
PhD Dissertation: Generalized Independent Components Analysis Over Finite Alphabets
Independent component analysis (ICA) is a statistical method for transfo...
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Linear Independent Component Analysis over Finite Fields: Algorithms and Bounds
Independent Component Analysis (ICA) is a statistical tool that decompos...
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Stochastic Approximation for Online Tensorial Independent Component Analysis
Independent component analysis (ICA) has been a popular dimension reduct...
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The least favorable noise
Suppose that a random variable X of interest is observed perturbed by in...
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Asymptotically optimal test for dependent multiple testing set up
In this paper we explore the behaviour of dependent test statistics for ...
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Linear Network Coding: Effects of Varying the Message Dimension on the Set of Characteristics
It is known a vector linear solution may exist if and only if the charac...
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On Optimal Operational Sequence of Components in a Warm Standby System
We consider an open problem of optimal operational sequence for the 1ou...
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Generalized Independent Components Analysis Over Finite Alphabets
Independent component analysis (ICA) is a statistical method for transforming an observable multidimensional random vector into components that are as statistically independent as possible from each other. Usually the ICA framework assumes a model according to which the observations are generated (such as a linear transformation with additive noise). ICA over finite fields is a special case of ICA in which both the observations and the independent components are over a finite alphabet. In this thesis we consider a formulation of the finitefield case in which an observation vector is decomposed to its independent components (as much as possible) with no prior assumption on the way it was generated. This generalization is also known as Barlow's minimal redundancy representation and is considered an open problem. We propose several theorems and show that this hard problem can be accurately solved with a branch and bound search tree algorithm, or tightly approximated with a series of linear problems. Moreover, we show that there exists a simple transformation (namely, order permutation) which provides a greedy yet very effective approximation of the optimal solution. We further show that while not every random vector can be efficiently decomposed into independent components, the vast majority of vectors do decompose very well (that is, within a small constant cost), as the dimension increases. In addition, we show that we may practically achieve this favorable constant cost with a complexity that is asymptotically linear in the alphabet size. Our contribution provides the first efficient set of solutions to Barlow's problem with theoretical and computational guarantees. Finally, we demonstrate our suggested framework in multiple source coding applications.
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