From 586716cf3082af0fe4b9280d81c112aae323eacf Mon Sep 17 00:00:00 2001 From: Daniel Falbel Date: Tue, 11 Jul 2023 12:58:01 +0200 Subject: [PATCH 1/2] Add denoising diffusion blog post --- .../denoising-diffusion-impl.Rmd | 792 ++++++++++++++++++ .../images/book.jpg | Bin 0 -> 122813 bytes .../images/overview.png | Bin 0 -> 900262 bytes .../images/reverse-diffusion.png | Bin 0 -> 531214 bytes .../images/schedules-1.png | Bin 0 -> 877625 bytes .../images/schedules.png | Bin 0 -> 38188 bytes .../images/sinusoidal-1.png | Bin 0 -> 41023 bytes .../images/squirrel.png | Bin 0 -> 363349 bytes .../references.bib | 114 +++ 9 files changed, 906 insertions(+) create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/denoising-diffusion-impl.Rmd create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/book.jpg create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/overview.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/reverse-diffusion.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/schedules-1.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/schedules.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/sinusoidal-1.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/images/squirrel.png create mode 100644 _posts/2023-04-21-denoising-diffusion-impl/references.bib diff --git a/_posts/2023-04-21-denoising-diffusion-impl/denoising-diffusion-impl.Rmd b/_posts/2023-04-21-denoising-diffusion-impl/denoising-diffusion-impl.Rmd new file mode 100644 index 00000000..40708a9c --- /dev/null +++ b/_posts/2023-04-21-denoising-diffusion-impl/denoising-diffusion-impl.Rmd @@ -0,0 +1,792 @@ +--- +title: "Implementing Denoising Diffusion with torch and luz" +description: > + In this blog post we walk you trough the implementation of denoising diffusion + probability models using the torch and luz package. +author: + - name: Daniel Falbel + affiliation: Posit + affiliation_url: https://www.posit.co/ +slug: falbeldenoisingdiff +date: 2023-04-21 +categories: + - Torch + - R + - Meta + - Concepts + - Packages/Releases +output: + distill::distill_article: + self_contained: false + toc: true +preview: images/overview.png +bibliography: references.bib +--- + +```{r setup, include=FALSE} +knitr::opts_chunk$set(echo = TRUE, eval = FALSE, fig.width = 6, fig.height = 6) +``` + +If you read the [previous blog post](https://blogs.rstudio.com/ai/posts/2023-04-13-denoising-diffusion/) you might be curious on how the denoising diffusion +models can be implemented in torch. You might have also read the [README](https://github.com/dfalbel/denoising-diffusion/blob/main/README.md) in the denoising +diffusion repository, but it doesn't go into much details on it's implemented. In this +blog post we will try to explain the code and how it relates to formulaes in the +reference papers. + +We are going to use `luz` to train the denoising diffusion model. Using luz allows +us to avoid much of the torch boilerplate code related to supporting training in different +devices like the cpu or the GPU, logging metrics, disabling autograd during evaluation, etc. + +However, since we are not building a standard supervised model, we will need to +implement a custom `step` method. + +## Denoising diffusion training overview + +We will start with a very high level overview of how denoising diffusion models +are trained. The theoretical explanations will come later as we detail each step. +In very practical terms, denoising diffusion models work in the following way: + +```{r fig.cap="Overview of denoising diffusion training procedure", echo=FALSE, eval=TRUE, out.width="100%"} +knitr::include_graphics("images/overview.png") +``` + +The general idea is to take real images, add Gaussian noise to them. The amount of +noise that is added depends on the step of the diffusion process, but during +training this is sampled. + +A neural network, taking as input the noisy image and the **noise rate**, is trained +to predict the amount of noise for each pixel in the noisy image by minimizing the +loss (usually the MSE) between the actual sampled noise and the predicted noise. + +## Implementing the nn_module + +In order to use luz, we need to define a torch `nn_module()`, which contains the +definition of the model we want to train. `nn_module`s in torch requires the user +to implement the `initialize` and `forward` methods. + +In `initialize` we declare the components of that model, what are it's parameters, +layers, etc. In `forward` we describe how the components relate to each other. + +We'll start by defining the initialize method. This method will initialize the following +attributes of the model: + +* `diffusion`: the neural network that is used to predict