  We propose a convex formulation of the fused lasso signal approximation
problem consisting of non-convex penalty functions. The fused lasso signal
model aims to estimate a sparse piecewise constant signal from a noisy
observation. Originally, the $\ell_1$ norm was used as a sparsity-inducing
convex penalty function for the fused lasso signal approximation problem.
However, the $\ell_1$ norm underestimates signal values. Non-convex
sparsity-inducing penalty functions better estimate signal values. In this
paper, we show how to ensure the convexity of the fused lasso signal
approximation problem with non-convex penalty functions. We further derive a
computationally efficient algorithm using the majorization-minimization
technique. We apply the proposed fused lasso method for the detection of
pulses.
