#

U-Net model for Denoising Diffusion Probabilistic Models (DDPM)

This is a U-Net based model to predict noise $ϵ_{θ} (x_{t}, t)$ .

U-Net is a gets it's name from the U shape in the model diagram. It processes a given image by progressively lowering (halving) the feature map resolution and then increasing the resolution. There are pass-through connection at each resolution.

U-Net diagram from paper

This implementation contains a bunch of modifications to original U-Net (residual blocks, multi-head attention) and also adds time-step embeddings $t$ .

24import math
25from typing import Optional, Tuple, Union, List
26
27import torch
28from torch import nn
29
30from labml_helpers.module import Module

#

Swish activation function

$x \cdot σ (x)$

33class Swish(Module):

#

40    def forward(self, x):
41        return x * torch.sigmoid(x)

#

Embeddings for $t$

44class TimeEmbedding(nn.Module):

#

n_channels is the number of dimensions in the embedding

49    def __init__(self, n_channels: int):

#

53        super().__init__()
54        self.n_channels = n_channels

#

First linear layer

56        self.lin1 = nn.Linear(self.n_channels // 4, self.n_channels)

#

Activation

58        self.act = Swish()

#

Second linear layer

60        self.lin2 = nn.Linear(self.n_channels, self.n_channels)

#

62    def forward(self, t: torch.Tensor):

#

Create sinusoidal position embeddings same as those from the transformer

P E_{t, i}^{(1)} P E_{t, i}^{(2)} = s in (\frac{t}{1000 0 ^{\frac{i}{d - 1}}}) = cos (\frac{t}{1000 0 ^{\frac{i}{d - 1}}})

where $d$ is half_dim

72        half_dim = self.n_channels // 8
73        emb = math.log(10_000) / (half_dim - 1)
74        emb = torch.exp(torch.arange(half_dim, device=t.device) * -emb)
75        emb = t[:, None] * emb[None, :]
76        emb = torch.cat((emb.sin(), emb.cos()), dim=1)

#

Transform with the MLP

79        emb = self.act(self.lin1(emb))
80        emb = self.lin2(emb)

#

83        return emb

#

Residual block

A residual block has two convolution layers with group normalization. Each resolution is processed with two residual blocks.

86class ResidualBlock(Module):

#

in_channels is the number of input channels
out_channels is the number of input channels
time_channels is the number channels in the time step ( $t$ ) embeddings
n_groups is the number of groups for group normalization
dropout is the dropout rate

94    def __init__(self, in_channels: int, out_channels: int, time_channels: int,
95                 n_groups: int = 32, dropout: float = 0.1):

#

103        super().__init__()

#

Group normalization and the first convolution layer

105        self.norm1 = nn.GroupNorm(n_groups, in_channels)
106        self.act1 = Swish()
107        self.conv1 = nn.Conv2d(in_channels, out_channels, kernel_size=(3, 3), padding=(1, 1))

#

Group normalization and the second convolution layer

110        self.norm2 = nn.GroupNorm(n_groups, out_channels)
111        self.act2 = Swish()
112        self.conv2 = nn.Conv2d(out_channels, out_channels, kernel_size=(3, 3), padding=(1, 1))

#

If the number of input channels is not equal to the number of output channels we have to project the shortcut connection

116        if in_channels != out_channels:
117            self.shortcut = nn.Conv2d(in_channels, out_channels, kernel_size=(1, 1))
118        else:
119            self.shortcut = nn.Identity()

#

Linear layer for time embeddings

122        self.time_emb = nn.Linear(time_channels, out_channels)
123        self.time_act = Swish()
124
125        self.dropout = nn.Dropout(dropout)

#

x has shape [batch_size, in_channels, height, width]
t has shape [batch_size, time_channels]

127    def forward(self, x: torch.Tensor, t: torch.Tensor):

#

First convolution layer

133        h = self.conv1(self.act1(self.norm1(x)))

#

Add time embeddings

135        h += self.time_emb(self.time_act(t))[:, :, None, None]

#

Second convolution layer

137        h = self.conv2(self.dropout(self.act2(self.norm2(h))))

#

Add the shortcut connection and return

140        return h + self.shortcut(x)

#

Attention block

This is similar to transformer multi-head attention.

