The Fixed Lambert Penalty and Scalar Update

Evaluate the bounded Lambert penalty or its exact scalar least-squares update with fixed shape \(c=1\).

Usage

lambert_penalty(t, lambda)
lambert_threshold(z, lambda)

Arguments

t

Finite, nonnegative coefficient magnitudes.

z

Finite signed scalar scores.

lambda

A finite positive penalty level.

Details

Write \(W_0\) for the principal real Lambert function, characterized by \(W_0(u)\exp(W_0(u))=u\) for \(u\geq0\). For a magnitude \(t\), let \(w=W_0(t/\lambda)\). The penalty is $$p_\lambda(t)=\frac{\lambda^2}{4} [e^{2w}(1+2w-2w^2)-1],\quad 0\leq t\leq e\lambda.$$ For \(t>e\lambda\), it is constant at \(\lambda^2(e^2-1)/4\). The expression is evaluated with cancellation-resistant arithmetic near zero.

The scalar update minimizes \((z-b)^2/2+p_\lambda(|b|)\) over \(b\in R\). It is zero when \(|z|\leq\lambda\), equals \(z\log(|z|/\lambda)\) for \(\lambda<|z|<e\lambda\), and equals \(z\) when \(|z|\geq e\lambda\). This normalization requires unit coordinate curvature, obtained by training-only RMS scaling in lambert.

Value

An unnamed numeric vector of the same length as t or z, containing penalty values or scalar updates, respectively. A scalar lambda is shared by all elements; elementwise penalty-level vectors are not supported. Zero-length input vectors return zero-length vectors.

See also

lambert

Examples

lambert_penalty(c(0, 1, exp(1), 4), lambda = 1)
#> [1] 0.0000000 0.9088501 1.5972640 1.5972640
lambert_threshold(c(-4, -1, 0, 1, 2, 4), lambda = 1)
#> [1] -4.000000  0.000000  0.000000  0.000000  1.386294  4.000000