Sliced Iterative Gaussianization (GIS) — Core Demo¶

Gaussianization via Iterative Slicing (GIS) — from Dai & Seljak (2020) — Gaussianizes data with a greedy stack of sliced layers. Where RBIG rotates all dimensions and Gaussianizes every marginal at each step, a GIS layer:

learns $K$ orthonormal projection directions $\mathbf{A}$ that maximize the sliced Wasserstein distance to a Gaussian reference (i.e. the directions along which the data is most non-Gaussian), and
Gaussianizes only those $K$ 1D slices with monotonic rational-quadratic (RQ) splines, leaving the orthogonal complement untouched.

This notebook fits GIS on 2D distributions, visualizes the learned directions layer by layer, verifies the output is Gaussian, and closes with a brief comparison against AnnealedRBIG on the same data.

For the mathematical details, see the theory page.

Colab / fresh environment? Run the cell below to install rbig from GitHub. Skip if already installed.

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!pip install "rbig[all] @ git+https://github.com/jejjohnson/rbig.git" -q
!pip install "rbig[all] @ git+https://github.com/jejjohnson/rbig.git" -q

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%matplotlib inline
import time

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

from rbig import GIS, AnnealedRBIG

plt.style.use("seaborn-v0_8-paper")
%matplotlib inline
import time

import matplotlib.pyplot as plt
import numpy as np
from scipy import stats

from rbig import GIS, AnnealedRBIG

plt.style.use("seaborn-v0_8-paper")

/home/user/rbig/.venv/lib/python3.11/site-packages/tqdm/auto.py:21: TqdmWarning: IProgress not found. Please update jupyter and ipywidgets. See https://ipywidgets.readthedocs.io/en/stable/user_install.html
  from .autonotebook import tqdm as notebook_tqdm

Data¶

We use two classic 2D benchmarks:

a banana distribution (a Gaussian bent along a parabola), and
a mixture of Gaussians (multi-modal).

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def make_banana(n_samples: int, seed: int = 0) -> np.ndarray:
    rng = np.random.default_rng(seed)
    x = rng.standard_normal((n_samples, 2))
    return np.stack([x[:, 0], x[:, 1] + 0.5 * x[:, 0] ** 2 - 1.0], axis=1)


def make_gmm(n_samples: int, seed: int = 0) -> np.ndarray:
    rng = np.random.default_rng(seed)
    means = np.array([[-2.0, 0.0], [2.0, 0.0], [0.0, 2.5]])
    labels = rng.integers(0, len(means), size=n_samples)
    return means[labels] + 0.6 * rng.standard_normal((n_samples, 2))


n_samples = 5_000
X_banana = make_banana(n_samples, seed=0)
X_gmm = make_gmm(n_samples, seed=1)

fig, axes = plt.subplots(1, 2, figsize=(9, 4))
for ax, X, title in zip(axes, [X_banana, X_gmm], ["Banana", "Mixture of Gaussians"]):
    ax.scatter(X[:, 0], X[:, 1], s=2, alpha=0.3, color="steelblue")
    ax.set_title(title)
    ax.set_aspect("equal")
plt.tight_layout()
plt.show()
def make_banana(n_samples: int, seed: int = 0) -> np.ndarray:
    rng = np.random.default_rng(seed)
    x = rng.standard_normal((n_samples, 2))
    return np.stack([x[:, 0], x[:, 1] + 0.5 * x[:, 0] ** 2 - 1.0], axis=1)


def make_gmm(n_samples: int, seed: int = 0) -> np.ndarray:
    rng = np.random.default_rng(seed)
    means = np.array([[-2.0, 0.0], [2.0, 0.0], [0.0, 2.5]])
    labels = rng.integers(0, len(means), size=n_samples)
    return means[labels] + 0.6 * rng.standard_normal((n_samples, 2))


n_samples = 5_000
X_banana = make_banana(n_samples, seed=0)
X_gmm = make_gmm(n_samples, seed=1)

fig, axes = plt.subplots(1, 2, figsize=(9, 4))
for ax, X, title in zip(axes, [X_banana, X_gmm], ["Banana", "Mixture of Gaussians"]):
    ax.scatter(X[:, 0], X[:, 1], s=2, alpha=0.3, color="steelblue")
    ax.set_title(title)
    ax.set_aspect("equal")
plt.tight_layout()
plt.show()

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Fit GIS¶

Defaults follow the paper / sinflow: n_directions=None selects $K = D/2$ (here $K=1$ per layer), directions are found by Stiefel-manifold optimization, and early stopping monitors the held-out log-likelihood.

