🌱 rkhs

rkhs is a small Python framework for marginal and conditional two-sample testing with kernels in JAX.

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⚙️ Installation

Coming soon

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🚀 Features

rkhs provides JAX-native two-sample tests (marginal, conditional, mixed) based on kernel embeddings with analytical or bootstrap confidence bounds, a simple API, and pluggable kernels.

Three test modes — one API

You can test the following two-sample hypotheses with one common set of primitives:

• Marginal: $H_0: P = Q$
• Conditional: $H_0(x_1,x_2): P(\cdot\mid X=x_1) = Q(\cdot\mid X=x_2)$
• Mixed: $H_0(x): P(\cdot\mid X=x) = Q$

The test compares kernel embeddings in RKHS norm and rejects $H_0$ at level $\alpha$ if

$$ |\hat\mu_P - \hat\mu_Q|_\mathcal{H} > \beta_P + \beta_Q \quad, $$

where $\beta_\ast$ are finite-sample confidence radii from the selected regime.

Confidence regimes

Analytical bounds. Finite-sample guarantees under the stated assumptions (conservative, little overhead).

Bootstrap bounds. Data-driven thresholds with typically higher power (cost scales with the number of resamples).

JAX integration

Works with jit/vmap, runs on CPU/GPU/TPU, and uses explicit PRNGKey for reproducibility.

Kernels

Popular kernels are built in: Gaussian, Matern, Laplacian, Polynomial, Linear.

Conditional tests use a scalar kernel on the input domain and a separate kernel on the output domain: VectorKernel(x=..., y=..., regularization=...).

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🧩 Usage

1) Marginal two-sample test (analytical bounds)

import jax
from rkhs.testing import TestEmbedding, TwoSampleTest
from rkhs.kernels import GaussianKernel

# toy data: two 3D Gaussians with different means
xs_1 = jax.random.normal(key=jax.random.key(1), shape=(200, 3))
xs_2 = jax.random.normal(key=jax.random.key(2), shape=(200, 3)) + 1.0

# kernel on the sample space
kernel = GaussianKernel(bandwidth=1.5, data_shape=(3,))

# embedding + analytical confidence radius
kme_1 = TestEmbedding.analytical(
    kme=kernel.kme(xs_1),   # embed dataset in RKHS
    kernel_bound=1.0        # sup_x k(x, x)
)
kme_2 = TestEmbedding.analytical(
    kme=kernel.kme(xs_2),   # embed dataset in RKHS
    kernel_bound=1.0        # sup_x k(x, x)
) 

# level-α test
test = TwoSampleTest.from_embeddings(kme_1, kme_2, level=0.05)

decision = test.reject      # boolean (reject H_0?)
distance = test.distance    # RKHS distance
threshold = test.threshold  # β_P + β_Q
print(decision, distance, threshold)

2) Marginal test (bootstrap bounds)

import jax
from rkhs.testing import TestEmbedding, TwoSampleTest
from rkhs.kernels import GaussianKernel

# toy data: two 3D Gaussians with different means
xs_1 = jax.random.normal(key=jax.random.key(1), shape=(200, 3))
xs_2 = jax.random.normal(key=jax.random.key(2), shape=(200, 3)) + 1.0

# kernel on the sample space
kernel = GaussianKernel(bandwidth=1.5, data_shape=(3,))

# embedding + analytical confidence radius
kme_1 = TestEmbedding.bootstrap(
    kme=kernel.kme(xs_1),   # embed dataset in RKHS
    key=jax.random.key(3),  # random key
    n_bootstrap=1000        # number of bootstrap resamples
)
kme_2 = TestEmbedding.bootstrap(
    kme=kernel.kme(xs_2),   # embed dataset in RKHS
    key=jax.random.key(4),  # random key
    n_bootstrap=1000        # number of bootstrap resamples
) 

# level-α test
test = TwoSampleTest.from_embeddings(kme_1, kme_2, level=0.05)

decision = test.reject      # boolean (reject H_0?)
distance = test.distance    # RKHS distance
threshold = test.threshold  # β_P + β_Q
print(decision, distance, threshold)

3) Conditional two-sample test at selected covariates

import jax
from rkhs import VectorKernel
from rkhs.testing import ConditionalTestEmbedding, TwoSampleTest
from rkhs.kernels import GaussianKernel

# synthetic x,y pairs with additive noise
xs_1 = jax.random.normal(key=jax.random.key(1), shape=(1000, 3))
ys_1 = xs_1 + jax.random.normal(key=jax.random.key(2), shape=(1000, 3)) * 0.05

xs_2 = jax.random.normal(key=jax.random.key(3), shape=(1000, 3))
ys_2 = xs_2 + jax.random.normal(key=jax.random.key(4), shape=(1000, 3)) * 0.05 + 0.5

# inputs at which to test for distributional equality: H_0(x): P(. | x) =? Q(. | x), for x in grid
covariates = jax.numpy.linspace(jax.numpy.array([-3, -3, -3]), jax.numpy.array([3, 3, 3]), num=100)

