torch-frft 0.5.1

PyTorch implementation of the fractional Fourier transform with trainable transform order.

Trainable Fractional Fourier Transform

A differentiable fractional Fourier transform (FRFT) implementation with layers that can be trained end-to-end with the rest of the network. This package provides implementations of both fast computation of continuous FRFT and discrete FRFT (DFRFT) and also provides pre-configured layers that can be used in neural networks.

The fast transform is an approximation of the continuous FRFT and is based on Digital computation of the fractional Fourier transform paper. The DFRFT is based on The discrete fractional Fourier transform paper. MATLAB implementations of both approaches are provided on Haldun M. Özaktaş's page as fracF.m and dFRT.m, respectively.

This package implements these approaches in PyTorch with certain optimazations and most importantly adds the ability to apply the transform along a certain dimension of a tensor.

With the implemented transforms basic layers that extend torch.nn.Module is provided, an example of the custom layer implementation is also provided in README.md file.

Table of Contents

• Trainable Fractional Fourier Transform
  • Table of Contents
  • Installation
    • For Usage
    • For Development
  • Usage
    • Transforms
    • Pre-configured Layers
    • Custom Layers

Installation

For Usage

You can install the package directly from PYPI using pip or poetry as follows:

pip install torch-frft

or

poetry add torch-frft

For Development

This codebase utilizes Poetry for package management. To install the dependencies:

poetry install

or if you do not want to use Poetry, you can install the dependencies provided in requirements.txt using pip or conda, e,g.,

pip install -r requirements.txt

Usage

Transforms

:warning: Transforms applied in the same device as the input tensor. If the input tensor is on GPU, the transform will be applied on GPU as well.

The package provides transform functions that operate on the $n^{th}$ dimension of an input tensor, frft() and dfrft(), which correspond to the fast computation of continuous fractional Fourier transform (FRFT) and discrete fractional Fourier transform (DFRFT), respectively. It also provides a function dfrftmtx() which computes the DFRFT matrix for a given length and order similar to MATLAB's dftmtx() function for ordinary DFT matrix. Note that the frft() only operates on even-sized lengths as in the original MATLAB implementation fracF.m.

import torch
from torch_frft.frft_module import frft
from torch_frft.dfrft_module import dfrft, dfrftmtx

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")

N = 128
a = 0.5
X = torch.rand(N, N, device=device)
Y1 = frft(X, a)  # equivalent to dim=-1
Y2 = frft(X, a, dim=0)

# 2D FRFT
a0, a1 = 1.25, 0.75
Y3 = frft(frft(X, a0, dim=0), a1, dim=1)

Pre-configured Layers

The package also provides two differentiable FRFT layers, FrFTLayer and DFrFTLayer, which can be used as follows:

import torch
import torch.nn as nn

from torch_frft.dfrft_module import dfrft
from torch_frft.layer import DFrFTLayer, FrFTLayer

# FRFT with initial order 1.25, operating on the last dimension
model = nn.Sequential(FrFTLayer(order=1.25, dim=-1))

# DFRFT with initial order 0.75, operating on the first dimension
model = nn.Sequential(DFrFTLayer(order=0.75, dim=0))

Then, the simplest toy example to train the layer is as follows:

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
num_samples, seq_length = 100, 16
a_original = 1.1
a_initial = 1.25
X = torch.randn(num_samples, seq_length, dtype=torch.float32, device=device)
Y = dfrft(X, a_original)

model = DFrFTLayer(order=a_initial).to(device)
optim = torch.optim.Adam(model.parameters(), lr=1e-3)
epochs = 1000

for epoch in range(1 + epochs):
    optim.zero_grad()
    loss = torch.norm(Y - model(X))
    loss.backward()
    optim.step()

print("Original  a:", a_original)
print("Estimated a:", model.order.item())

Note that you can also place these layers directly into torch.nn.Sequential. Remark that these transforms generate complex-valued outputs, so you may need to convert them to real-valued outputs, e.g., taking the real part, absolute value, etc. For example, the following code snippet implements a simple fully-connected network with real parts of FrFTLayer and DFrFTLayer in between:

class Real(nn.Module):
    def __init__(self) -> None:
        super().__init__()

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        return x.real


model = nn.Sequential(
    nn.Linear(16, 6),
    nn.ReLU(),
    DFrFTLayer(1.35, dim=-1),
    Real(),
    nn.ReLU(),
    nn.Linear(6, 1),
    FrFTLayer(0.65, dim=0),
    Real(),
    nn.ReLU(),
)

Custom Layers

It is also possible to create custom layers with the provided frft() and dfrft() functions. The below example contains in-between Linear and ReLU layers and the same fractional order for forward and backward DFRFT transforms is as follows:

import torch
import torch.nn as nn
from torch_frft.dfrft_module import dfrft


class CustomLayer(nn.Module):
    def __init__(self, in_feat: int, out_feat: int, *, order: float = 1.0, dim: int = -1) -> None:
        super().__init__()
        self.in_features = in_feat
        self.out_features = out_feat
        self.order = nn.Parameter(torch.tensor(order, dtype=torch.float32), requires_grad=True)
        self.dim = dim

    def forward(self, x: torch.Tensor) -> torch.Tensor:
        x1 = dfrft(x, self.order, dim=self.dim)
        a1 = nn.ReLU()(x1.real) + 1j * nn.ReLU()(x1.imag)
        x2 = nn.Linear(self.in_features, self.in_features, dtype=a1.dtype, device=x.device)(a1)
        a2 = nn.ReLU()(x2.real) + 1j * nn.ReLU()(x2.imag)
        x3 = dfrft(a2, -self.order, dim=self.dim)
        a3 = nn.ReLU()(x3.real) + 1j * nn.ReLU()(x3.imag)
        x4 = nn.Linear(self.in_features, self.out_features, dtype=a3.dtype, device=x.device)(a3)
        a4 = nn.ReLU()(x4.real)
        return a4

Then, a simple training example for the given CustomLayer can be given as follows:

device = torch.device("cuda" if torch.cuda.is_available() else "cpu")
num_samples, seq_length, out_length = 100, 32, 5
X = torch.rand(num_samples, seq_length, device=device)
y = torch.rand(num_samples, out_length, device=device)

model = CustomLayer(seq_length, out_length, order=1.25)
optim = torch.optim.Adam(model.parameters(), lr=1e-3)
epochs = 1000

for epoch in range(epochs):
    optim.zero_grad()
    loss = torch.nn.MSELoss()
    output = loss(model(X), y)
    output.backward()
    optim.step()
    if epoch % 100 == 0:
        print(f"Epoch {epoch:4d} | Loss {output.item():.4f}")
print("Final a:", model.order)