Problem: In one of my programs, I need to calculate a matrix multiplication A @ B where both are of size N by N for considerably large N. I’m conjecturing that approximating this product by using band_matrix(A, width) @ B could just suffice the needs, where band_matrix(A, width) denotes a band matrix part of A with width width. For example, width = 0 gives the diagonal matrix with diagonal elements taken from A and width = 1 gives the tridiagonal matrix taken in a similar manner.
My try: I’m trying to extract the tridiagonal matrix, for instance, in the following way:
# Step 1: Extract the main diagonal
main_diag = torch.diagonal(A, dim1=-2, dim2=-1) # Shape: [d1, d2, N]
# Step 2: Extract the upper diagonal (offset=1)
upper_diag = torch.diagonal(A, offset=1, dim1=-2, dim2=-1) # Shape: [d1, d2, N-1]
# Step 3: Extract the lower diagonal (offset=-1)
lower_diag = torch.diagonal(A, offset=-1, dim1=-2, dim2=-1) # Shape: [d1, d2, N-1]
# Step 4: Reconstruct the tridiagonal matrix
# Main diagonal
tridiag = torch.diag_embed(main_diag) # Shape: [d1, d2, N, N]
# Upper diagonal (shift the values to create the first upper diagonal)
tridiag += torch.diag_embed(upper_diag, offset=1)
# Lower diagonal (shift the values to create the first lower diagonal)
tridiag += torch.diag_embed(lower_diag, offset=-1)
but I’m not sure if tridiag @ B would be much more efficient than the original A @ B or just the same complexity since Torch may not know the specific structure to tridiag. In theory, computation with a tridiagonal matrix should be N times faster.
Any help with understanding PyTorch’s behaviour in this type of scenario or implementing some alternative GPU optimized approaches would be greatly appreciated.
1
To perform efficient matrix multiplication with a tridiagonal matrix on the GPU using PyTorch, you can try those leads :
- Use batched tridiagonal matrix multiplication: PyTorch provides some support for batched operations on banded matrices using torch.bmm (batched matrix-matrix multiplication), but it requires you to handle the band structure yourself. For a tridiagonal matrix, we only need to store the main, upper, and lower diagonals.
- Custom implementation with CUDA kernels or use specialized libraries: For more advanced optimization, consider writing custom CUDA kernels for tridiagonal matrix multiplication or use specialized libraries like cuSPARSE or SciPy.
I try an approach of it :
import torch
def band_matrix_mult(A: torch.Tensor, B: torch.Tensor) -> torch.Tensor:
"""
Multiplies a tridiagonal matrix A with matrix B, assuming A is tridiagonal.
Args:
- A (torch.Tensor): Tridiagonal matrix of shape (N, N).
- B (torch.Tensor): Matrix to multiply with A, of shape (N, M).
Returns:
- torch.Tensor: Resulting matrix of shape (N, M).
"""
N, M = B.shape
# Extract diagonals from A
main_diag = torch.diagonal(A) # Shape: [N]
upper_diag = torch.diagonal(A, offset=1) # Shape: [N-1]
lower_diag = torch.diagonal(A, offset=-1) # Shape: [N-1]
# Initialize result matrix
result = torch.zeros_like(B)
# Multiply main diagonal
result += main_diag.unsqueeze(-1) * B
# Multiply upper diagonal
result[:-1] += upper_diag.unsqueeze(-1) * B[1:]
# Multiply lower diagonal
result[1:] += lower_diag.unsqueeze(-1) * B[:-1]
return result
# Example usage
N = 5
A = torch.tensor([[2., 3., 0., 0., 0.],
[1., 2., 3., 0., 0.],
[0., 1., 2., 3., 0.],
[0., 0., 1., 2., 3.],
[0., 0., 0., 1., 2.]])
B = torch.randn((5, 4))
# Efficient tridiagonal multiplication
result = band_matrix_mult(A, B)
print(result)
if we talk about performance :
This custom function leverages the tridiagonal structure for efficiency and should have a time complexity of
𝑂(𝑁^2⋅𝑀), which is significantly better than the 𝑂(𝑁^3) complexity of dense matrix multiplication.
If you’re working with extremely large matrices or need even more performance optimization, consider using GPU-based libraries or writing a custom CUDA kernel to take full advantage of GPU parallelism for sparse matrix operations.