Informative Data Projections: A Framework and Two Examples

Methods for Projection Pursuit aim to facilitate the visual exploration of high-dimensional data by identifying interesting low-dimensional projections. A major challenge is the design of a suitable quality metric of projections, commonly referred to as the projection index, to be maximized by the Projection Pursuit algorithm. In this paper, we introduce a new information-theoretic strategy for tackling this problem, based on quantifying the amount of information the projection conveys to a user given their prior beliefs about the data. The resulting projection index is a subjective quantity, explicitly dependent on the intended user. As a useful illustration, we developed this idea for two particular kinds of prior beliefs. The first kind leads to PCA (Principal Component Analysis), shining new light on when PCA is (not) appropriate. The second kind leads to a novel projection index, the maximization of which can be regarded as a robust variant of PCA. We show how this projection index, though non-convex, can be effectively maximized using a modified power method as well as using a semidefinite programming relaxation. The usefulness of this new projection index is demonstrated in comparative empirical experiments against PCA and a popular Projection Pursuit method

Hardware Based Projection onto The Parity Polytope and Probability Simplex

This paper is concerned with the adaptation to hardware of methods for Euclidean norm projections onto the parity polytope and probability simplex. We first refine recent efforts to develop efficient methods of projection onto the parity polytope. Our resulting algorithm can be configured to have either average computational complexity $\mathcal{O}\left(d\right)$ or worst case complexity $\mathcal{O}\left(d\log{d}\right)$ on a serial processor where $d$ is the dimension of projection space. We show how to adapt our projection routine to hardware. Our projection method uses a sub-routine that involves another Euclidean projection; onto the probability simplex. We therefore explain how to adapt to hardware a well know simplex projection algorithm. The hardware implementations of both projection algorithms achieve area scalings of $\mathcal{O}(d\left(\log{d}\right)^2)$ at a delay of $\mathcal{O}(\left(\log{d}\right)^2)$. Finally, we present numerical results in which we evaluate the fixed-point accuracy and resource scaling of these algorithms when targeting a modern FPGA

Comments on orientifold projection in the conifold and SO x USp duality cascade

We study the O3-plane in the conifold. On the D3-brane world-volume we obtain SO x USp gauge theory that exhibits a duality cascade phenomenon. The orientifold projection is determined on the type IIB string side, and corresponds to that of O4-plane on the dual type IIA side. We show that SUGRA solutions of Klebanov-Tseytlin and Klebanov-Strassler survive under the projection. We also investigate the orientifold projection in the generalized conifolds, and verify desired features of the O4-projection in the type IIA picture.Comment: 1+27 pages, 9 figures, references added; version to appear in Phys. Rev.

Alignment Theory of Parallel-beam CT Image Reconstruction for Elastic-type Objects using Virtual Focusing Method

X-ray tomography has been studied in various fields. Although a great deal of effort has been directed at reconstructing the projection image set from a rigid-type specimen, little attention has been addressed to the reconstruction of projected images from an object showing elastic motion. Here, we present a mathematical solution to reconstruct the projection image set obtained from an object with specific elastic motions: periodically, regularly, and elliptically expanded or contracted specimens. To reconstruct the projection image set from expanded or contracted specimens, we introduce new methods; detection of sample's motion modes, mathematical re-scaling pixel values and converting projection angle for a common layerComment: 30 pages, 11 figure