the noise for each pixel + given the noisy image. To initialize this neural net we need to specify the image + size and other hyper-parameters like the embeddings size, the widths and depths of + the neural net blocks. More details on this neural net later. For now, it's suffiencient + to know that it takes a noisy image and the noise rate as input and tries to predict + the noise for each pixel. + +* `schedule`: the schedule determines the amount of noise that should be added to + the real image depending on the diffusion step we are in. The schedule has a few + configurations related to how it changes the amount of noise for each step. + +* `loss`: The loss function used to train the model. The `diffusion` network will + be trained to minimize that loss. + +* `loss_on`: In the diagram from the first section we showed that the neural network + learns to predict the amount of noise for each pixel, and indeed this seems to lead + to better results. However, the network could be trained to predict the real image + and in this case the loss would be computed between the real image and the predicted + image. This parameter allows us to switch between these two approaches. + +The code for defining the initialize method is shown below. Note that this not the +full module code, we are just showing a small portion so it's easier to navigate. + +```{r} +diffusion_model <- nn_module( + initialize = function(image_size, + embedding_dim = 32, + widths = c(32, 64, 96, 128), + block_depth = 2, + schedule = NULL, + loss = "noise", + loss_on = NULL) { + + self$diffusion <- diffusion(image_size, embedding_dim, widths, block_depth) + + schedule <- if(is.null(schedule)) + diffusion_schedule_config("cosine", 0.02, 0.95) + + self$diffusion_schedule <- do.call(diffusion_schedule, schedule) + self$image_size <- image_size + self$normalize <- normalize(image_size[1]) + + self$loss <- if (is.null(loss)) nnf_l1_loss else loss + self$loss_on <- loss_on + } + # ... implementation of the other methods +) +``` + +We will now define the forward method, which defines how the components initialized +in the above method relate to each other. We start by defining the `denoise` method, +the `forward` method will be just a wrapper around it, but for readability later it's +nice to have a method called `denoise`. + +The denoise method takes as input a batch of noisy image (`images` parameter) and +a batch of rates (the amount of noise/signal for each image). It then computes the +predicted real image and the predicted amount of noise. The way it computes depends +on the `loss_on` configuration, i.e. if the neural network is trained to predict the +noise or the image. Given the prediction for one of them, the rates and the noisy images, +it's possible to compute the other. + +In simpler terms, the `denoise` method is responsible for separating the noise and the signal +from noisy images. And since `forward` is just a wrapper around `denoise`, this is what +happens when we call `model(images, rates)`. + +```{r} +diffusion_model <- nn_module( + # ... other methods + denoise = function(images, rates) { + if (self$loss_on == "noise") { + pred_noises <- self$diffusion(images, rates$noise^2) + pred_images <- (images - rates$noise * pred_noises) / rates$signal + } else { + pred_images <- self$diffusion(images, rates$signal^2) + pred_noises <- (images - rates$signal * pred_images) / rates$noise + } + + list( + pred_noises = pred_noises, + pred_images = pred_images + ) + }, + forward = function(images, rates) { + self$denoise(images, rates) + } + # other methods +) +``` + +We'll now get into how this model is trained, but first a quick introduction on +how to customize the training loop with luz. + +## Quick intro to luz custom steps + +luz provides different levels of abstraction of the training loop and depending on +the type of model you are implementing, you might one to use one or another. + +On the highest level, and the simpler models to implement with luz, are the supervised models. +In this setup, for each batch during training, luz will execute it's **default step** which looks like like the following: + +```{r} +loss <- ctx$model$loss(ctx$model(ctx$input), ctx$target) +loss$backward() +ctx$opt$step() +ctx$opt$zero_grad() +``` + +The `ctx` object in luz is holding the context of that training step, which includes +the model that is being trained, the input and target batches and the optimizer. +You can see more information about the context [here](https://mlverse.github.io/luz/reference/ctx.html). + +Users can override the **default step** by implementing the `step` method in their +`nn_module()`. This is useful when the model in case needs a different