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t0 = time.perf_counter()
gis_banana = GIS(n_layers=100, random_state=0).fit(X_banana)
t_banana = time.perf_counter() - t0

t0 = time.perf_counter()
gis_gmm = GIS(n_layers=100, random_state=0).fit(X_gmm)
t_gmm = time.perf_counter() - t0

print(f"Banana: {gis_banana.n_layers_} layers kept, {t_banana:.1f}s")
print(f"GMM:    {gis_gmm.n_layers_} layers kept, {t_gmm:.1f}s")
t0 = time.perf_counter()
gis_banana = GIS(n_layers=100, random_state=0).fit(X_banana)
t_banana = time.perf_counter() - t0

t0 = time.perf_counter()
gis_gmm = GIS(n_layers=100, random_state=0).fit(X_gmm)
t_gmm = time.perf_counter() - t0

print(f"Banana: {gis_banana.n_layers_} layers kept, {t_banana:.1f}s")
print(f"GMM:    {gis_gmm.n_layers_} layers kept, {t_gmm:.1f}s")

Banana: 20 layers kept, 9.9s
GMM:    15 layers kept, 8.5s

Early stopping kicked in far below the n_layers=100 budget — a handful of targeted layers suffices in 2D.

Visualizing the Learned Directions¶

Each fitted layer stores its orthonormal directions in layer.A_ (shape (D, K)). We replay the data through the flow — whitening first, then layer by layer — and draw each layer's direction on top of the representation it saw. The arrow is the 1D slice the layer Gaussianizes; everything orthogonal to it passes through unchanged.

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def plot_layer_directions(model: GIS, X: np.ndarray, max_layers: int = 6):
    """Scatter the running representation with each layer's direction."""
    Xt = model.whitener_.transform(X) if model.whitener_ is not None else X.copy()
    n_show = min(max_layers, model.n_layers_)
    fig, axes = plt.subplots(2, (n_show + 1) // 2, figsize=(12, 7))
    axes = np.atleast_1d(axes).ravel()
    for i in range(n_show):
        layer = model.layers_[i]
        ax = axes[i]
        ax.scatter(Xt[:, 0], Xt[:, 1], s=2, alpha=0.25, color="steelblue")
        # A_: (D, K); draw each direction through the data mean.
        center = Xt.mean(axis=0)
        for k in range(layer.A_.shape[1]):
            a = layer.A_[:, k]
            scale = 3.0
            ax.annotate(
                "",
                xy=center + scale * a,
                xytext=center - scale * a,
                arrowprops={"arrowstyle": "-|>", "color": "crimson", "lw": 2},
            )
        ax.set_title(f"Layer {i + 1} input + direction")
        ax.set_aspect("equal")
        Xt, _ = layer.transform(Xt)
    for ax in axes[n_show:]:
        ax.axis("off")
    fig.tight_layout()
    return fig


fig = plot_layer_directions(gis_banana, X_banana)
fig.suptitle("Banana — learned slice per layer", y=1.02)
plt.show()
def plot_layer_directions(model: GIS, X: np.ndarray, max_layers: int = 6):
    """Scatter the running representation with each layer's direction."""
    Xt = model.whitener_.transform(X) if model.whitener_ is not None else X.copy()
    n_show = min(max_layers, model.n_layers_)
    fig, axes = plt.subplots(2, (n_show + 1) // 2, figsize=(12, 7))
    axes = np.atleast_1d(axes).ravel()
    for i in range(n_show):
        layer = model.layers_[i]
        ax = axes[i]
        ax.scatter(Xt[:, 0], Xt[:, 1], s=2, alpha=0.25, color="steelblue")
        # A_: (D, K); draw each direction through the data mean.
        center = Xt.mean(axis=0)
        for k in range(layer.A_.shape[1]):
            a = layer.A_[:, k]
            scale = 3.0
            ax.annotate(
                "",
                xy=center + scale * a,
                xytext=center - scale * a,
                arrowprops={"arrowstyle": "-|>", "color": "crimson", "lw": 2},
            )
        ax.set_title(f"Layer {i + 1} input + direction")
        ax.set_aspect("equal")
        Xt, _ = layer.transform(Xt)
    for ax in axes[n_show:]:
        ax.axis("off")
    fig.tight_layout()
    return fig


fig = plot_layer_directions(gis_banana, X_banana)
fig.suptitle("Banana — learned slice per layer", y=1.02)
plt.show()

