RIEnet Torch: A Rotational Invariant Estimator Network for GMV Optimization

This library implements the neural estimators introduced in:

• Bongiorno, C., Manolakis, E., & Mantegna, R. N. (2026). End-to-End Large Portfolio Optimization for Variance Minimization with Neural Networks through Covariance Cleaning. The Journal of Finance and Data Science: 100179. 10.1016/j.jfds.2026.100179
• Bongiorno, C., Manolakis, E., & Mantegna, R. N. (2025). Neural Network-Driven Volatility Drag Mitigation under Aggressive Leverage. In Proceedings of the 6th ACM International Conference on AI in Finance (ICAIF ’25). 10.1145/3768292.3770370

RIEnet Torch is a PyTorch implementation for end-to-end global minimum-variance portfolio construction.

Given a tensor of asset returns, the model estimates a structured covariance / precision representation and produces analytic GMV portfolio weights in a single forward pass.

This repository is intended for:

• research and methodological replication,
• experimentation on large equity universes,
• integration into quantitative portfolio construction workflows,
• deployment as a standalone PyTorch package.

For the original TensorFlow implementation, see the RIEnet repository.

Release Notes

0.1.5

This release fixes three correctness issues found during parity review against the TensorFlow implementation:

• EigenWeightsLayer now computes exact GMV weights for both orthonormal and positive row-scaled eigenvectors when inverse_std is provided.
• The dimension-aware q feature now uses the random-matrix ratio n_stocks / n_days instead of its reciprocal. This affects the default RIEnetLayer configuration.
• LagTransformLayer now evaluates the continuous tanh(x) / x form near zero, preventing the compact branch from exploding near beta = -epsilon and removing the factor-of-two bias in the per-lag small-beta limit.

Users should upgrade to 0.1.5. Existing checkpoints trained with the previous q definition should be evaluated carefully because their dimensional input semantics have changed.

0.1.4

This release fixes a correctness bug in EigenWeightsLayer when inverse_std is provided. Earlier versions applied inverse standard deviations only to the left side of the inverse-covariance product, so the normalized result was not generally the exact unconstrained GMV portfolio. The module now applies the scaling on both sides and matches the direct solution obtained from the full covariance matrix while retaining the efficient spectral computation. Users who call EigenWeightsLayer with inverse_std should upgrade to 0.1.4.

The variance loss now also rejects incompatible shapes and non-finite inputs instead of silently broadcasting them or propagating invalid values.

What this package provides

• End-to-end training on a realized-variance objective for GMV portfolios
• Access to portfolio weights, cleaned covariance matrices, and precision matrices
• A dimension-agnostic architecture suitable for large cross-sectional universes
• A PyTorch implementation aligned with the published methodology

Evidence in published experiments

The empirical properties of the method are documented in the associated papers.

In particular, the published experiments evaluate the model on large equity universes under a global minimum-variance objective and compare it against standard covariance-based benchmarks.

For details on datasets, training protocol, benchmark definitions, and evaluation metrics, please refer to the papers listed above.

Module Organization

• rienet_torch.trainable_layers: modules with trainable parameters (RIEnetLayer, LagTransformLayer, DeepLayer, DeepRecurrentLayer, CorrelationEigenTransformLayer).
• rienet_torch.ops_layers: deterministic operation modules (statistics, normalization, eigensystem algebra, weight post-processing).
• rienet_torch.losses: the GMV variance objective.
• rienet_torch.serialization: Torch-native save/load helpers.

Installation

Install from PyPI:

pip install rienet-torch

Or install from source:

cd /path/to/RIEnet-torch
pip install -e .

For development:

pip install -e ".[dev]"

Quick Start

Basic Usage

import torch
from rienet_torch import RIEnetLayer, variance_loss_function

# Defaults reproduce the compact GMV architecture
# (bidirectional GRU with 16 units, 8-unit volatility head)
rienet_layer = RIEnetLayer(output_type=["weights", "precision"])

# Sample data: (batch_size, n_stocks, n_days)
returns = torch.randn(32, 10, 60) * 0.02

# Retrieve GMV weights and cleaned precision in one pass
outputs = rienet_layer(returns)
weights = outputs["weights"]      # (32, 10, 1)
precision = outputs["precision"]  # (32, 10, 10)

# GMV training objective
covariance = torch.randn(32, 10, 10)
covariance = covariance @ covariance.transpose(-1, -2)
loss = variance_loss_function(covariance, weights)
print(loss.shape)  # torch.Size([32, 1, 1])

Training with the GMV Variance Loss

import torch
from rienet_torch import RIEnetLayer, variance_loss_function

model = RIEnetLayer(output_type="weights")
optimizer = torch.optim.Adam(model.parameters(), lr=1e-4)

