Axial Attention - minimal runnable implementation
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"""
Axial Attention - minimal runnable implementation
=================================================
Reference: Ho et al. 2019, "Axial Attention in Multidimensional Transformers"
(arxiv 1912.12180)
Core idea:
Vanilla 2D self-attention has O((HW)^2 * d) cost which is prohibitive
for high-resolution feature maps.
Axial attention factorizes it: one attention pass along the H axis,
another along the W axis.
H-axial: fold W into batch; each column attends within its H tokens
W-axial: fold H into batch; each row attends within its W tokens
Complexity drops from O((HW)^2 d) to O(HW(H+W) d), saving HW/(H+W) factor.
Stacking one H-axial then one W-axial gives an attention path between any
two positions (h1, w1) -> (h2, w2), but it's a factorized two-step
propagation, NOT equivalent to a single dense 2D attention.
Run:
python axial_attention.py
"""
import math
import torch
import torch.nn as nn
import torch.nn.functional as F
class AxialAttention(nn.Module):
"""1D axial attention over a specific spatial dim.
Args:
channels: input/output channel count C
d_k: attention head dim
axis: 'H' or 'W' - which spatial axis to attend along
"""
def __init__(self, channels: int, d_k: int, axis: str):
super().__init__()
assert axis in ("H", "W")
self.axis = axis
self.d_k = d_k
self.W_q = nn.Linear(channels, d_k, bias=False)
self.W_k = nn.Linear(channels, d_k, bias=False)
self.W_v = nn.Linear(channels, d_k, bias=False)
self.out = nn.Linear(d_k, channels)
def forward(self, x: torch.Tensor) -> torch.Tensor:
"""
Args:
x: [B, C, H, W]
Returns:
[B, C, H, W]
"""
B, C, H, W = x.shape
if self.axis == "H":
# Fold W into batch: [B, C, H, W] -> [B*W, H, C]
x_in = x.permute(0, 3, 2, 1).reshape(B * W, H, C)
else: # axis == 'W'
# Fold H into batch: [B, C, H, W] -> [B*H, W, C]
x_in = x.permute(0, 2, 3, 1).reshape(B * H, W, C)
Q = self.W_q(x_in) # [B*?, seq, d_k]
K = self.W_k(x_in)
V = self.W_v(x_in)
score = (Q @ K.transpose(-2, -1)) / math.sqrt(self.d_k)
attn = F.softmax(score, dim=-1)
out = attn @ V # [B*?, seq, d_k]
out = self.out(out) # [B*?, seq, C]
# Restore spatial shape.
if self.axis == "H":
# [B*W, H, C] -> [B, W, H, C] -> [B, C, H, W]
out = out.reshape(B, W, H, C).permute(0, 3, 2, 1)
else:
# [B*H, W, C] -> [B, H, W, C] -> [B, C, H, W]
out = out.reshape(B, H, W, C).permute(0, 3, 1, 2)
return out
class AxialBlock(nn.Module):
"""H-axial + W-axial stacked.
Two layers in series create an attention path between any (h1, w1)
and (h2, w2) -- the H-axial pass moves information along column w1 to
(h2, w1), then the W-axial pass moves it to (h2, w2). But this is a
factorized two-step propagation, NOT equivalent to one dense 2D attention.
The combined "weights" are a product of two 1D softmaxes; expressivity is
strictly weaker than full 2D attention. The win is O(HW(H+W)d) vs
O(H^2 W^2 d).
"""
def __init__(self, channels: int, d_k: int):
super().__init__()
self.h_attn = AxialAttention(channels, d_k, axis="H")
self.w_attn = AxialAttention(channels, d_k, axis="W")
self.norm1 = nn.GroupNorm(8, channels)
self.norm2 = nn.GroupNorm(8, channels)
def forward(self, x):
x = x + self.h_attn(self.norm1(x)) # residual
x = x + self.w_attn(self.norm2(x))
return x
# --- complexity comparison ----------------------------------------------------
def complexity_vanilla(H: int, W: int, d: int) -> int:
"""Vanilla 2D self-attention FLOPs ~ 2 * (HW)^2 * d (QK^T + attn @ V)."""
return 2 * (H * W) ** 2 * d
def complexity_axial(H: int, W: int, d: int) -> int:
"""Axial: H attention W times + W attention H times, each O(seq^2 d)."""
return 2 * W * H * H * d + 2 * H * W * W * d
def print_complexity_table():
print(f"{'H':>4} {'W':>4} | {'vanilla':>14} | {'axial':>14} | {'speedup':>8}")
print("-" * 60)
for H, W in [(8, 8), (32, 32), (64, 64), (128, 128), (256, 256)]:
d = 64
v = complexity_vanilla(H, W, d)
a = complexity_axial(H, W, d)
print(f"{H:>4} {W:>4} | {v:>14,} | {a:>14,} | {v / a:>8.1f}x")
# --- sanity check -------------------------------------------------------------
def shape_check():
B, C, H, W = 2, 32, 16, 16
x = torch.randn(B, C, H, W)
h_only = AxialAttention(C, d_k=32, axis="H")
w_only = AxialAttention(C, d_k=32, axis="W")
block = AxialBlock(C, d_k=32)
y_h = h_only(x)
y_w = w_only(x)
y_b = block(x)
print(f"[shape] x : {tuple(x.shape)}")
print(f"[shape] H-axial : {tuple(y_h.shape)}")
print(f"[shape] W-axial : {tuple(y_w.shape)}")
print(f"[shape] H+W block: {tuple(y_b.shape)}")
assert y_h.shape == y_w.shape == y_b.shape == x.shape
def receptive_field_check():
"""Verify a single H-axial layer keeps columns isolated.
Perturbing one pixel at (row=0, col=2) should only change outputs in
column 2 (across all rows), leaving every other column untouched.
"""
torch.manual_seed(0)
B, C, H, W = 1, 4, 6, 6
x = torch.randn(B, C, H, W)
layer = AxialAttention(C, d_k=8, axis="H")
y_orig = layer(x)
# Perturb one pixel in column 2.
x_perturbed = x.clone()
x_perturbed[0, :, 0, 2] += 1.0
y_perturbed = layer(x_perturbed)
diff = (y_orig - y_perturbed).abs().sum(dim=1) # [B, H, W]
print(f"[recep] diff per spatial position after perturbing (row=0, col=2):")
print(diff[0])
col2_sum = diff[0, :, 2].sum().item()
other_sum = (diff[0].sum() - diff[0, :, 2].sum()).item()
print(f"[recep] col=2 total diff = {col2_sum:.4f} (should be > 0)")
print(f"[recep] other cols diff = {other_sum:.6f} (should be 0)")
assert col2_sum > 1e-4 and other_sum < 1e-6
print("[recep] PASS - H-axial isolates columns as expected")
if __name__ == "__main__":
print("== Axial Attention ==\n")
print("--- complexity comparison (FLOPs, d=64) ---")
print_complexity_table()
print("\n--- shape check ---")
shape_check()
print("\n--- receptive field check ---")
receptive_field_check()
print("\nAll checks passed.")