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fig = plot_layer_directions(gis_gmm, X_gmm)
fig.suptitle("GMM — learned slice per layer", y=1.02)
plt.show()
fig = plot_layer_directions(gis_gmm, X_gmm)
fig.suptitle("GMM — learned slice per layer", y=1.02)
plt.show()

The Stiefel optimizer picks the direction along which the current representation deviates most from a Gaussian — the parabolic ridge of the banana, the mode-separation axes of the mixture — rather than a fixed PCA or random axis. After each layer, the residual non-Gaussianity lives along new directions, which the next layer then targets.

Is the Output Gaussian?¶

The forward transform should map the data to (approximately) $\mathcal{N}(\mathbf{0}, \mathbf{I}_2)$.

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fig, axes = plt.subplots(2, 3, figsize=(12, 7))
for row, (X, model, name) in enumerate(
    [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]
):
    Z = model.transform(X)
    ax = axes[row, 0]
    ax.scatter(Z[:, 0], Z[:, 1], s=2, alpha=0.3, color="seagreen")
    circle = plt.Circle((0, 0), 3.0, color="black", fill=False, ls="--", lw=1)
    ax.add_patch(circle)
    ax.set_title(f"{name}: transformed (3σ circle)")
    ax.set_aspect("equal")

    grid = np.linspace(-4, 4, 200)
    for j in range(2):
        ax = axes[row, j + 1]
        ax.hist(Z[:, j], bins=60, density=True, alpha=0.6, color="seagreen")
        ax.plot(grid, stats.norm.pdf(grid), "k--", lw=1.5, label="N(0, 1)")
        ax.set_title(f"{name}: latent dim {j + 1}")
        ax.legend()
plt.tight_layout()
plt.show()
fig, axes = plt.subplots(2, 3, figsize=(12, 7))
for row, (X, model, name) in enumerate(
    [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]
):
    Z = model.transform(X)
    ax = axes[row, 0]
    ax.scatter(Z[:, 0], Z[:, 1], s=2, alpha=0.3, color="seagreen")
    circle = plt.Circle((0, 0), 3.0, color="black", fill=False, ls="--", lw=1)
    ax.add_patch(circle)
    ax.set_title(f"{name}: transformed (3σ circle)")
    ax.set_aspect("equal")

    grid = np.linspace(-4, 4, 200)
    for j in range(2):
        ax = axes[row, j + 1]
        ax.hist(Z[:, j], bins=60, density=True, alpha=0.6, color="seagreen")
        ax.plot(grid, stats.norm.pdf(grid), "k--", lw=1.5, label="N(0, 1)")
        ax.set_title(f"{name}: latent dim {j + 1}")
        ax.legend()
plt.tight_layout()
plt.show()

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for X, model, name in [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]:
    Z = model.transform(X)
    # Normality of each latent marginal (D'Agostino-Pearson).
    pvals = [stats.normaltest(Z[:, j]).pvalue for j in range(Z.shape[1])]
    cov = np.cov(Z.T)
    print(
        f"{name:7s} latent normaltest p-values: "
        f"{np.array2string(np.array(pvals), precision=3)}  "
        f"| cov diag: {np.diag(cov).round(2)} off-diag: {cov[0, 1]:+.3f}"
    )
for X, model, name in [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]:
    Z = model.transform(X)
    # Normality of each latent marginal (D'Agostino-Pearson).
    pvals = [stats.normaltest(Z[:, j]).pvalue for j in range(Z.shape[1])]
    cov = np.cov(Z.T)
    print(
        f"{name:7s} latent normaltest p-values: "
        f"{np.array2string(np.array(pvals), precision=3)}  "
        f"| cov diag: {np.diag(cov).round(2)} off-diag: {cov[0, 1]:+.3f}"
    )

Banana  latent normaltest p-values: [0.886 0.554]  | cov diag: [0.96 0.98] off-diag: -0.002
GMM     latent normaltest p-values: [8.739e-01 7.043e-04]  | cov diag: [0.96 0.98] off-diag: -0.017

Invertibility¶

Every layer is an exact bijection (the RQ spline inverse is analytic), so the full flow inverts to numerical precision.

