Numerical experimentation · one dimension
Split the curvature, keep the function
Every positive bend goes into F; every negative bend goes into G. Their difference recovers the same target Φ. Move the oscillations and watch the convex pieces change.
The target stays the same when λ changes
A quadratic with movable oscillations.
Two convex functions
FG Both use the same vertical scale.
The split is visible in the second derivatives
Φ″F″ = max(Φ″, 0) + λG″ = max(−Φ″, 0) + λ
Vertical guides mark the exact zeros of Φ″. Adding λ raises both convex curvatures equally; their difference is unchanged.
How the decomposition is computed
On [a, b] = [−2, 2], integrate the positive and negative parts of Φ″ twice. The affine normalization is F(a) = Φ(a), F′(a) = Φ′(a), G(a) = G′(a) = 0. This gives the exact identity F − G = Φ.
When λ = 0, the two curvatures never overlap: F″G″ = 0. This minimizes their sum pointwise among all twice differentiable convex decompositions. A positive λ adds the same quadratic λ(x − a)²/2 to both pieces, so it adds unnecessary curvature without changing their difference.
The plots use analytic curvature and its exact antiderivatives between known zero crossings. The displayed identity residual measures floating-point arithmetic at 801 plotting points; it is not a certificate for an optimization algorithm. No network requests or training are needed.