct a\nprojection that maximally avoids secant directions and hence ensures that\ndistinct data points are not mapped too close together in the reduced space.\nThis type of algorithm is based on a mathematical framework inspired by the\nconstructive proof of Whitney's embedding theorem from differential topology.\nComputing all (unit) secants for a set of points is by nature computationally\nexpensive, thus opening the door for exploitation of GPU architecture for\nachieving fast versions of these algorithms. We present a polynomial-time\ndata-reduction algorithm that produces a meaningful low-dimensional\nrepresentation of a data set by iteratively constructing improved projections\nwithin the framework described above. Key to our algorithm design and\nimplementation is the use of GPUs which, among other things, minimizes the\ncomputational time required for the calculation of all secant lines. One goal\nof this report is to share ideas with GPU experts and to discuss a class of\nmathematical algorithms that may be of interest to the broader GPU community.\n