Tighter Bounds on Johnson Lindenstrauss Transforms

Johnson and Lindenstrauss (1984) proved that any finite set of data in a high dimensional space can be projected into a low dimensional space with the Euclidean metric information of the set being preserved within any desired accuracy. Such dimension reduction plays a critical role in many applications with massive data. There has been extensive effort in the literature on how to find explicit constructions of Johnson-Lindenstrauss projections. In this poster, we show how algebraic codes over finite fields can be used for fast Johnson-Lindenstrauss projections of data in high dimensional Euclidean spaces

Johnson-Lindenstrauss Transformations

With the quick progression of technology and the increasing need to process large data, there has been an increased interest in data-dependent and data-independent dimension reduction techniques such as principle component analysis (PCA) and Johnson\-Lindenstrauss (JL) transformations, respectively. In 1984, Johnson and Lindenstrauss proved that any finite set of data in a high-dimensional space can be projected into a low-dimensional space while preserving the pairwise Euclidean distance within any desired accuracy, provided the projected dimension is sufficiently large; however, if the desired projected dimension is too small, Woodruff and Jayram, and Kane, Nelson, and Meka in 2011 separately proved such a projection does not exist. In this thesis, we answer an open problem by providing a precise threshold for the projected dimension, above which, there exists a projection approximately preserving the Euclidean distance, but below which, there does not exist such a projection. We, also, give a brief survey of JL constructions, covering the initial constructions and those based on fast-Fourier transforms and codes, and discuss applications in which JL transformations have been implemented

Secret Sharing and Network Coding

In this thesis, we consider secret sharing schemes and network coding. Both of these fields are vital in today\u27s age as secret sharing schemes are currently being implemented by government agencies and private companies, and as network coding is continuously being used for IP networks. We begin with a brief overview of linear codes. Next, we examine van Dijk\u27s approach to realize an access structure using a linear secret sharing scheme; then we focus on a much simpler approach by Tang, Gao, and Chen. We show how this method can be used to find an optimal linear secret sharing scheme for an access structure with six participants. In the last chapter, we examine network coding and point out some similarities between secret sharing schemes and network coding. We present results from a paper by Silva and Kschischang; in particular, we present the concept of universal security and their coset coding scheme to achieve universal security

Johnson-Lindenstrauss projection of high dimensional data

Johnson and Lindenstrauss (1984) proved that any finite set of data in a high dimensional space can be projected into a low dimensional space with the Euclidean metric information of the set being preserved within any desired accuracy. Such dimension reduction plays a critical role in many applications with massive data. There have been extensive effort in the literature on how to find explicit constructions of Johnson-Lindenstrauss projections. In this poster, we show how algebraic codes over finite fields can be used for fast Johnson-Lindenstrauss projections of data in high dimensional Euclidean spaces. This is joint work with Shuhong Gao and Yue Mao

Strategies for Mastering Uncertainty

This chapter describes three general strategies to master uncertainty in technical systems: robustness, flexibility and resilience. It builds on the previous chapters about methods to analyse and identify uncertainty and may rely on the availability of technologies for particular systems, such as active components. Robustness aims for the design of technical systems that are insensitive to anticipated uncertainties. Flexibility increases the ability of a system to work under different situations. Resilience extends this characteristic by requiring a given minimal functional performance, even after disturbances or failure of system components, and it may incorporate recovery. The three strategies are described and discussed in turn. Moreover, they are demonstrated on specific technical systems