LapKDE Flow: Exact Invertible Transports Between Kernel Density Estimates

Kernel density estimation and normalizing flows address opposite sides of the same problem: KDE is directly data-driven and effective in low-data regimes, but static; flows compose, sample, and train, but their scalar maps must be fitted. We close this gap with *LapKDE Flow*, a normalizing flow whose scalar map is $T=F_2^{-1}\circ F_1$, the monotone transport between two Laplace-kernel density estimates. The map is exactly invertible in closed form and admits an exact log-determinant. Anchoring the knots at data recovers KDE as a flow layer; adding rotations and depth yields a multivariate estimator; and adding gradient steps turns the same construction into a trained flow - three operating points of one construction rather than three separate models. Because its bandwidths act as statistical temperatures rather than free slopes, LapKDE Flow provides two independent routes to quantization and propagates gradient to every mixture component from every data point, unlike fixed-knot splines. With zero gradient steps, the anchored estimator outperforms classical small-sample estimators by up to $0.49$ bits per dimension on MNIST. When trained, the same layer improves on published Gaussianization-flow results on all seven standard benchmarks, with the advantage persisting under a matched parameter budget on three of the four comparable benchmarks. It also remains competitive on the circle and in linear ICA. The contribution is not another scalar transform, but a model family in which estimating a density and fitting a density become the same operation.