cplx-gmm

Complex-valued Gaussian mixture models with structured covariance matrices.

cplx-gmm provides a scikit-learn-style estimator for fitting Gaussian mixture models (GMMs) to complex-valued data using expectation-maximization (EM). It supports circularly symmetric complex Gaussian components and structured covariance types that are not available in the standard scikit-learn GaussianMixture, including circulant, block-circulant, Toeplitz, and block-Toeplitz covariance matrices.

✨ Highlights

• Complex-valued Gaussian mixture models for data in complex vector spaces
• scikit-learn-style estimator API
• Structured covariance models beyond standard scikit-learn GMMs
• Full, diagonal, spherical, circulant, block-circulant, Toeplitz, and block-Toeplitz covariance types
• Optional zero-mean component constraint
• Sampling from fitted complex-valued mixture models
• Retained FFT-domain fitted parameters for circulant and block-circulant models
• Tested covariance-structure correctness for non-trivial dimensions
• Modern Python packaging with pyproject.toml, uv, pytest, and ruff

📌 Citation

If you use cplx-gmm in academic work, please cite the package directly:

@software{fesl_cplx_gmm,
  author = {Fesl, Benedikt},
  title = {{cplx-gmm}: Complex-valued Gaussian mixture models with structured covariance matrices},
  year = {2026},
  url = {https://github.com/benediktfesl/cplx-gmm},
  version = {0.2.0}
}

Plain-text citation:

B. Fesl, cplx-gmm: Complex-valued Gaussian mixture models with structured covariance matrices, version 0.2.0. Available: https://github.com/benediktfesl/cplx-gmm

📦 Installation

Install from PyPI:

pip install cplx-gmm

or with uv:

uv add cplx-gmm

🧩 Covariance Structures

The covariance models are one of the main reasons to use this package. In addition to the usual full, diag, and spherical covariance types, cplx-gmm supports structured covariance matrices that are common in signal processing and wireless channel modeling.

covariance_type	Description
"full"	Full covariance matrix for each component.
"diag"	Diagonal covariance for each component.
"spherical"	One scalar variance per component.
"circulant"	Circulant covariance matrix for each component. Fitted in the Fourier domain and exposed in the original domain.
"block-circulant"	Block-circulant covariance matrix. Requires blocks=(n_1, n_2). Fitted with a two-dimensional FFT representation and exposed in the original domain.
"toeplitz"	Toeplitz covariance matrix for each component.
"block-toeplitz"	Block-Toeplitz covariance matrix. Requires blocks=(n_1, n_2).

For block-structured covariance types, pass the block dimensions in the constructor:

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="block-circulant",
    blocks=(4, 8),
    random_state=0,
)

model.fit(X)

Here, blocks=(4, 8) means that the feature dimension must satisfy n_features = 4 * 8.

🚀 Quick Start

import numpy as np

from cplx_gmm import GaussianMixtureCplx

rng = np.random.default_rng(0)

X = (
    rng.normal(size=(1_000, 8))
    + 1j * rng.normal(size=(1_000, 8))
) / np.sqrt(2)

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="full",
    random_state=0,
    max_iter=100,
    n_init=1,
)

model.fit(X)

labels = model.predict(X)
responsibilities = model.predict_proba(X)
log_likelihood = model.score(X)

samples, component_labels = model.sample(n_samples=100)

The estimator follows the usual scikit-learn pattern: model configuration is passed to the constructor, and fit(X) receives the data.

🧠 Estimator API

The main class is:

from cplx_gmm import GaussianMixtureCplx

Core methods:

Method	Description
fit(X, y=None)	Fit the complex-valued GMM.
fit_predict(X, y=None)	Fit the model and return component labels.
predict(X)	Predict the most likely component for each sample.
predict_proba(X)	Return posterior component probabilities.
score_samples(X)	Return per-sample log-likelihoods.
score(X, y=None)	Return the mean log-likelihood.
sample(n_samples=1)	Draw samples from the fitted mixture model.

Fitted parameters follow scikit-learn-style trailing-underscore names such as weights_, means_, covariances_, precisions_, precisions_cholesky_, converged_, n_iter_, and lower_bound_.

🔁 Fourier-Domain Parameters

For covariance_type="circulant" and covariance_type="block-circulant", fitting is performed in a Fourier-domain representation where the structured covariance becomes diagonal. After fitting, the public parameters means_ and covariances_ are transformed back to the original data domain.

Starting with version 0.2.0, the fitted Fourier-domain parameters are retained as additional fitted attributes.

For covariance_type="circulant":

Attribute	Description
means_fft_	Fitted component means in the FFT domain.
covariances_fft_	Diagonal FFT-domain covariance entries.
means_fft	Copy-returning property for means_fft_.
covariances_fft	Copy-returning property for covariances_fft_.

For covariance_type="block-circulant":

Attribute	Description
means_fft2_	Fitted component means in the two-dimensional FFT domain.
covariances_fft2_	Diagonal two-dimensional FFT-domain covariance entries.
means_fft2	Copy-returning property for means_fft2_.
covariances_fft2	Copy-returning property for covariances_fft2_.

