VolterraSys-PyTorch: Multidimensional linear and nonlinear Volterra kernel layers in wavelet and natural bases

VolterraSys-PyTorch provides PyTorch modules for trainable 1D linear and quadratic Volterra kernels in wavelet and natural bases. Tensors use channels-last layout: (batch, length, channels).

Capabilities

• Linear shift-invariant 1D kernels: LSIVolterra1D.
• Quadratic shift-invariant 1D kernels: QSIVolterra1D.
• Linear shift-variant 1D kernels: LinearVolterra1D.
• Orthogonal and biorthogonal wavelet-domain computation through fdwt.
• Natural-domain computation with wave=None.
• Trainable kernels with PyTorch autograd, optimizers, and state_dict serialization.

Installation

pip install VolterraSys-PyTorch

Minimal example

import torch

from volterrasys.LSIVolterra1D import LSIVolterra1D
from volterrasys.QSIVolterra1D import QSIVolterra1D
from volterrasys.LinearVolterra1D import LinearVolterra1D

x = torch.randn(2, 32, 1)

lsi = LSIVolterra1D(filters=2, kernel_size=4, wave="bior1.3")
qsi = QSIVolterra1D(filters=2, kernel_size=4, wave="haar")
linear = LinearVolterra1D(filters=2, Ny=16, wave=None)

y_lsi = lsi(x)       # (2, 32, 2)
y_qsi = qsi(x)       # (2, 32, 2)
y_linear = linear(x) # (2, 16, 2)

The wavelet-domain and natural-domain implementations use the same channels-last tensor layout. filters is the number of output channels.

TensorFlow provenance

This repository is an operation-by-operation PyTorch port of kkt-ee/VolterraSys. The port is based on TensorFlow commit 4ef3877 from bugfix/biortho-linear1d, marked by the tf-final-before-pytorch-port tag. The original TensorFlow Git history is retained as this repository's ancestry.

The port preserves the TensorFlow kernel construction, circular indexing, wavelet-basis transformations, and einsum contractions. The corresponding PyTorch dependency is fdwt.

Citation

If this package proves useful in related work, please cite the following thesis, whose Chapter 2 presents the underlying theory and computational details:

@misc{tarafdar2026interpretablefrugallearningsystems,
      title={Interpretable and Frugal Learning Systems Employing Multiresolution Pyramids and Volterra Kernels},
      author={Kishore Kumar Tarafdar},
      year={2026},
      eprint={2606.15011},
      archivePrefix={arXiv},
      primaryClass={eess.SP},
      url={https://arxiv.org/abs/2606.15011},
}

License

Apache License 2.0. See LICENSE.

VolterraSys-PyTorch (C) 2026 Kishore Kumar Tarafdar, भारत 🇮🇳