# vector-valued kernel over inputs X and outputs Y
kernel = VectorKernel(
    x=GaussianKernel(bandwidth=0.5, data_shape=(3,)),  # kernel used for ridge regression
    y=GaussianKernel(bandwidth=1.0, data_shape=(3,)),  # kernel used for embedding marginal distribution at each covariate
    regularization=0.1
)

# conditional embedding + bootstrap confidence radius
cme_1 = ConditionalTestEmbedding.bootstrap(
    cme=kernel.cme(xs_1, ys_1), # embed dataset in vector-valued RKHS
    grid=covariates,            # covariates used in bootstrap of threshold parameters
    key=jax.random.key(5),      # random key
    n_bootstrap=100             # number of bootstrap resamples
)

cme_2 = ConditionalTestEmbedding.bootstrap(
    cme=kernel.cme(xs_2, ys_2), # embed dataset in vector-valued RKHS
    grid=covariates,            # covariates used in bootstrap of threshold parameters
    key=jax.random.key(6),      # random key
    n_bootstrap=100             # number of bootstrap resamples
)

# evaluate CMEs at covariates -> embeds each distribution over Y in RKHS of `kernel.y`
kme_1 = cme_1(covariates)
kme_2 = cme_2(covariates)

# batched test across all covariates
test = TwoSampleTest.from_embeddings(kme_1, kme_2, level=0.05)
reject_per_x = test.reject   # Boolean array

decision = test.reject      # boolean (reject H_0(x)?, individually for each covariate). shape: covariates.shape
distance = test.distance    # RKHS distance. shape: covariates.shape
threshold = test.threshold  # β_P + β_Q
print(decision, distance, threshold)

4) Mixed test: $P(\cdot\mid X=x)$ vs. $Q$

import jax
from rkhs import VectorKernel
from rkhs.testing import TestEmbedding, ConditionalTestEmbedding, TwoSampleTest
from rkhs.kernels import GaussianKernel

# dataset from marginal distribution over Y
ys_1 = jax.random.normal(key=jax.random.key(1), shape=(1000, 3)) * 0.05

# synthetic x,y pairs with additive noise
xs_2 = jax.random.normal(key=jax.random.key(2), shape=(1000, 3))
ys_2 = xs_2 + jax.random.normal(key=jax.random.key(3), shape=(1000, 3)) * 0.05

# inputs at which to test for distributional equality: H_0(x): P =? Q(. | x), for x in grid
covariates = jax.numpy.linspace(jax.numpy.array([-3, -3, -3]), jax.numpy.array([3, 3, 3]), num=200)

y_kernel = GaussianKernel(bandwidth=1.0, data_shape=(3,))

# vector-valued kernel over inputs X and outputs Y
vector_kernel = VectorKernel(
    x=GaussianKernel(bandwidth=0.5, data_shape=(3,)),  # kernel used for ridge regression
    y=y_kernel,                                        # kernel used for embedding marginal distribution at each covariate
    regularization=0.1
)

# embedding + analytical confidence radius (can be drop-in replaced with bootstrap radius)
kme_1 = TestEmbedding.analytical(
    kme=y_kernel.kme(ys_1), # embed dataset in RKHS
    kernel_bound=1.0        # sup_x k(x, x)
)

# conditional embedding + analytical confidence radius
cme_2 = ConditionalTestEmbedding.bootstrap(
    cme=vector_kernel.cme(xs_2, ys_2),  # embed dataset in vector-valued RKHS
    grid=covariates,                    # covariates used in bootstrap of threshold parameters
    key=jax.random.key(4),              # random key
    n_bootstrap=100                     # number of bootstrap resamples
)

# evaluate CME at covariates -> embeds each distribution over Y in RKHS of `kernel.y`
kme_2 = cme_2(covariates)

# batched test across all covariates
test = TwoSampleTest.from_embeddings(kme_1, kme_2, level=0.05)
reject_per_x = test.reject   # Boolean array

decision = test.reject      # boolean (reject H_0(x)?, individually for each covariate). shape: covariates.shape
distance = test.distance    # RKHS distance. shape: covariates.shape
threshold = test.threshold  # β_P + β_Q
print(decision, distance, threshold)

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🔍 Kernel quick reference

• LinearKernel — compares means (first moment).
• PolynomialKernel(degree=d) — compares moments up to degree $d$.
• Gaussian, Matern, Laplacian — characteristic; compare full distributions.

For conditional tests:

• Input kernel (x): used to learn the conditional embedding (not for comparison).
• Output kernel (y): determines what aspects of the conditional law are compared.

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🧠 Notes

• Embeddings preserve batch axes; passing a batch of covariates returns a batch of embeddings.
• All randomness is explicit via jax.random.PRNGKey.
• You can use your own custom kernel by extending rkhs.Kernel:

from jax import Array
from rkhs import Kernel
import jax

class MyCustomKernel(Kernel):
    def __init__(self, data_shape: tuple[int, ...]):
        super().__init__(data_shape)
        ...
    
    def _dot(self, x1: Array, x2: Array) -> Array:
        ...  # your logic here (must be jit-compilable)

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📚 References

• Marginal test: Gretton, A., et al. (2012). A Kernel Two-Sample Test. JMLR page · PDF

• Conditional test: Massiani, P.-F., et al. (2025). A Kernel Conditional Two-Sample Test. arXiv · PDF