Example:

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="circulant",
    random_state=0,
)

model.fit(X)

means_original_domain = model.means_
covariances_original_domain = model.covariances_

means_fourier_domain = model.means_fft_
covariances_fourier_domain = model.covariances_fft_

The original-domain attributes remain the primary public representation. The Fourier-domain attributes are useful for downstream estimators or algorithms that can exploit the diagonal covariance structure in the Fourier domain.

🔒 Zero-Mean Components

Some signal processing models assume zero-mean Gaussian components. This can be enforced with:

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="full",
    zero_mean=True,
)

model.fit(X)

When zero_mean=True, all component means are fixed to zero during fitting.

🔁 Circulant Covariances

Circulant covariance matrices are diagonalized by the discrete Fourier transform (DFT). For "circulant" and "block-circulant", the estimator fits a diagonal covariance model in the Fourier domain and transforms the fitted parameters back to the original domain.

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="circulant",
    random_state=0,
)

model.fit(X)

For block-circulant covariances, a two-dimensional FFT representation is used.

📐 Toeplitz Covariances

Toeplitz and block-Toeplitz covariance fitting uses an EM-based inverse covariance update inspired by:

T. A. Barton and D. R. Fuhrmann, “Covariance Estimation for Multidimensional Data using the EM Algorithm,” Proceedings of the 27th Asilomar Conference on Signals, Systems and Computers, 1993, pp. 203–207.

Example:

model = GaussianMixtureCplx(
    n_components=4,
    covariance_type="toeplitz",
    random_state=0,
)

model.fit(X)

🔥 Warm Start

warm_start=True is currently supported for the generic covariance types:

• "full"
• "diag"
• "spherical"

For structured covariance types, warm start is currently not enabled. This is planned as a future update.

📚 Research Background

This implementation was developed in the context of complex-valued Gaussian mixture modeling for wireless channel estimation and related signal processing applications.

The results of the following works are, in parts, based on the complex-valued implementation:

• M. Koller, B. Fesl, N. Turan, and W. Utschick, “An Asymptotically MSE-Optimal Estimator Based on Gaussian Mixture Models,” IEEE Transactions on Signal Processing, vol. 70, pp. 4109–4123, 2022.
  [IEEE] [arXiv]

• N. Turan, B. Fesl, M. Grundei, M. Koller, and W. Utschick, “Evaluation of a Gaussian Mixture Model-based Channel Estimator using Measurement Data,” International Symposium on Wireless Communication Systems (ISWCS), 2022.
  [IEEE] [arXiv]

• B. Fesl, M. Joham, S. Hu, M. Koller, N. Turan, and W. Utschick, “Channel Estimation based on Gaussian Mixture Models with Structured Covariances,” 56th Asilomar Conference on Signals, Systems, and Computers, 2022, pp. 533–537.
  [IEEE] [arXiv]

• B. Fesl, N. Turan, M. Joham, and W. Utschick, “Learning a Gaussian Mixture Model from Imperfect Training Data for Robust Channel Estimation,” IEEE Wireless Communications Letters, 2023.
  [IEEE] [arXiv]

• M. Koller, B. Fesl, N. Turan, and W. Utschick, “An Asymptotically Optimal Approximation of the Conditional Mean Channel Estimator Based on Gaussian Mixture Models,” IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP), 2022, pp. 5268–5272.
  [IEEE] [arXiv]

• B. Fesl, A. Faika, N. Turan, M. Joham, and W. Utschick, “Channel Estimation with Reduced Phase Allocations in RIS-Aided Systems,” IEEE 24th International Workshop on Signal Processing Advances in Wireless Communications (SPAWC), 2023, pp. 161–165.
  [IEEE] [arXiv]

• N. Turan, B. Fesl, M. Koller, M. Joham, and W. Utschick, “A Versatile Low-Complexity Feedback Scheme for FDD Systems via Generative Modeling,” IEEE Transactions on Wireless Communications, 2023.
  [IEEE] [arXiv]

• N. Turan, B. Fesl, and W. Utschick, “Enhanced Low-Complexity FDD System Feedback with Variable Bit Lengths via Generative Modeling,” 57th Asilomar Conference on Signals, Systems, and Computers, 2023.
  [IEEE] [arXiv]

• N. Turan, M. Koller, B. Fesl, S. Bazzi, W. Xu, and W. Utschick, “GMM-based Codebook Construction and Feedback Encoding in FDD Systems,” 56th Asilomar Conference on Signals, Systems, and Computers, 2022, pp. 37–42.
  [IEEE] [arXiv]

• ... and more

🧪 Development

Clone the repository and install the development environment with uv:

git clone https://github.com/benediktfesl/cplx-gmm.git
cd cplx-gmm
uv sync

Run tests:

uv run pytest

Run linting:

uv run ruff check .

✅ Test Coverage

The test suite covers:

• package imports
• scikit-learn-style estimator API
• validation behavior
• all supported covariance types
• structural covariance correctness
• retained FFT-domain fitted parameters
• EM lower-bound monotonicity
• zero-mean fitting
• initialization options
• warm-start behavior for generic covariance types
• explicit rejection of warm start for structured covariance types
• reproducibility with fixed random_state
• sampling behavior
• real-valued compatibility checks
• doubled real-valued likelihood equivalence
• example execution

📄 License

This project is licensed under the BSD 3-Clause License.

The implementation is based on ideas and portions of the original scikit-learn Gaussian mixture implementation, which is also distributed under the BSD 3-Clause License.

See LICENSE for details.