3 research outputs found
Polar -Complex and -Bicomplex Singular Value Decomposition and Principal Component Pursuit
Informed by recent work on tensor singular value decomposition and circulant
algebra matrices, this paper presents a new theoretical bridge that unifies the
hypercomplex and tensor-based approaches to singular value decomposition and
robust principal component analysis. We begin our work by extending the
principal component pursuit to Olariu's polar -complex numbers as well as
their bicomplex counterparts. In so doing, we have derived the polar
-complex and -bicomplex proximity operators for both the - and
trace-norm regularizers, which can be used by proximal optimization methods
such as the alternating direction method of multipliers. Experimental results
on two sets of audio data show that our algebraically-informed formulation
outperforms tensor robust principal component analysis. We conclude with the
message that an informed definition of the trace norm can bridge the gap
between the hypercomplex and tensor-based approaches. Our approach can be seen
as a general methodology for generating other principal component pursuit
algorithms with proper algebraic structures.Comment: 12 pages, 2 figure
Amostragem compressiva em análise de Clifford
Doutoramento em Matemática e Aplicações (PDMA)Compressed sensing is a new paradigm in signal processing which
states that for certain matrices sparse representations can be obtained by a simple l1-minimization. In this thesis we explore this paradigm for higher-dimensional signal. In particular three cases are being studied: signals taking values in a bicomplex algebra, quaternionic signals, and complex signals which are representable by a nonlinear Fourier basis, a so-called Takenaka-Malmquist system.Amostragem Compressiva é um novo paradigma em processamento
de sinal, no qual se assegura, para determinadas matrizes, que as representações esparsas de sinais podem ser obtidas por intermédio de um simples procedimento de l1-minimização. Nesta tese, exploramos este paradigma para sinais em dimensões superiores. Estudaremos três casos particulares: sinais com valores na álgebra bi-complexa, sinais quaterniónicos e, finalmente, sinais complexos representáveis por uma base de Fourier não-linear, dito sistema de Takenaka-Malmquist
Theoretical Concepts of Quantum Mechanics
Quantum theory as a scientific revolution profoundly influenced human thought about the universe and governed forces of nature. Perhaps the historical development of quantum mechanics mimics the history of human scientific struggles from their beginning. This book, which brought together an international community of invited authors, represents a rich account of foundation, scientific history of quantum mechanics, relativistic quantum mechanics and field theory, and different methods to solve the Schrodinger equation. We wish for this collected volume to become an important reference for students and researchers