high-order-layers-torch 1.1.0

High order layers in pytorch

Functional Layers in PyTorch

This is a PyTorch implementation of my tensorflow repository and is more complete due to the flexibility of PyTorch.

Lagrange Polynomial, Piecewise Lagrange Polynomial, Discontinuous Piecewise Lagrange Polynomial, Fourier Series, sum and product layers in PyTorch. The sparsity of using piecewise polynomial layers means that by adding new segments the representational power of your network increases, but the time to complete a forward step remains constant. Implementation includes simple fully connected layers and convolution layers using these models. More details to come. This is a PyTorch implementation of this paper including extension to Fourier Series and convolutional neural networks.

The layers used here do not require additional activation functions and use a simple sum or product in place of the activation. Product is performed in this manner

The 1 is added to each function output to as each of the sub products is also computed. The linear part is controlled by the alpha parameter.

Fully Connected Layer Types

All polynomials are Lagrange polynomials with Chebyshev interpolation points.

A helper function is provided in selecting and switching between these layers

from high_order_layers_torch.layers import *
layer1 = high_order_fc_layers(
    layer_type=layer_type,
    n=n, 
    in_features=784,
    out_features=100,
    segments=segments,
    alpha=linear_part
)

where layer_type is one of

layer_type	representation
continuous	piecewise polynomial using sum at the neuron
continuous_prod	piecewise polynomial using products at the neuron
discontinuous	discontinuous piecewise polynomial with sum at the neuron
discontinuous_prod	discontinous piecewise polynomial with product at the neuron
polynomial	single polynomial (non piecewise) with sum at the neuron
polynomial_prod	single polynomial (non piecewise) with product at the neuron
product	Product
fourier	fourier series with sum at the neuron

n is the number of interpolation points per segment for polynomials or the number of frequencies for fourier series, segments is the number of segments for piecewise polynomials, alpha is used in product layers and when set to 1 keeps the linear part of the product, when set to 0 it subtracts the linear part from the product.

Product Layers

Product layers

Convolutional Layer Types

conv_layer = high_order_convolution_layers(layer_type=layer_type, n=n, in_channels=3, out_channels=6, kernel_size=5, segments=segments, rescale_output=rescale_output, periodicity=periodicity)

All polynomials are Lagrange polynomials with Chebyshev interpolation points.

layer_type	representation
continuous(1d,2d)	piecewise continuous polynomial
discontinuous(1d,2d)	piecewise discontinuous polynomial
polynomial(1d,2d)	single polynomial
fourier(1d,2d)	fourier series convolution

Installing

Installing locally

This repo uses poetry, so run

poetry install

and then

poetry shell

Installing from pypi

pip install high-order-layers-torch

or

poetry add high-order-layers-torch

Examples

Simple function approximation

Approximating a simple function using a single input and single output (single layer) with no hidden layers to approximate a function using continuous and discontinuous piecewise polynomials (with 5 pieces) and simple polynomials and fourier series. The standard approach using ReLU is non competitive. To see more complex see the implicit representation page here.

XOR : 0.5 for x*y > 0 else -0.5

Simple XOR problem, the function is discontinuous along the axis and we try and fit that function

mnist (convolutional)

python mnist.py max_epochs=1 train_fraction=0.1 layer_type=continuous n=4 segments=2

cifar100 (convolutional)

python cifar100.py -m max_epochs=20 train_fraction=1.0 layer_type=polynomial segments=2 n=7 nonlinearity=False rescale_output=False periodicity=2.0 lr=0.001 linear_output=False

invariant mnist (fully connected)

Without polynomial refinement

python invariant_mnist.py max_epochs=100 train_fraction=1 layer_type=polynomial n=5 p_refine=False

with polynomial refinement (p-refinement)

python invariant_mnist.py max_epochs=100 train_fraction=1 layer_type=continuous n=2 p_refine=False target_n=5 p_refine=True

Implicit Representation

An example of implicit representation can be found here

Test

After installing and running

poetry shell

run

pytest 

Reference

@misc{Loverich2020,
  author = {Loverich, John},
  title = {High Order Layers Torch},
  year = {2020},
  publisher = {GitHub},
  journal = {GitHub repository},
  howpublished = {\url{https://github.com/jloveric/high-order-layers-torch}},
}