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

VolterraSys provides TensorFlow/Keras layers for trainable multidimensional linear and quadratic Volterra kernels in wavelet and natural bases.

Capabilities

• linear Volterra kernel layer for 1D data (shift variant): linearVolterra1D.
• Linear shift invariant wavelet and natural basis Volterra kernel layers for 1D, 2D, 3D data: LSIVolterra1D, LSIVolterra2D, LSIVolterra3D.
• Quadratic shift invariant wavelet and natural basis Volterra kernel layers for 1D, 2D, 3D data: QSIVolterra1D, QSIVolterra2D, QSIVolterra3D.
• Multiresolution Volterra kernels with orthogonal wavelets such as wave="haar" (default) and biorthogonal wavelets wave="bior1.3". Supports wavelet families: "db", "sym", "coif", "bior", "rbio"
• Natural-domain kernels have wave=None.

Dependencies

• TensorFlow (>=2.15)
• Inputs are TensorFlow tensors with channel-last layout.
• Wavelet domain requires the TFDWT package which can be installed via pip install TFDWT.

Installation

pip install VolterraSys

Minimal Example

import tensorflow as tf

# Linear shift invariant wavelet bases Volterra kernels
from volterrasys.LSIVolterra1D import LSIVolterra1D
from volterrasys.LSIVolterra2D import LSIVolterra2D
from volterrasys.LSIVolterra3D import LSIVolterra3D

# Quadratic shift invariant wavelet bases Volterra kernels
from volterrasys.QSIVolterra1D import QSIVolterra1D
from volterrasys.QSIVolterra2D import QSIVolterra2D
from volterrasys.QSIVolterra3D import QSIVolterra3D

# Natural-domain linear and quadratic kernels layer examples
# Linear
x1d = tf.random.normal([1, 32, 1])
y1d = LSIVolterra1D(filters=2, kernel_size=3, wave=None)(x1d)

# Quadratic
x2d = tf.random.normal([1, 8, 8, 1])
yq2d = QSIVolterra2D(filters=2, kernel_size=2, wave=None)(x2d)

x3d = tf.random.normal([1, 6, 6, 6, 1])
yq3d = QSIVolterra3D(filters=2, kernel_size=2, wave=None)(x3d)

print(y1d.shape, yq2d.shape, yq3d.shape)

Wavelet domain kernel layer examples

# Linear layers
layer1d = LSIVolterra1D(filters=1, kernel_size=4, wave="haar")
layer2d = LSIVolterra2D(filters=1, kernel_size=4, wave="haar")
layer3d = LSIVolterra3D(filters=1, kernel_size=4, wave="haar")

# Quadratic layers
qlayer1d = QSIVolterra1D(filters=1, kernel_size=4, wave='haar')
qlayer2d = QSIVolterra2D(filters=1, kernel_size=4, wave='haar')
qlayer3d = QSIVolterra3D(filters=1, kernel_size=4, wave='haar')

Citation

This software is released for broad research, educational, and engineering use. 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.

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