procedure +for training than the default supervised procedure. More details on customizing the +`step` can be found [here](https://mlverse.github.io/luz/articles/custom-loop.html#fully-flexible-step). + +## Implementing the step + +We now implement the custom training step for our module. We want it to take a batch +of input images, successively add noise to them, then train the neural net to +be good at denoising those noisy images by minimizing a loss and applyting gradient descent. + +Here's the implementation of `step()`: + +```{r} +diffusion_model <- nn_module( + # ... other methods + step = function() { + ctx$input <- images <- ctx$model$normalize(ctx$input) + + diffusion_times <- torch_rand(images$shape[1], 1, 1, 1, device = images$device) + rates <- self$diffusion_schedule(diffusion_times) + + ctx$buffers$noises <- noises <- torch_randn_like(images) + noisy_images <- rates$signal * images + rates$noise * noises + + ctx$pred <- ctx$model(noisy_images, rates) + + loss <- if (self$loss_on == "noise") { + self$loss(noises, ctx$pred$pred_noises) + } else if (self$loss_on == "image") { + self$loss(images, ctx$pred$pred_images) + } + + if (ctx$training) { + ctx$opt$zero_grad() + loss$backward() + ctx$opt$step() + } + + ctx$loss[[ctx$opt_name]] <- loss$detach() + } + # ... other methods +) +``` + +Note the first 5 lines of code (repeated below), where we add noise to the images. +This is the so called **forward diffusion process*. + +The first technical detail here is that luz will make the data available to the +`step()` trough the context, for example, `ctx$input` will contain the batch of +input images. We first normalize the images, so they have zero mean and a standard +deviation of 1 channel-wise. + +We then sample `diffusion_times` uniformly between 0 and 1. The diffusion time +represents the current step of the diffusion model. 0 means we are in the begging +of the diffusion process. 1 means that we are in the end of the diffusion process. +During training we randomly select diffusion times so the model sees many different +parts of the diffusion process, later, when we sample images we will start with pure +noise (i.e. `diffusion_times=1`) and slowly reduce it until it's 0 - which would +give us a image with no noise added. + +The diffusion times are passed to the denoising schedule, which regulates the amount +of noise depending on the diffusion time. We expect the schedule to give higher noise +rates the later we are in the diffusion process, i.e. the closer diffusion times is to +one. + +We then sample noises from the standard Normal distribution and combine them with +the original images to create the noisy images - which will be the input to the +neural net. Here another technical detail is that we also save the noises to +`ctx$buffer$noises`, this is only to allow us to get the noise values in other +methods, for example when using a luz callback. + +```{r} +ctx$input <- images <- ctx$model$normalize(ctx$input) + +diffusion_times <- torch_rand(images$shape[1], 1, 1, 1, device = images$device) +rates <- self$diffusion_schedule(diffusion_times) + +ctx$buffers$noises <- noises <- torch_randn_like(images) +noisy_images <- rates$signal * images + rates$noise * noises +``` + +One thing that might be strange at this point is that we said that we successively +add noise to the image for a few diffusion steps, but here, we just add noise once. +The next section gives you the theoretical background that allows us to do it in +just one computation. + +## Forward diffusion + +As we mentioned in the previous blog post, this process consists of successively +transforming an input image into noise. In mathematical terms, +we define $x_0$ is the true image. We then define the distribution of images at +the **diffusion step** $t$ is: + +$$q(x_{t} | x_{t-1}) = \mathcal{N}(x_{t-1} \sqrt{1 - \beta_t}, I\beta_t )$$ + +with $\beta_t \in (0,1)$, the diffusion rate. During training we have to be able +to sample from sample from $q(x_t | x_0)$. Using the above formula we would need +a `for` loop that successively adds noise, in pseudo code: + +```{r} +for (t in 1:n_steps) { + image[t] <- images[t-1] + beta[t] * torch_randn_like(images[t-1]) +} +``` + +To make computations easier, we reparametrize this as: + +$$x_t = \sqrt{\bar{\alpha_t}}x_0 + \sqrt{1-\bar{\alpha_t}}\epsilon$$ + +with $\epsilon = \mathcal{N}(0, I)$. We define $\bar{\alpha}_t$ with a **schedule** that +is a strictly decreasing function of $t$ - and also usually defined in terms of $T$ the +maximum number of diffusion steps - so that $\bar{\alpha}_T$ is very close to 0. + +This means that, with a reparametrization trick, we can simplify the code so we don't +need to use the `for` loop, which would also require us to use discrete diffusion times +which complicates further. + +## step() continued + +We now explain the last few lines of the `step()` method that will execute the +training procedure. For easier navigation, we will repeat the last lines below. + +As we saw earlier, `noisy_images` now contains the noisy images created by adding noise +to them. `rates` is a measure of the noise/signal there's in each image. + +We now use `ctx$model()`, which in turns call's `forward` and consequently `denoise` +on the noisy images. This applies the neural network and generates the predictions +for both the real image and the noise. + +Next we compute the loss, depending on the `loss_on` configuration, we will compute +it on images or on noise. `step()` in luz is called both during the training and +the validation loop, thus only if `ctx$training` we will update the weights of the +neural net by using the usual torch `zero_grad(); backward(); step()` procedure. + +Finally we log the loss in the `ctx$loss` context attribute, which is used by luz +to display the loss in the progress bar and save it in the training history. + +```{r} +ctx$pred <- ctx$model(noisy_images, rates) + +loss <- if (self$loss_on == "noise") { + self$loss(noises, ctx$pred$pred_noises) +} else if (self$loss_on == "image") { + self$loss(images, ctx$pred$pred_images) +} + +if (ctx$training) { + ctx$opt$zero_grad() + loss$backward() + ctx$opt$step() +} + +ctx$loss[[ctx$opt_name]] <- loss$detach() +``` + +We have now finished the first pass trough the diffusion model and we this overview +we could start training it. But before that, let's dive a little deeper into the +architecture of the neural net that helps separating noise from signal from the +noisy images. + +## Neural net architecture + +This neural net is basically a [U-Net](https://en.wikipedia.org/wiki/U-Net) (see also previous [blog post](https://blogs.rstudio.com/ai/posts/2019-08-23-unet/)) with a very small change +in how it deals with inputs in order to be able to accommodate the `rates` input. + +We chose using a U-Net here as we are trying to learn the distribution of an image +dataset. If a different kind of data were to be used, e.g. text data or tabular data, +we would need a different combination of layers that makes more sense for the data domain. + +### Unet + +A U-Net is a convolutional neural network that successively downsamples the image resolution while increasing its depth. After a few downsampling blocks, it starts upsampling the representation and decreasing the channels depth. The main idea in U-Net, is that much like Residual Networks, the upsampling blocks take as input both the representation from the previous upsampling block and the representation from a previous downsampling block. + +```{r fig.cap="U-Net model", eval = TRUE, echo=FALSE, out.width="50%"} +knitr::include_graphics("https://blogs.rstudio.com/ai/posts/2019-08-23-unet/images/unet.png") +``` + +Unlike the original U-Net implementation, we use ResNet Blocks in the downsampling and upsampling blocks of the U-Net. Each down or upsampling blocks contain *block_depth* of those ResNet blocks. We also use the [Swish activation](https://en.wikipedia.org/wiki/Swish_function) function. We report those +choises even though we don't think they are necessary for the model performance. + +### Sinusoidal embeddings + +A sinusoidal embedding [@vaswani2017] is used to encode the diffusion times (or +the noise variance) into the model. The visualization below shows how diffusion +times are mapped to the embedding - assuming the dimension size of 32. Each row +is a embedding vector given the diffusion time. Sinusoidal embedding have nice +properties, like preserving the relative distances [@kazemnejad2019:pencoding]. + +```{r fig.cap="Sinusoidal embeddings vector", eval=TRUE, echo=FALSE, out.width="50%"} +knitr::include_graphics("images/sinusoidal-1.png") +``` + +The noise variance (or the weight of the noise in the resulting noisy image) is +encoded into a vector which is then upsampled to the same size of the input image. +They are then concatenated in the channels dimension and passed to the U-Net. + +### Implementing the architecture + +Below we show the code that does this work. This is the neural net module that is +initialized in the `diffusion_model` module in the second section of this article. + +We could go into more details on the `unet` and the `sinusoidal_embedding` but we +will leave up to the reader to search for the code in the [GitHub repository](https://github.com/dfalbel/denoising-diffusion). + +```{r} +diffusion <- nn_module( + initialize = function(image_size, embedding_dim = 32, + widths = c(32, 64, 96, 128), block_depth = 2) { + self$unet <- unet(2*embedding_dim, embedding_dim, widths = widths, block_depth) + self$embedding <- sinusoidal_embedding(embedding_dim = embedding_dim) + + self$conv <- nn_conv2d(image_size[1], embedding_dim, kernel_size = 1) + self$upsample <- nn_upsample(size = image_size[2:3]) + + self$conv_out <- nn_conv2d(embedding_dim, image_size[1], kernel_size = 1) + purrr::walk(self$conv_out$parameters, nn_init_zeros_) + }, + forward = function(noisy_images, noise_variances) { + embedded_variance <- noise_variances |> + self$embedding() |> + self$upsample() + + embedded_image <- noisy_images |> + self$conv() + + unet_input <- torch_cat(list(embedded_image, embedded_variance), dim = 2) + unet_output <- unet_input %>% + self$unet() %>% + self$conv_out() + } +) +``` + +## Schedule + +As mentioned earlier, the schedule determines the amount of noise that should be +added to the real image depending on the diffusion step we are in. For instance, +let's consider the 'cosine' schedule that we use by default. + +We implement it as `nn_module()` even though it doesn't have any trainable +'weights'. Technically at `diffusion_time=0`, the signal rate should be zero +and at `diffusion_time=1` it should be 0, however this leads to problems when +adjusting the model, thus we use the `min_signal_rate` and `max_signal_rates` +that avoids the schedule to return values in the extreme. + +```{r} +cosine_schedule <- nn_module( + initialize = function(min_signal_rate = 0.02, max_signal_rate = 0.98) { + self$start_angle <- nn_buffer(torch_acos(max_signal_rate)) + self$end_angle <- nn_buffer(torch_acos(min_signal_rate)) + }, + forward = function(diffusion_times) { + angles <- self$start_angle + diffusion_times*(self$end_angle - self$start_angle) + + list( + signal = torch_cos(angles), + noise = torch_sin(angles) + ) + } +) +``` + +The graph below shows the values of the signal and noise rates and we progress +in the diffusion process: + +```{r, echo=FALSE} +# script to get the plot below +library(torch) +library(tidyverse) +cosine_schedule <- nn_module( + initialize = function(min_signal_rate = 0.02, max_signal_rate = 0.98) { + self$start_angle <- nn_buffer(torch_acos(max_signal_rate)) + self$end_angle <- nn_buffer(torch_acos(min_signal_rate)) + }, + forward = function(diffusion_times) { + angles <- self$start_angle + diffusion_times*(self$end_angle - self$start_angle) + + list( + signal = torch_cos(angles), + noise = torch_sin(angles) + ) + } +) + +x <- cosine_schedule() +result <- x(seq(0, 1, length.out = 100)) + +df <- as.data.frame(lapply(result, as.numeric)) +df$diffusion_time <- seq(0, 1, length.out = 100) +df %>% + pivot_longer(cols = c(signal, noise), "type") %>% + ggplot(aes(x = diffusion_time, y = value, color = type)) + + geom_line() + + labs( + title = "Signal and noise rates depending on the diffusion time", + x = "Diffusion time", + y = "Rate", + color = "Rate of:" + ) + + theme_minimal() +``` + +```{r, eval=TRUE, fig.cap="Noise/Signal rates per diffusion time", echo=FALSE, out.width="100%"} +knitr::include_graphics("images/schedules.png") +``` + +The image below illustrates what it means to apply an *amount of noise* to the +images. We took a real image and added noise to them according to the cosine schedule +(first row) and linear schedule (second row) for lienarly spaced diffusion times ranging +ranging from 0 to 1. + +```{r eval=TRUE, layout="l-body-outset", fig.cap="Images transformed by the forward diffusion process at different diffusion times", echo=FALSE, out.height="130px"} +knitr::include_graphics("images/schedules-1.png") +``` + +## Training the model + +We have defined the model training loop in `step()` and the neural net architecture +that will be used to make predictions. We can now train the model using this procedure. + +Before training though we need training data. In our experiments we trained the model +with two different datasets: the [Oxford Pets](https://www.robots.ox.ac.uk/~vgg/data/pets/) +dataset and the [Oxford Flowers](https://www.robots.ox.ac.uk/~vgg/data/flowers/). + +### Dataset + +Fortunately those datasets are readily available in the torchdatasets package, thus +we don't need to figure out how to download and read the images from it. We had only +to implement a transform that would center crop and resize the images so they all +have the same size. + +We created a wrapper dataset, that takes either the pets dataset or the flowers dataset +and applies the transforms to the input data. Center cropping the image before resizing +is a good idea for generative models so you preserve the original scale of the image. + +```{r} +diffusion_dataset <- dataset( + "DiffusionDataset", + initialize = function(dataset, image_size, ...) { + + self$image_size <- image_size + + self$transform <- function(x) { + img <- x |> + transform_to_tensor() + + c(ch, height, width) %<-% img$size() + crop_size <- min(height, width) + + img |> + transform_center_crop(c(crop_size, crop_size)) |> + transform_resize(self$image_size) + } + + self$data <- dataset( + dir, + transform = self$transform, + ... + ) + }, + .getitem = function(i) { + self$data[i] + }, + .length = function() { + length(self$data) + } +) +``` + +For instance, this is called with the pets dataset like so: + +```{r} +train_dataset <- diffusion_dataset( + torchdatasets::oxford_pet_dataset, + target_type = "species", + image_size, + split = "train", + download = TRUE +) +``` + +The flowers and pets dataset have return for each item a `list()` with an image +and its target. We will simply ignore the target, as we don't use it when training +our model. We don't show further details about the dataset, feel free to look +at the `datasets.R` file for more details. + +### Fitting + +A simplified version of the code from `train.R` is shown below. Here we setup the +model optimizer and set the values for the model and the optimizer hyper-parameters. + +```{r} +model <- diffusion_model |> + setup( + optimizer = torchopt::optim_adamw + ) |> + set_hparams( + image_size = c(3, 64, 64), + block_depth = 2, + loss = torch::nn_mae_loss(), + widths = c(32, 64, 96, 128), + embedding_dim = 32, + schedule = diffusion_schedule_config("cosine", 0.02, 0.95), + loss_on = "noise" + ) |> + set_opt_hparams(lr = 1e-4, weight_decay = 1e-4) +``` + +We then use `fit` to train the model. + +```{r} +train_dataset <- diffusion_dataset( + torchdatasets::oxford_pet_dataset, + target_type = "species", + image_size, + split = "train", + download = TRUE +) + +fitted <- model |> + fit( + dataset, + epochs = 50, + dataloader_options = list(batch_size = batch_size, num_workers = num_workers) + ) +``` + +Training on the pets dataset takes ~40 minutes using a modern GPU. The model +is trained for 50 epochs. + +## Sampling images + +Once the model has been trained we can sample images by using to reverse the +diffusion process. In slightly more formal terms, with our training procedure +the model learned $p(x_0 | x_t, \bar{\alpha_t})$, i.e. it learned to predict the +initial image $x_0$ given $x_t$ (a noisy image) and $\bar{\alpha_t}$ (the 'amount of noise' in the noisy image.) + +Now suppose we have a image $x_T$, an image at the time step $T$ - which would be equivalent +to `diffusion_time=1`. At the `diffusion_time=1` the noise rate is much higher compared +to the signal rate, thus the image is indistinguishable from pure noise, meaning that +to sample from the distribution $x_T$ we can just sample random noise. + +Given an instance $x_T$ and knowing that $\bar{\alpha_t}$ is 1 we can use our model +to predict $x_0$ by denoising $x_T$. That prediction of $x_0$ might be reasonable, +but it works better if instead of denoising in a single model pass, we denoised +it slowly - just like the forward diffusion process adds noise slowly to the images. +Luckily, given a prediction of $x_0$ and the noise in $x_T$, it's easy to sample +from $x_{T-1}$ as we just need to re-add the noise with the intensity given by +the schedule for that step. + +Below we show the code for the reverse diffusion process, which is implemented +as another method in the `diffusion_model` module. The `diffusion_steps` parameter +is equivalent to $T$ in the equations above - it represents the number of steps in +the diffusion process. The diffusion step is mapped to diffusion times by simply +making $\frac{step}{T}$. + +```{r} +diffusion_model <- nn_module( + # ... + reverse_diffusion = function(initial_noise, diffusion_steps) { + noisy_images <- initial_noise + + diffusion_times <- torch_ones(c(initial_noise$shape[1], 1, 1, 1), device = initial_noise$device) + rates <- self$diffusion_schedule(diffusion_times) + + # we want to combine with 'next' value in mind, thus we remove the first + # value here + for (step in seq(1, 0, length.out = diffusion_steps)[-1]) { + c(pred_noises, pred_images) %<-% self$denoise(noisy_images, rates) + + # remix the predicted components using the next signal and noise rates + diffusion_times <- torch_ones_like(diffusion_times)*step + rates <- self$diffusion_schedule(diffusion_times) + + noisy_images <- rates$signal * pred_images + rates$noise * pred_noises + } + + self$normalize$denormalize(pred_images) + } + # ... +) +``` + +In conclusion, the sampling process does the following: + +- Starts with a noisy image that is just random Gaussian