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"""Nyströmer encoder.
Efficient attention module that reduces the complexity of the attention module from
O(n**2) to O(n). The attention matrix is assumed low rank and thus the information
can be represented by a smaller matrix.
Stolen from:
https://github.com/lucidrains/nystrom-attention/blob/main/nystrom_attention/nystrom_attention.py
"""
from math import ceil
from typing import Optional, Tuple, Union
from einops import rearrange, reduce
import torch
from torch import einsum, nn, Tensor
from torch.nn import functional as F
def moore_penrose_inverse(x: Tensor, iters: int = 6) -> Tensor:
"""Moore-Penrose pseudoinverse."""
x_abs = torch.abs(x)
col = x_abs.sum(dim=-1)
row = x_abs.sum(dim=-2)
z = rearrange(x, "... i j -> ... j i") / (torch.max(col) * torch.max(row))
I = torch.eye(x.shape[-1], device=x.device)
I = rearrange(I, "i j -> () i j")
for _ in range(iters):
xz = x @ z
z = 0.25 * z @ (13 * I - (xz @ (15 * I - (xz @ (7 * I - xz)))))
return z
class NystromAttention(nn.Module):
def __init__(
self,
dim: int,
dim_head: int = 64,
num_heads: int = 8,
num_landmarks: int = 256,
inverse_iter: int = 6,
residual: bool = True,
residual_conv_kernel: int = 13,
eps: float = 1.0e-8,
dropout_rate: float = 0.0,
):
super().__init__()
self.residual = None
self.eps = eps
self.num_heads = num_heads
inner_dim = self.num_heads * dim_head
self.num_landmarks = num_landmarks
self.inverse_iter = inverse_iter
self.scale = dim_head ** -0.5
self.qkv_fn = nn.Linear(dim, 3 * inner_dim, bias=False)
self.fc_out = nn.Sequential(nn.Linear(inner_dim, dim), nn.Dropout(dropout_rate))
if residual:
self.residual = nn.Conv2d(
in_channels=num_heads,
out_channels=num_heads,
kernel_size=(residual_conv_kernel, 1),
padding=(residual_conv_kernel // 2, 0),
groups=num_heads,
bias=False,
)
@staticmethod
def _pad_sequence(
x: Tensor, mask: Optional[Tensor], n: int, m: int
) -> Tuple[Tensor, Tensor]:
"""Pad sequence."""
padding = m - (n % m)
x = F.pad(x, (0, 0, padding, 0), value=0)
mask = F.pad(mask, (padding, 0), value=False) if mask is not None else mask
return x, mask
def _compute_landmarks(
self, q: Tensor, k: Tensor, mask: Optional[Tensor], n: int, m: int
) -> Tuple[Tensor, Tensor, Optional[Tensor]]:
"""Compute landmarks of the attention matrix."""
divisor = ceil(n / m)
landmark_einops_eq = "... (n l) d -> ... n d"
q_landmarks = reduce(q, landmark_einops_eq, "sum", l=divisor)
k_landmarks = reduce(k, landmark_einops_eq, "sum", l=divisor)
mask_landmarks = None
if mask is not None:
mask_landmarks_sum = reduce(mask, "... (n l) -> ... n", "sum", l=divisor)
divisor = mask_landmarks_sum[..., None] + self.eps
mask_landmarks = mask_landmarks_sum > 0
q_landmarks /= divisor
k_landmarks /= divisor
return q_landmarks, k_landmarks, mask_landmarks
@staticmethod
def _compute_similarities(
q: Tensor,
k: Tensor,
q_landmarks: Tensor,
k_landmarks: Tensor,
mask: Optional[Tensor],
mask_landmarks: Optional[Tensor],
) -> Tuple[Tensor, Tensor, Tensor]:
einops_eq = "... i d, ... j d -> ... i j"
sim1 = einsum(einops_eq, q, k_landmarks)
sim2 = einsum(einops_eq, q_landmarks, k_landmarks)
sim3 = einsum(einops_eq, q_landmarks, k)
if mask is not None and mask_landmarks is not None:
mask_value = -torch.finfo(q.type).max
sim1.masked_fill_(
~(mask[..., None] * mask_landmarks[..., None, :]), mask_value
)
sim2.masked_fill_(
~(mask_landmarks[..., None] * mask_landmarks[..., None, :]), mask_value
)
sim3.masked_fill_(
~(mask_landmarks[..., None] * mask[..., None, :]), mask_value
)
return sim1, sim2, sim3
def _nystrom_attention(
self,
q: Tensor,
k: Tensor,
v: Tensor,
mask: Optional[Tensor],
n: int,
m: int,
return_attn: bool,
) -> Tuple[Tensor, Optional[Tensor]]:
q_landmarks, k_landmarks, mask_landmarks = self._compute_landmarks(
q, k, mask, n, m
)
sim1, sim2, sim3 = self._compute_similarities(
q, k, q_landmarks, k_landmarks, mask, mask_landmarks
)
# Compute attention
attn1, attn2, attn3 = map(lambda t: t.softmax(dim=-1), (sim1, sim2, sim3))
attn2_inv = moore_penrose_inverse(attn2, self.inverse_iter)
out = (attn1 @ attn2_inv) @ (attn3 @ v)
if return_attn:
return out, attn1 @ attn2_inv @ attn3
return out, None
def forward(
self, x: Tensor, mask: Optional[Tensor] = None, return_attn: bool = False
) -> Union[Tensor, Tuple[Tensor, Tensor]]:
"""Compute the Nystrom attention."""
_, n, _, h, m = *x.shape, self.num_heads, self.num_landmarks
if n % m != 0:
x, mask = self._pad_sequence(x, mask, n, m)
q, k, v = self.qkv_fn(x).chunk(3, dim=-1)
q, k, v = map(lambda t: rearrange(t, "b n (h d) -> b h n d", h=h), (q, k, v))
q = q * self.scale
out, attn = self._nystrom_attention(q, k, v, mask, n, m, return_attn)
# Add depth-wise convolutional residual of values
if self.residual is not None:
out += self.residual(out)
out = rearrange(out, "b h n d -> b n (h d)", h=h)
out = self.fc_out(out)
out = out[:, -n:]
if return_attn:
return out, attn
return out
|