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for X, model, name in [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]:
    X_rec = model.inverse_transform(model.transform(X))
    err = np.max(np.abs(X_rec - X))
    print(f"{name:7s} max |inverse(transform(X)) - X| = {err:.2e}")
for X, model, name in [(X_banana, gis_banana, "Banana"), (X_gmm, gis_gmm, "GMM")]:
    X_rec = model.inverse_transform(model.transform(X))
    err = np.max(np.abs(X_rec - X))
    print(f"{name:7s} max |inverse(transform(X)) - X| = {err:.2e}")

Banana  max |inverse(transform(X)) - X| = 4.88e-14
GMM     max |inverse(transform(X)) - X| = 5.11e-15

Brief Comparison: GIS vs RBIG¶

Same data, same evaluation: held-out mean log-likelihood, number of layers, and wall-clock fit time. (A detailed convergence study lives in the SIG vs RBIG convergence notebook.)

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X_test_banana = make_banana(n_samples, seed=42)
X_test_gmm = make_gmm(n_samples, seed=43)

results = {}
for X_tr, X_te, gis_model, name in [
    (X_banana, X_test_banana, gis_banana, "Banana"),
    (X_gmm, X_test_gmm, gis_gmm, "GMM"),
]:
    t0 = time.perf_counter()
    rbig_model = AnnealedRBIG(n_layers=100, random_state=0).fit(X_tr)
    t_rbig = time.perf_counter() - t0
    results[name] = {
        "GIS": (gis_model.score(X_te), gis_model.n_layers_),
        "RBIG": (rbig_model.score(X_te), len(rbig_model.layers_)),
        "t_rbig": t_rbig,
    }

print(f"{'data':8s} {'model':6s} {'test LL':>9s} {'layers':>7s}")
for name, res in results.items():
    for m in ["GIS", "RBIG"]:
        ll, nl = res[m]
        print(f"{name:8s} {m:6s} {ll:9.3f} {nl:7d}")
X_test_banana = make_banana(n_samples, seed=42)
X_test_gmm = make_gmm(n_samples, seed=43)

results = {}
for X_tr, X_te, gis_model, name in [
    (X_banana, X_test_banana, gis_banana, "Banana"),
    (X_gmm, X_test_gmm, gis_gmm, "GMM"),
]:
    t0 = time.perf_counter()
    rbig_model = AnnealedRBIG(n_layers=100, random_state=0).fit(X_tr)
    t_rbig = time.perf_counter() - t0
    results[name] = {
        "GIS": (gis_model.score(X_te), gis_model.n_layers_),
        "RBIG": (rbig_model.score(X_te), len(rbig_model.layers_)),
        "t_rbig": t_rbig,
    }

print(f"{'data':8s} {'model':6s} {'test LL':>9s} {'layers':>7s}")
for name, res in results.items():
    for m in ["GIS", "RBIG"]:
        ll, nl = res[m]
        print(f"{name:8s} {m:6s} {ll:9.3f} {nl:7d}")

data     model    test LL  layers
Banana   GIS       -2.867      20
Banana   RBIG      -3.349      22
GMM      GIS       -2.985      15
GMM      RBIG      -3.363      21

On these 2D targets GIS reaches a better held-out log-likelihood with a comparable (or smaller) layer count: each layer attacks the most non-Gaussian direction directly, and the RQ-spline marginals generalize more gracefully than histogram CDFs near the support boundary. The gap grows with dimension — see the convergence notebook.

Summary¶

GIS stacks layers that learn where to slice (Stiefel-manifold K-SWD maximization) and Gaussianize only those slices with RQ splines.
A handful of targeted layers Gaussianizes classic 2D benchmarks, and the flow inverts to numerical precision.
Compared to RBIG, GIS uses far fewer (but individually more expensive) layers for similar held-out log-likelihood.