143class AttentionBlock(Module):

#

n_channels is the number of channels in the input
n_heads is the number of heads in multi-head attention
d_k is the number of dimensions in each head
n_groups is the number of groups for group normalization

150    def __init__(self, n_channels: int, n_heads: int = 1, d_k: int = None, n_groups: int = 32):

#

157        super().__init__()

#

Default d_k

160        if d_k is None:
161            d_k = n_channels

#

Normalization layer

163        self.norm = nn.GroupNorm(n_groups, n_channels)

#

Projections for query, key and values

165        self.projection = nn.Linear(n_channels, n_heads * d_k * 3)

#

Linear layer for final transformation

167        self.output = nn.Linear(n_heads * d_k, n_channels)

#

Scale for dot-product attention

169        self.scale = d_k ** -0.5

#

171        self.n_heads = n_heads
172        self.d_k = d_k

#

x has shape [batch_size, in_channels, height, width]
t has shape [batch_size, time_channels]

174    def forward(self, x: torch.Tensor, t: Optional[torch.Tensor] = None):

#

t is not used, but it's kept in the arguments because for the attention layer function signature to match with ResidualBlock .

181        _ = t

#

Get shape

183        batch_size, n_channels, height, width = x.shape

#

Change x to shape [batch_size, seq, n_channels]

185        x = x.view(batch_size, n_channels, -1).permute(0, 2, 1)

#

Get query, key, and values (concatenated) and shape it to [batch_size, seq, n_heads, 3 * d_k]

187        qkv = self.projection(x).view(batch_size, -1, self.n_heads, 3 * self.d_k)

#

Split query, key, and values. Each of them will have shape [batch_size, seq, n_heads, d_k]

189        q, k, v = torch.chunk(qkv, 3, dim=-1)

#

Calculate scaled dot-product $\frac{Q K ^{⊤}}{d _{k}}$

191        attn = torch.einsum('bihd,bjhd->bijh', q, k) * self.scale

#

Softmax along the sequence dimension $se q so f t ma x (\frac{Q K ^{⊤}}{d _{k}})$

193        attn = attn.softmax(dim=2)

#

Multiply by values

195        res = torch.einsum('bijh,bjhd->bihd', attn, v)

#

Reshape to [batch_size, seq, n_heads * d_k]

197        res = res.view(batch_size, -1, self.n_heads * self.d_k)

#

Transform to [batch_size, seq, n_channels]

199        res = self.output(res)

#

Add skip connection

202        res += x

#

Change to shape [batch_size, in_channels, height, width]

205        res = res.permute(0, 2, 1).view(batch_size, n_channels, height, width)

#

208        return res

#

Down block

This combines ResidualBlock and AttentionBlock . These are used in the first half of U-Net at each resolution.

211class DownBlock(Module):

#

218    def __init__(self, in_channels: int, out_channels: int, time_channels: int, has_attn: bool):
219        super().__init__()
220        self.res = ResidualBlock(in_channels, out_channels, time_channels)
221        if has_attn:
222            self.attn = AttentionBlock(out_channels)
223        else:
224            self.attn = nn.Identity()

#

226    def forward(self, x: torch.Tensor, t: torch.Tensor):
227        x = self.res(x, t)
228        x = self.attn(x)
229        return x

#

Up block

This combines ResidualBlock and AttentionBlock . These are used in the second half of U-Net at each resolution.

232class UpBlock(Module):

#

239    def __init__(self, in_channels: int, out_channels: int, time_channels: int, has_attn: bool):
240        super().__init__()

#

The input has in_channels + out_channels because we concatenate the output of the same resolution from the first half of the U-Net

243        self.res = ResidualBlock(in_channels + out_channels, out_channels, time_channels)
244        if has_attn:
245            self.attn = AttentionBlock(out_channels)
246        else:
247            self.attn = nn.Identity()

#

249    def forward(self, x: torch.Tensor, t: torch.Tensor):
250        x = self.res(x, t)
251        x = self.attn(x)
252        return x

#

Middle block

It combines a ResidualBlock , AttentionBlock , followed by another ResidualBlock . This block is applied at the lowest resolution of the U-Net.

255class MiddleBlock(Module):

#

263    def __init__(self, n_channels: int, time_channels: int):
264        super().__init__()
265        self.res1 = ResidualBlock(n_channels, n_channels, time_channels)
266        self.attn = AttentionBlock(n_channels)
267        self.res2 = ResidualBlock(n_channels, n_channels, time_channels)

#

269    def forward(self, x: torch.Tensor, t: torch.Tensor):
270        x = self.res1(x, t)
271        x = self.attn(x)
272        x = self.res2(x, t)
273        return x

#

Scale up the feature map by $2 \times$

276class Upsample(nn.Module):

#

281    def __init__(self, n_channels):
282        super().__init__()
283        self.conv = nn.ConvTranspose2d(n_channels, n_channels, (4, 4), (2, 2), (1, 1))

#

285    def forward(self, x: torch.Tensor, t: torch.Tensor):

#

t is not used, but it's kept in the arguments because for the attention layer function signature to match with ResidualBlock .

