A New Sparsity Promoting Residual Transform Operator for Lasso Regression

Abstract. Lasso regression is a widely employed approach within the [Formula: see text] regularization framework used to promote sparsity and recover piecewise smooth signals [Formula: see text] when the given observations are obtained from noisy, blurred, and/or incomplete data environments. In choosing the regularizing sparsity-promoting operator, it is assumed that the particular type of variability of the underlying signal, for example, piecewise constant or piecewise linear behavior across the entire domain, is both known and fixed. Such an assumption is problematic in more general cases, e.g., when a signal exhibits piecewise oscillatory behavior with varying wavelengths and magnitudes. To address the limitations of assuming a fixed (and typically low order) variability when choosing a sparsity-promoting operator, this investigation proposes a novel residual transform operator that can be used within the Lasso regression formulation. In a nutshell, the idea is that for a general piecewise smooth signal [Formula: see text], it is possible to design two operators [Formula: see text] and [Formula: see text] such that [Formula: see text], where [Formula: see text] is a discretized approximation of [Formula: see text], but [Formula: see text]. The corresponding residual transform operator, [Formula: see text], yields a result that (1) effectively reduces the variability dependent error that occurs when applying either [Formula: see text] or [Formula: see text] to [Formula: see text], a property that holds even when [Formula: see text] is not a good approximation to the true sparse domain vector of [Formula: see text], and (2) does not require [Formula: see text] or [Formula: see text] to have prior information regarding the variability of the underlying signal. Numerical experiments demonstrate the effectiveness of the new residual transform operator when compared to standard sparsity-promoting operators used in Lasso regression for recovering piecewise smooth signals.