w1Tnorm

w1Tnorm stands for weighted transformed L1 norm, a regularization term used in sparse modeling and signal processing. It is defined for a vector x in R^n and a weight vector w in R^n_+ by the expression ||x||_{w1T} = sum_{i=1}^n w_i T(|x_i|), where T: [0, ∞) → [0, ∞) is a fixed, nonnegative transformation function. The function T is typically chosen to be increasing and concave, with T(0) = 0, so that the penalty promotes sparsity more aggressively than the standard L1 norm. If T is the identity, the w1Tnorm reduces to the weighted L1 norm.

As a regularizer, the w1Tnorm is generally nonconvex when T is concave, which can offer stronger sparsity-inducing properties than L1 but at the cost of more challenging optimization. In many common implementations, the regularizer is not a true norm because it is not positively homogeneous of degree one unless T is linear. Nevertheless, w1Tnorm is valued for its ability to balance sparsity and model fit in a flexible, coordinate-separable form.

Common choices for T include T(t) = t/(a + t) (the transformed L1 or TL1 regularizer) and T(t) = log(1 + a t), with a > 0. These choices yield nonconvex penalties that encourage sparse solutions while remaining amenable to iterative optimization techniques such as majorization-minimization, reweighted L1 schemes, or proximal methods that exploit coordinate separability. The w1Tnorm is used in linear and logistic regression, matrix completion, and compressed sensing to promote sparsity in coefficients or latent factors.

Related concepts include the standard L1 norm, transformed L1 Regularizers, and other nonconvex sparsity-inducing penalties that share the goal of achieving sparser solutions with tractable computation.