Equiangular Tight Frames as Dictionaries in Sparse Representations

2 Jul 2021, 15:35


Sergei Novikov (Samara National Research University)


Let $\mathbf{\Phi}$ be $d\times n$-matrix with real or complex numbers, and the columns of $\mathbf{\Phi}$ are $\ell_2$-normalized.
Consider a linear under-determined set of equations
$$ \mathbf{\Phi}\mathbf{\alpha}=\mathbf{x}. $$ We shall refer hereafter to $\mathbf{x}$ as a signal to be processed, and $\mathbf{\alpha}$ will stand for its {\it representation}. The matrix $\mathbf{\Phi}$ will be referred to as the {\it dictionary}, and its columns $\left{\mathbf{\varphi}i\right}{i=1}^n$ will be called {\it atoms}.

Equiangular tight frames have the important advantage over other dictionaries. In particular, it's possible to calculate the spark for such dictionaries.

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Sergei Novikov (Samara National Research University)

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