288        _ = t
289        return self.conv(x)

#

Scale down the feature map by $\frac{1}{2} \times$

292class Downsample(nn.Module):

#

297    def __init__(self, n_channels):
298        super().__init__()
299        self.conv = nn.Conv2d(n_channels, n_channels, (3, 3), (2, 2), (1, 1))

#

301    def forward(self, x: torch.Tensor, t: torch.Tensor):

#

t is not used, but it's kept in the arguments because for the attention layer function signature to match with ResidualBlock .

304        _ = t
305        return self.conv(x)

#

U-Net

308class UNet(Module):

#

image_channels is the number of channels in the image. $3$ for RGB.
n_channels is number of channels in the initial feature map that we transform the image into
ch_mults is the list of channel numbers at each resolution. The number of channels is ch_mults[i] * n_channels
is_attn is a list of booleans that indicate whether to use attention at each resolution
n_blocks is the number of UpDownBlocks at each resolution

313    def __init__(self, image_channels: int = 3, n_channels: int = 64,
314                 ch_mults: Union[Tuple[int, ...], List[int]] = (1, 2, 2, 4),
315                 is_attn: Union[Tuple[bool, ...], List[bool]] = (False, False, True, True),
316                 n_blocks: int = 2):

#

324        super().__init__()

#

Number of resolutions

327        n_resolutions = len(ch_mults)

#

Project image into feature map

330        self.image_proj = nn.Conv2d(image_channels, n_channels, kernel_size=(3, 3), padding=(1, 1))

#

Time embedding layer. Time embedding has n_channels * 4 channels

333        self.time_emb = TimeEmbedding(n_channels * 4)

#

First half of U-Net - decreasing resolution

336        down = []

#

Number of channels

338        out_channels = in_channels = n_channels

#

For each resolution

340        for i in range(n_resolutions):

#

Number of output channels at this resolution

342            out_channels = in_channels * ch_mults[i]

#

Add n_blocks

344            for _ in range(n_blocks):
345                down.append(DownBlock(in_channels, out_channels, n_channels * 4, is_attn[i]))
346                in_channels = out_channels

#

Down sample at all resolutions except the last

348            if i < n_resolutions - 1:
349                down.append(Downsample(in_channels))

#

Combine the set of modules

352        self.down = nn.ModuleList(down)

#

Middle block

355        self.middle = MiddleBlock(out_channels, n_channels * 4, )

#

Second half of U-Net - increasing resolution

358        up = []

#

Number of channels

360        in_channels = out_channels

#

For each resolution

362        for i in reversed(range(n_resolutions)):

#

n_blocks at the same resolution

364            out_channels = in_channels
365            for _ in range(n_blocks):
366                up.append(UpBlock(in_channels, out_channels, n_channels * 4, is_attn[i]))

#

Final block to reduce the number of channels

368            out_channels = in_channels // ch_mults[i]
369            up.append(UpBlock(in_channels, out_channels, n_channels * 4, is_attn[i]))
370            in_channels = out_channels

#

Up sample at all resolutions except last

372            if i > 0:
373                up.append(Upsample(in_channels))

#

Combine the set of modules

376        self.up = nn.ModuleList(up)

#

Final normalization and convolution layer

379        self.norm = nn.GroupNorm(8, n_channels)
380        self.act = Swish()
381        self.final = nn.Conv2d(in_channels, image_channels, kernel_size=(3, 3), padding=(1, 1))

#

x has shape [batch_size, in_channels, height, width]
t has shape [batch_size]

383    def forward(self, x: torch.Tensor, t: torch.Tensor):

#

Get time-step embeddings

390        t = self.time_emb(t)

#

Get image projection

393        x = self.image_proj(x)

#

h will store outputs at each resolution for skip connection

396        h = [x]

#

First half of U-Net

398        for m in self.down:
399            x = m(x, t)
400            h.append(x)

#

Middle (bottom)

403        x = self.middle(x, t)

#

Second half of U-Net

406        for m in self.up:
407            if isinstance(m, Upsample):
408                x = m(x, t)
409            else:

#

Get the skip connection from first half of U-Net and concatenate

411                s = h.pop()
412                x = torch.cat((x, s), dim=1)

#

414                x = m(x, t)

#

Final normalization and convolution

417        return self.final(self.act(self.norm(x)))

U-Net model for Denoising Diffusion Probabilistic Models (DDPM)

Swish activation function

Embeddings for t

Residual block

Attention block

Down block

Up block

Middle block

Scale up the feature map by 2×

Scale down the feature map by 21​×

U-Net

First half of U-Net - decreasing resolution

Second half of U-Net - increasing resolution

Embeddings for $t$

Scale up the feature map by $2 \times$

Scale down the feature map by $\frac{1}{2} \times$