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Clifford and Geometric Algebra with TensorFlow

# TFGA - TensorFlow Geometric Algebra

Python package for Geometric / Clifford Algebra with TensorFlow 2.

This project is a work in progress. Its API may change and the examples aren't polished yet.

Pull requests and suggestions either by opening an issue or by sending me an email are welcome.

## Installation

Install using pip: pip install tfga

Requirements:

• Python 3
• tensorflow 2
• numpy

## Basic usage

There are two ways to use this library. In both ways we first create a GeometricAlgebra instance given a metric. Then we can either work on tf.Tensor instances directly where the last axis is assumed to correspond to the algebra's blades.

import tensorflow as tf
from tfga import GeometricAlgebra

# Create an algebra with 3 basis vectors given their metric.
# Contains geometric algebra operations.
ga = GeometricAlgebra(metric=[1, 1, 1])

# Create geometric algebra tf.Tensor for vector blades (ie. e_0 + e_1 + e_2).
# Represented as tf.Tensor with shape [8] (one value for each blade of the algebra).
# tf.Tensor: [0, 1, 1, 1, 0, 0, 0, 0]
ordinary_vector = ga.from_tensor_with_kind(tf.ones(3), kind="vector")

# 5 + 5 e_01 + 5 e_02 + 5 e_12
quaternion = ga.from_tensor_with_kind(tf.fill(dims=4, value=5), kind="even")

# 5 + 1 e_0 + 1 e_1 + 1 e_2 + 5 e_01 + 5 e_02 + 5 e_12
multivector = ordinary_vector + quaternion

# Inner product e_0 | (e_0 + e_1 + e_2) = 1
# ga.print is like print, but has extra formatting for geometric algebra tf.Tensor instances.
ga.print(ga.inner_prod(ga.e0, ordinary_vector))

# Exterior product e_0 ^ e_1 = e_01.
ga.print(ga.ext_prod(ga.e0, ga.e1))

# Grade reversal ~(5 + 5 e_01 + 5 e_02 + 5 e_12)
# = 5 + 5 e_10 + 5 e_20 + 5 e_21
# = 5 - 5 e_01 - 5 e_02 - 5 e_12
ga.print(ga.reversion(quaternion))

# tf.Tensor 5
ga.print(quaternion[0])

# tf.Tensor of shape [1]: -5 (ie. reversed sign of e_01 component)
ga.print(ga.select_blades_with_name(quaternion, "10"))

# tf.Tensor of shape [8] with only e_01 component equal to 5
ga.print(ga.keep_blades_with_name(quaternion, "10"))


Alternatively we can convert the geometric algebra tf.Tensor instance to MultiVector instances which wrap the operations and provide operator overrides for convenience. This can be done by using the __call__ operator of the GeometricAlgebra instance.

# Create geometric algebra tf.Tensor instances
a = ga.e123
b = ga.e1

# Wrap them as MultiVector instances
mv_a = ga(a)
mv_b = ga(b)

# Reversion ((~mv_a).tensor equivalent to ga.reversion(a))
print(~mv_a)

# Geometric / inner / exterior product
print(mv_a * mv_b)
print(mv_a | mv_b)
print(mv_a ^ mv_b)


## Keras layers

TFGA also provides Keras layers which provide layers similar to the existing ones but using multivectors instead. For example the GeometricProductDense layer is exactly the same as the Dense layer but uses multivector-valued weights and biases instead of scalar ones. The exact kind of multivector-type can be passed too. Example:

import tensorflow as tf
from tfga import GeometricAlgebra
from tfga.layers import TensorToGeometric, GeometricToTensor, GeometricProductDense

# 4 basis vectors (e0^2=+1, e1^2=-1, e2^2=-1, e3^2=-1)
sta = GeometricAlgebra([1, -1, -1, -1])

# We want our dense layer to perform a matrix multiply
# with a matrix that has vector-valued entries.
vector_blade_indices = sta.get_kind_blade_indices(BladeKind.VECTOR),

# Create our input of shape [Batch, Units, BladeValues]
tensor = tf.ones([20, 6, 4])

# The matrix-multiply will perform vector * vector
# so our result will be scalar + bivector.
# Use the resulting blade type for the bias too which is
# added to the result.
result_indices = tf.concat([
sta.get_kind_blade_indices(BladeKind.SCALAR), # 1 index
sta.get_kind_blade_indices(BladeKind.BIVECTOR) # 6 indices
], axis=0)

sequence = tf.keras.Sequential([
# Converts the last axis to a dense multivector
# (so, 4 -> 16 (total number of blades in the algebra))
TensorToGeometric(sta, blade_indices=vector_blade_indices),
# Perform matrix multiply with vector-valued matrix
GeometricProductDense(
algebra=sta, units=8, # units is analagous to Keras' Dense layer
blade_indices_kernel=vector_blade_indices,
blade_indices_bias=result_indices
),
# Extract our wanted blade indices (last axis 16 -> 7 (1+6))
GeometricToTensor(sta, blade_indices=result_indices)
])

# Result will have shape [20, 8, 7]
result = sequence(tensor)


### Available layers

Class Description
GeometricProductDense Analagous to Keras' Dense with multivector-valued weights and biases. Each term in the matrix multiplication does the geometric product x * w.
GeometricSandwichProductDense Analagous to Keras' Dense with multivector-valued weights and biases. Each term in the matrix multiplication does the geometric product w *x * ~w.
GeometricProductElementwise Performs multivector-valued elementwise geometric product of the input units with a different weight for each unit.
GeometricSandwichProductElementwise Performs multivector-valued elementwise geometric sandwich product of the input units with a different weight for each unit.
GeometricProductConv1D Analagous to Keras' Conv1D with multivector-valued kernels and biases. Each term in the kernel multiplication does the geometric product x * k.
TensorToGeometric Converts from a tf.Tensor to the geometric algebra tf.Tensor with as many blades on the last axis as basis blades in the algebra where blade indices determine which basis blades the input's values belong to.
GeometricToTensor Converts from a geometric algebra tf.Tensor with as many blades on the last axis as basis blades in the algebra to a tf.Tensor where blade indices determine which basis blades we extract for the output.
TensorWithKindToGeometric Same as TensorToGeometric but using BladeKind (eg. "bivector", "even") instead of blade indices.
GeometricToTensorWithKind Same as GeometricToTensor but using BladeKind (eg. "bivector", "even") instead of blade indices.
GeometricAlgebraExp Calculates the exponential function of the input. Input must square to a scalar.

## Notebooks

Generic examples

Using Keras layers to estimate triangle area

Classical Electromagnetism using Geometric Algebra

Quantum Electrodynamics using Geometric Algebra

Projective Geometric Algebra

1D Multivector-valued Convolution Example

## Tests

Tests using Python's built-in unittest module are available in the tests directory. All tests can be run by executing python -m unittest discover tests from the root directory of the repository.

## Citing

See our Zenodo page. For citing all versions the following BibTeX can be used

@software{python_tfga,
author       = {Kahlow, Robin},
title        = {TensorFlow Geometric Algebra},
publisher    = {Zenodo},
doi          = {10.5281/zenodo.3902404},
url          = {https://doi.org/10.5281/zenodo.3902404}
}


## Disclaimer

TensorFlow, the TensorFlow logo and any related marks are trademarks of Google Inc.

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