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Parallel Scan (Blelloch algorithm) in pure PyTorch for ARG selective SSM recurrence.
Eliminates all external dependencies (mamba-ssm, selective_scan_cuda).
Runs in 2*log2(T) parallel steps (11 steps for T=2048) on any GPU.
100% mathematically exact equivalence to sequential SSM recurrence.
"""
from __future__ import annotations
import math
import torch
import torch.nn.functional as F
def npo2(length: int) -> int:
"""Returns the next power of 2 above length."""
return 2 ** math.ceil(math.log2(length))
def pad_npo2(x: torch.Tensor) -> torch.Tensor:
"""Pads input length dimension (dim 1) to the next power of 2."""
length = x.size(1)
len_p2 = npo2(length)
if len_p2 == length:
return x
pad_tuple = (0, 0, 0, 0, 0, len_p2 - length)
return F.pad(x, pad_tuple, "constant", 0)
class PScan(torch.autograd.Function):
"""
Computes the parallel associative scan:
H[t] = A[t] * H[t-1] + X[t] with H[-1] = 0
in O(log2(T)) parallel steps via Blelloch tree reduction.
"""
@staticmethod
def pscan(A: torch.Tensor, X: torch.Tensor) -> None:
# A: (B, D, L, N)
# X: (B, D, L, N)
B, D, L, _ = A.size()
num_steps = int(math.log2(L))
# ── Up sweep ──────────────────────────────────────────────────────────
Aa = A
Xa = X
for _ in range(num_steps - 2):
T = Xa.size(2)
Aa = Aa.view(B, D, T // 2, 2, -1)
Xa = Xa.view(B, D, T // 2, 2, -1)
Xa[:, :, :, 1].add_(Aa[:, :, :, 1].mul(Xa[:, :, :, 0]))
Aa[:, :, :, 1].mul_(Aa[:, :, :, 0])
Aa = Aa[:, :, :, 1]
Xa = Xa[:, :, :, 1]
if Xa.size(2) == 4:
Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 0]))
Aa[:, :, 1].mul_(Aa[:, :, 0])
Xa[:, :, 3].add_(Aa[:, :, 3].mul(Xa[:, :, 2] + Aa[:, :, 2].mul(Xa[:, :, 1])))
elif Xa.size(2) == 2:
Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 0]))
return
else:
return
# ── Down sweep ────────────────────────────────────────────────────────
Aa = A[:, :, 2 ** (num_steps - 2) - 1 : L : 2 ** (num_steps - 2)]
Xa = X[:, :, 2 ** (num_steps - 2) - 1 : L : 2 ** (num_steps - 2)]
Xa[:, :, 2].add_(Aa[:, :, 2].mul(Xa[:, :, 1]))
Aa[:, :, 2].mul_(Aa[:, :, 1])
for k in range(num_steps - 3, -1, -1):
stride = 2 ** k
Aa = A[:, :, stride - 1 : L : stride]
Xa = X[:, :, stride - 1 : L : stride]
T = Xa.size(2)
Aa = Aa.view(B, D, T // 2, 2, -1)
Xa = Xa.view(B, D, T // 2, 2, -1)
Xa[:, :, 1:, 0].add_(Aa[:, :, 1:, 0].mul(Xa[:, :, :-1, 1]))
Aa[:, :, 1:, 0].mul_(Aa[:, :, :-1, 1])
@staticmethod
def pscan_rev(A: torch.Tensor, X: torch.Tensor) -> None:
# Backward scan in reverse time order
B, D, L, _ = A.size()
num_steps = int(math.log2(L))
# Up sweep
Aa = A
Xa = X
for _ in range(num_steps - 2):
T = Xa.size(2)
Aa = Aa.view(B, D, T // 2, 2, -1)
Xa = Xa.view(B, D, T // 2, 2, -1)
Xa[:, :, :, 0].add_(Aa[:, :, :, 0].mul(Xa[:, :, :, 1]))
Aa[:, :, :, 0].mul_(Aa[:, :, :, 1])
Aa = Aa[:, :, :, 0]
Xa = Xa[:, :, :, 0]
if Xa.size(2) == 4:
Xa[:, :, 2].add_(Aa[:, :, 2].mul(Xa[:, :, 3]))
Aa[:, :, 2].mul_(Aa[:, :, 3])
Xa[:, :, 0].add_(Aa[:, :, 0].mul(Xa[:, :, 1].add(Aa[:, :, 1].mul(Xa[:, :, 2]))))
elif Xa.size(2) == 2:
Xa[:, :, 0].add_(Aa[:, :, 0].mul(Xa[:, :, 1]))
return
else:
return
# Down sweep
stride_top = 2 ** (num_steps - 2)
Aa = A[:, :, 0 : L : stride_top]
Xa = X[:, :, 0 : L : stride_top]
Xa[:, :, 1].add_(Aa[:, :, 1].mul(Xa[:, :, 2]))
Aa[:, :, 1].mul_(Aa[:, :, 2])
for k in range(num_steps - 3, -1, -1):
stride = 2 ** k
Aa = A[:, :, 0 : L : stride]
Xa = X[:, :, 0 : L : stride]
T = Xa.size(2)
Aa = Aa.view(B, D, T // 2, 2, -1)
Xa = Xa.view(B, D, T // 2, 2, -1)
Xa[:, :, :-1, 1].add_(Aa[:, :, :-1, 1].mul(Xa[:, :, 1:, 0]))
Aa[:, :, :-1, 1].mul_(Aa[:, :, 1:, 0])
@staticmethod
def forward(ctx, A_in: torch.Tensor, X_in: torch.Tensor) -> torch.Tensor:
# A_in: (B, L, D, N)
# X_in: (B, L, D, N)
L = X_in.size(1)
if L == npo2(L):
A = A_in.clone()
X = X_in.clone()
else:
A = pad_npo2(A_in).clone()
X = pad_npo2(X_in).clone()
A = A.transpose(2, 1).contiguous() # (B, D, L_p2, N)
X = X.transpose(2, 1).contiguous() # (B, D, L_p2, N)
PScan.pscan(A, X)
ctx.save_for_backward(A_in, X)
return X.transpose(2, 1)[:, :L].contiguous()
@staticmethod
def backward(ctx, grad_output_in: torch.Tensor):
A_in, X = ctx.saved_tensors
L = grad_output_in.size(1)
if L == npo2(L):
grad_output = grad_output_in.clone()
else:
grad_output = pad_npo2(grad_output_in).clone()
A_in = pad_npo2(A_in)
grad_output = grad_output.transpose(2, 1).contiguous() # (B, D, L_p2, N)
A_in_t = A_in.transpose(2, 1).contiguous() # (B, D, L_p2, N)
# Shift A 1 step to the left for adjoint recurrence
A_shifted = F.pad(A_in_t[:, :, 1:], (0, 0, 0, 1))
PScan.pscan_rev(A_shifted, grad_output)
grad_A = torch.zeros_like(X)
grad_A[:, :, 1:].add_(X[:, :, :-1] * grad_output[:, :, 1:])
return grad_A.transpose(2, 1)[:, :L].contiguous(), grad_output.transpose(2, 1)[:, :L].contiguous()
pscan = PScan.apply
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