optimal-transport-ica 0.1.1

Optimal Transport Independent Component Analysis

ot-ica: Linear Independent Component Analysis via Optimal Transport

ot-ica is a Python library that recovers independent source signals from linear mixtures by maximizing the 1D squared Wasserstein distance ($W_2^2$) between each projected component and a standard Gaussian.

Unlike classical ICA algorithms (FastICA, JADE, InfoMax) that use proxies (such as kurtosis or parametric approximations) that can fail on heterogeneous source distributions, ot-ica computes the exact 1D Wasserstein distance via quantile sorting. This provides an assumption-free, exact non-Gaussianity contrast function that is computed efficiently without density estimation or binning.

---

Key Features

• Exact Contrast Evaluation: Bypasses heuristic approximations by utilizing the exact 1D squared Wasserstein distance ($W_2^2$) computed analytically via sorted quantiles.
• Flexible Optimization Modes:
  • Deflationary: Extract independent components one by one.
  • Symmetric: Solve for all components simultaneously under orthogonal constraints.
• Diverse Optimizers:
  • Riemannian gradient ascent on the Stiefel manifold.
  • Standard Stochastic Gradient Descent (SGD) and L-BFGS.
  • A fast Fixed-Point update rule (OT-Mapping) for extremely quick convergence.
• Robust Costs: Supports standard L2 Wasserstein distance and a robust Huber-like logcosh Wasserstein cost to handle extreme outliers.
• Continuous Smoothing (Dithering): Inject continuous noise to smooth discrete CDF steps and avoid flat gradients.
• Fully Vectorized: Written in PyTorch, supporting GPU acceleration out of the box.

---

Installation

Install from PyPI:

pip install optimal-transport-ica

Or install from source:

git clone https://github.com/ashutoshjha3103/ot_ica.git
cd ot_ica
pip install .

Verify your installation:

python -c 'from ot_ica import WassersteinICA; print("OK")'

---

Quick Start

Here are three ways to use the ot_ica package for source separation:

1. Deflationary Optimization (Extracting One Component)

import numpy as np
import torch
from ot_ica import WassersteinICA

# Generate synthetic mixed data
rng = np.random.default_rng(0)
S = rng.laplace(0, 1, size=(3, 2000))          # 3 Laplace sources
A = rng.standard_normal((3, 3))                # Mixing matrix
X = torch.tensor(A @ S, dtype=torch.float32)   # Mixed signals

# Initialize and Whiten
ica = WassersteinICA(X)
ica.whiten()

# Optimize one component using deflationary gradient ascent
w_est, w2_score = ica.optimize_wasserstein2(continuous=True)

print("Estimated unmixing row:", w_est)
print("Wasserstein distance score:", float(w2_score))

2. Symmetric Optimization (Riemannian SGD on the Stiefel Manifold)

# Solves for all components simultaneously using Stiefel manifold SGD
W_est = ica.optimize_symmetric(n_components=3, optimizer='stiefel', lr=0.5)

# Reconstruct independent sources
S_est = torch.matmul(W_est, ica.X_white)
print("Estimated unmixing matrix W:\n", W_est)

3. Fast Fixed-Point Optimization (OT-Mapping)

# Runs a fast fixed-point rule by stepping away from the ideal Gaussian target
W_est_fp = ica.optimize_fixed_point(n_components=3, step_size=0.5)

# Reconstruct independent sources
S_est_fp = torch.matmul(W_est_fp, ica.X_white)
print("Fixed-Point unmixing matrix W:\n", W_est_fp)

---

Visualizing the Contrast

1. Mixing Two Sources (Central Limit Theorem)

Mixing two independent sources (e.g., Laplace and Uniform) results in a more Gaussian distribution due to the Central Limit Theorem. The 1D squared Wasserstein distance ($W_2^2$) provides the contrast by dipping toward the Gaussian mixture and peaking at the pure, non-Gaussian sources.

2. Contrast Function Race

Comparing how FastICA (logcosh), JADE, InfoMax, and OT-ICA ($W_2^2$) scan a 2D mixture. All four rotate the projection and scan for a non-Gaussianity peak, landing on the independent components with varying sharpness.

---

Citation

If you use this package or method in your research, please cite our paper:

@misc{jha2026linearindependentcomponentanalysis,
      title={Linear Independent Component Analysis via Optimal Transport}, 
      author={Ashutosh Jha and Michel Besserve and Simon Buchholz},
      year={2026},
      eprint={2607.14081},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2607.14081}, 
}

---

License

This project is licensed under the GNU General Public License v3 (GPLv3) - see the LICENSE file for details.