# Synthetic training data
X_train = torch.randn(256, 10, 60) * 0.02
Sigma_train = torch.randn(256, 10, 10)
Sigma_train = Sigma_train @ Sigma_train.transpose(-1, -2)

model.train()
for step in range(100):
    optimizer.zero_grad()
    weights = model(X_train)
    loss = variance_loss_function(Sigma_train, weights).mean()
    loss.backward()
    torch.nn.utils.clip_grad_norm_(model.parameters(), max_norm=1.0)
    optimizer.step()

Tip: When you intend to deploy RIEnet on portfolios of varying size, train on batches that span different asset universes. The RIE-based architecture is dimension agnostic and benefits from heterogeneous training shapes.

Using Different Output Types

import torch
from rienet_torch import RIEnetLayer

returns = torch.randn(32, 10, 60) * 0.02

# GMV weights only
weights = RIEnetLayer(output_type="weights")(returns)

# Precision matrix only
precision_matrix = RIEnetLayer(output_type="precision")(returns)

# Precision, covariance, and lag-transformed inputs in one pass
outputs = RIEnetLayer(
    output_type=["precision", "covariance", "input_transformed"]
)(returns)
precision_matrix = outputs["precision"]
covariance_matrix = outputs["covariance"]
lagged_inputs = outputs["input_transformed"]

# Spectral components (non-inverse)
spectral = RIEnetLayer(
    output_type=["eigenvalues", "eigenvectors", "transformed_std"]
)(returns)
cleaned_eigenvalues = spectral["eigenvalues"]   # (batch, n_stocks, 1)
eigenvectors = spectral["eigenvectors"]         # (batch, n_stocks, n_stocks)
transformed_std = spectral["transformed_std"]   # (batch, n_stocks, 1)

# Optional: disable variance normalisation
raw_covariance = RIEnetLayer(
    output_type="covariance",
    normalize_transformed_variance=False
)(returns)

When RIEnet is trained end-to-end on the GMV variance loss, leave normalize_transformed_variance=True (the default). The loss is invariant to global covariance rescalings and the layer keeps the implied variance scale centred around one. Disable the normalization only when using alternative objectives where the absolute volatility scale must be preserved.

Using LagTransformLayer Directly

LagTransformLayer is exposed both at package root and in the dedicated module:

import torch
from rienet_torch import LagTransformLayer
# or: from rienet_torch.lag_transform import LagTransformLayer

# Dynamic lookback (T can change call-by-call)
compact = LagTransformLayer(variant="compact")
y1 = compact(torch.randn(4, 12, 20))
y2 = compact(torch.randn(4, 12, 40))

# Fixed lookback inferred at first build/call (requires static T)
per_lag = LagTransformLayer(variant="per_lag")
z1 = per_lag(torch.randn(4, 12, 20))
z2 = per_lag(torch.randn(4, 8, 20))  # n_assets can change

Using EigenWeightsLayer Directly

EigenWeightsLayer is part of the public API and can be imported directly:

import torch
from rienet_torch import EigenWeightsLayer

layer = EigenWeightsLayer(name="gmv_weights")

eigenvectors = torch.linalg.qr(torch.randn(8, 20, 20)).Q
inverse_eigenvalues = torch.rand(8, 20, 1)
inverse_std = torch.rand(8, 20, 1)

# Exact GMV branch from a correlation eigensystem and inverse standard deviations
weights = layer(eigenvectors, inverse_eigenvalues, inverse_std)

# Covariance-eigensystem branch (inverse_std omitted)
weights_cov = layer(eigenvectors, inverse_eigenvalues)

Notes:

• inverse_std is optional by design.
• When inverse_std is provided, the eigenvectors and inverse eigenvalues must describe the corresponding correlation matrix. Eigenvectors may be orthonormal or positive row-scaled orthonormal vectors. The module computes exact unconstrained GMV weights without constructing or inverting the covariance matrix.
• If inverse_std is omitted, the eigensystem is interpreted as belonging directly to the covariance matrix.
• Output shape is always (..., n_assets, 1), normalized to sum to one along assets.

