  This paper is devoted to theoretical aspects on optimality of sparse
approximation. We undertake a quantitative study of new types of greedy-like
bases that have recently arisen in the context of nonlinear $m$-term
approximation in Banach spaces as a generalization of the properties that
characterize almost greedy bases, i.e., quasi-greediness and democracy. As a
means to compare the efficiency of these new bases with already existing ones
in regards to the implementation of the Thresholding Greedy Algorithm, we place
emphasis on obtaining estimates for their sequence of unconditionality
parameters. Using an enhanced version of the original method from [S. J.
Dilworth, N. J. Kalton, and D. Kutzarova, On the existence of almost greedy
bases in Banach spaces, Studia Math. 159 (2003), no. 1, 67-101] for building
almost greedy bases, we manage to construct bidemocratic bases whose
unconditionality parameters satisfy significantly worse estimates than almost
greedy bases even in Hilbert spaces.
