TorchFuncBasis

A PyTorch-based library for functional basis representations and smooth function approximation.

Features

• B-spline basis functions with arbitrary order
• Fourier basis functions
• Smooth function approximation with penalized regression
• Batch processing support
• GPU acceleration through PyTorch

Installation

https://github.com/mynanshan/TorchFuncBasis.git
cd TorchFuncBasis
pip install torchfuncbasis

Quick Start

import torch
from torchfuncbasis.basis import BSplineBasis
from torchfuncbasis.smoother import points2basiscoefs

Create a B-spline basis

basis = BSplineBasis(n_basis=11, domain_range=(0, 1), order=4)
print(basis)

Evaluate Basis matrix or Gram matrix

A Basis object can be called to evaluate the basis matrix or its derivatives. The input is allowed to be a batch of points with shape (*batch, n_points, *dim_domain):

• batch is arbitrary leading dimensions, can be empty
• n_points is the number of points in the domain
• dim_domain is the dimension of the domain, can be dropped if the domain is one-dimensional

x = torch.linspace(0, 1, 101).unsqueeze(0).repeat(5, 7, 1)
print(f"x's shape: {x.shape}")
basis_matrix = basis(x)
print(f"basis_matrix.shape: {basis_matrix.shape}")
basis_deriv_matrix = basis(x, derivative=1)
gram_matrix = basis.gram_matrix()
print(f"gram_matrix.shape: {gram_matrix.shape}")
gram_matrix_deriv = basis.gram_matrix(derivative=1)

Fit smooth function

Suppose we have a set of points (x, y) and a basis object basis. The y is expected to have shape (*batch, n_points, *dim_response), where

• *batch and n_points are the same as those in x
• dim_response is the dimension of the response variable, can be dropped if the response is one-dimensional

We can fit a smooth function by solving a penalized least-squares problem with points2basiscoefs. The returned value is the basis coefficients of shape (*batch, dim_response, n_basis).

x = torch.linspace(0, 1, 101).unsqueeze(0).repeat(5, 7, 1)
print(f"x's shape: {x.shape}")
y = torch.cat([
    (torch.sin(2 * torch.pi * x) + torch.randn_like(x) * 0.05).unsqueeze(-1),
    (torch.cos(2 * torch.pi * x) + torch.randn_like(x) * 0.05).unsqueeze(-1)
], dim=-1)
print(f"y's shape: {y.shape}")
coefs = points2basiscoefs(x, y, basis, smoothing_param=1e-4)
print(f"coefs.shape: {coefs.shape}")

References

• scikit-fda: a Numpy-based library for comprehensive functional data analysis, including basis functions and smoothing methods. scikit-fda is our main reference for code architecture.
• pykan/spline.py for PyTorch-based B-spline implementation.

To Be Added...

• Other univariate basis functions
• Tensor-product basis functions
• A wrapper for smoothing