Using CorrelationEigenTransformLayer Directly

import torch
from rienet_torch import CorrelationEigenTransformLayer

layer = CorrelationEigenTransformLayer(name="corr_cleaner")

# Correlation matrix: (batch, n_assets, n_assets)
corr = torch.eye(6).expand(4, 6, 6).clone()

# Optional attributes: (batch, k) e.g. q, lookback, regime flags, etc.
attrs = torch.tensor([
    [0.5, 60.0],
    [0.7, 60.0],
    [1.2, 30.0],
    [0.9, 90.0],
], dtype=torch.float32)

# With attributes (default output_type='correlation')
cleaned_corr = layer(corr, attributes=attrs)

# Request multiple outputs
details = layer(
    corr,
    attributes=attrs,
    output_type=[
        "correlation",
        "inverse_correlation",
        "eigenvalues",
        "eigenvectors",
        "inverse_eigenvalues",
    ],
)
cleaned_eigvals = details["eigenvalues"]              # (batch, n_assets, 1)
cleaned_inv_eigvals = details["inverse_eigenvalues"]  # (batch, n_assets, 1)
inv_corr = details["inverse_correlation"]             # (batch, n_assets, n_assets)

# Without attributes
cleaned_corr_no_attr = CorrelationEigenTransformLayer(name="corr_cleaner_no_attr")(corr)

Notes:

• attributes is optional and can have shape (batch, k) or (batch, n_assets, k).
• The output is a cleaned correlation matrix (batch, n_assets, n_assets).
• If you change attribute width k, use a new layer instance.

Loss Function

Variance Loss Function

from rienet_torch import variance_loss_function

loss = variance_loss_function(
    covariance_true=true_covariance,    # (batch_size, n_assets, n_assets)
    weights_predicted=predicted_weights # (batch_size, n_assets, 1)
)

Mathematical Formula:

Loss = n_assets x w^T Sigma w

Where w are the portfolio weights and Sigma is the realised covariance matrix.

Serialization

The package uses PyTorch-native serialization:

from rienet_torch import RIEnetLayer
from rienet_torch.serialization import save_module, load_module

model = RIEnetLayer(output_type="weights")
save_module(model, "rienet.pt")

restored = load_module(RIEnetLayer, "rienet.pt")

This stores:

• the state_dict,
• the module config when get_config() is available,
• the lazy-build metadata needed to materialize shape-dependent parameters.

Architecture Details

The RIEnet pipeline consists of:

1. Input Scaling: annualise returns by 252
2. Lag Transformation: five-parameter memory kernel for temporal weighting
3. Covariance Estimation: sample covariance across assets
4. Eigenvalue Decomposition: spectral analysis of the covariance matrix
5. Recurrent Cleaning: bidirectional GRU/LSTM processing of eigen spectra
6. Marginal Volatility Head: dense network forecasting inverse standard deviations
7. Matrix Reconstruction: RIE-based synthesis of Sigma^{-1} and GMV weight normalisation

Paper defaults use a single bidirectional GRU layer with 16 units per direction and a marginal-volatility head with 8 hidden units, matching the compact network described in Bongiorno et al. (2025).

Requirements

• Python >= 3.10
• PyTorch >= 2.5.0
• NumPy >= 1.26.0

Development

cd /path/to/RIEnet-torch
pip install -e ".[dev]"
python -m pytest tests/

Release Automation

This repository is ready for automated publishing with GitHub Actions:

• CI workflow: .github/workflows/ci.yml
• PyPI publish workflow: .github/workflows/publish.yml
• TestPyPI publish workflow: .github/workflows/publish-testpypi.yml

The release checklist is documented in RELEASING.md.

Citation

Please cite the following references when using RIEnet:

@article{bongiorno2026end,
  title={End-to-end large portfolio optimization for variance minimization with neural networks through covariance cleaning},
  author={Bongiorno, Christian and Manolakis, Efstratios and Mantegna, Rosario Nunzio},
  journal={The Journal of Finance and Data Science},
  pages={100179},
  year={2026},
  publisher={Elsevier}
}

@inproceedings{bongiorno2025Neural,
  author = {Bongiorno, Christian and Manolakis, Efstratios and Mantegna, Rosario Nunzio},
  title = {Neural Network-Driven Volatility Drag Mitigation under Aggressive Leverage},
  year = {2025},
  isbn = {9798400722202},
  publisher = {Association for Computing Machinery},
  address = {New York, NY, USA},
  url = {https://doi.org/10.1145/3768292.3770370},
  doi = {10.1145/3768292.3770370},
  booktitle = {Proceedings of the 6th ACM International Conference on AI in Finance},
  pages = {449–455},
  numpages = {7},
  location = {},
  series = {ICAIF '25}
}

For software citation:

@software{rienet_torch2026,
  title={RIEnet Torch: A Rotational Invariant Estimator Network for Global Minimum-Variance Optimisation},
  author={Bongiorno, Christian},
  year={2026},
  version={0.1.0},
  url={https://github.com/your-user/rienet-torch}
}

You can print citation information programmatically:

import rienet_torch
rienet_torch.print_citation()

Support

For questions, issues, or contributions, please:

• Open an issue on GitHub
• Check the documentation
• Contact Prof. Christian Bongiorno (christian.bongiorno@centralesupelec.fr) for calibrated model weights or collaboration requests