Part 1 — 1D Marginal Transforms

The atomic operation of Gaussianization: turn one coordinate’s distribution into a standard normal via $z = \Phi^{-1}(F(x))$ for some monotone CDF estimator $F$ . Every method in the curriculum stacks these 1D maps between rotations, so Part 1 builds them every way that matters — estimating $F$ , computing the Jacobian the flow needs, inverting it, and training it — grounded in rbig and gauss_flows.

Each notebook keeps the Part 0 pattern: derive the idea, then confirm it against the packages.

Notebooks¶

#	notebook	master list	what you take away
00	ECDF & histograms	1.1–1.3	rank→uniform→normal; Glivenko–Cantelli; the ECDF’s degenerate Jacobian
01	KDE & Gaussian-mixture CDFs	1.4–1.6	smooth CDFs; analytic mixture log-det; choosing $h$ / $K$ (BIC)
02	Monotone-spline CDFs	1.7–1.8	monotonicity (PCHIP vs overshoot); RQS with exact inverse + analytic log-det
03	Learnable mixture-CDF	1.9	the marginal as a trainable layer; end-to-end MLE
04	Inversion strategies	1.10–1.12	bisection vs Newton; safeguarded hybrid; differentiating the inverse (unroll/one-step/adjoint); batched `vmap`

Each estimator notebook also carries a Jacobian / log-determinant section ( $\mathrm{d}z/\mathrm{d}x = f(x)/\varphi(z)$ , $\log|T'| = \log f(x) - \log\varphi(z)$ ), since that per-coordinate gradient is the term a flow sums in log_prob.

Threads from Part 0¶

The change-of-variables log-det (Part 0, 00) becomes concrete here: each estimator’s $f(x)$ is its log-det.
The forward/inverse trade-off (Part 0, 02) is realised: smooth CDFs invert by root-find, splines invert in closed form.
Differentiating the root-find inverse — and the live gauss_flows#111 zero-gradient pitfall — is covered in notebook 04.

Running¶

Same uv environment as Part 0 (rbig + gauss_flows + a Jupyter stack), with interpax added for gauss_flows.HistogramCDF:

cd projects/gaussianization
uv pip install --python .venv-tutorials/bin/python interpax
.venv-tutorials/bin/jupyter nbconvert --to notebook --execute --inplace \
  notebooks/01_marginal_transforms/0*.ipynb --ExecutePreprocessor.timeout=600

Notebooks are paired (jupytext, py:percent) and set jax_enable_x64.