noise. +- Repeats the following for the number of diffusion steps: + + - Uses the model to 'denoise' the noisy image + - Re-creates a noisy image with from the denoised image and a reduced amount of noise specified by the schedule. + +The image below illustrates the process forward diffusion process (first row) +then from random noise, the reverse diffusion process (second row). + +```{r fig.cap="Forward (first row) and reverse (second row) diffusion process for 10 diffusion steps", echo=FALSE, eval=TRUE, layout="1-page", out.height="130px"} +knitr::include_graphics("https://github.com/dfalbel/denoising-diffusion/raw/main/README_files/figure-commonmark/reverse-1.png") +``` + +The `generate` method is implemented a simple wraper around `reverse_diffusion`: + +```{r} +diffusion_model <- nn_module( + # ... + generate = function(num_images, diffusion_steps = 20) { + initial_noise <- torch_randn(c(num_images, self$image_size), device=self$device) + self$reverse_diffusion(initial_noise, diffusion_steps = diffusion_steps) + } + # ... +) +``` + +Images can be generated using `model$generate(num_images = 20)`. + +## Measuring performance + +Measuring performance of image generation models is not a simple process. In general +we want the images to be similar from the data distribution it was trained on, but +they still need to have some novelty. + +There are several metrics that can be used for that end. In this article we used the +Kernel Inception Distance which is relatively easy to compute and faster than the +Frechet Inception Distance - another commonly used metric. + +The KID metric compares feature representations of the generated and real images. +It uses the Inception neural network to extract deep features from both sets of +images and computes the [Kernel Maximum Mean Discrepancy (MMD)](https://stats.stackexchange.com/a/276618) +between the feature representations. + +This metric is not implemented by default, but we can implement a custom luz +metric to compute the KID. We don't go into details of how KID works and the full +implementation can be found in the `kid.R` file in the GitHub repository. We have chosen +to focus on the parts that require technical knowledge of luz. + +### luz custom metrics 101 + +Luz can also be extended by implementing custom metrics. Custom metrics are created using +the `luz_metric` class constructor. Implementing a custom luz metric requires implementing: + +- the `initialize()` method, called at the start of each epoch. It's usually used to initialize + buffers, and possibly other states that might be required for the metric computation. +- `update(preds, target)`: called once per batch iteration. It takes a batch of predictions + and targets and is in general used to update buffers for the metric. +- `compute()`: called at the end of each epoch. Uses the internal state to compute metric values. + This function is called whenever we need to obtain the current metric value. Eg, it’s called every + training step for metrics displayed in the progress bar, but only called once per epoch to record it’s + value when the progress bar is not displayed. + +See the [luz metric documentation](https://mlverse.github.io/luz/reference/luz_metric.html) for more information. + +### Implementing KID + +Let's assume that `metric_kid_base` is a luz metric that takes a batch of generated +images and a batch of real images to compute the KID. + +Note that neither 'preds' or 'targets' are used in the update method. Instead, we +generate new images using the `generate` method and compare with the original input +images, that we stored in our custom `step` method. + +```{r} +metric_kid <- luz_metric( + "kid_metric", + inherit = metric_kid_base, + initialize = function(diffusion_steps = 5) { + super$initialize() + self$diffusion_steps <- diffusion_steps + }, + update = function(preds, target) { + ctx$model$eval() + with_no_grad({ + images <- ctx$model$normalize$denormalize(ctx$input) + generated_images <- ctx$model$generate( + num_images=images$shape[1], + diffusion_steps=self$diffusion_steps + ) + super$update(images, generated_images) + }) + ctx$model$train() + } +) +``` + +Once a custom metric is implemented, it can be passed to the `setup()` function, +just like any other builtin metric. For example, the setup section shown in the 'Fitting' section, +can be replaced by: + +```{r} +# ... +model <- diffusion_model |> + setup( + optimizer = optimizer, + metrics = luz_metric_set( + metrics = list(image_loss(), noise_loss()) + ) + ) +# ... +``` + +## Conclusion + +This blog post is hopefully a good way to get started with diffusion models using torch. +We tried to describe as much as possible the code available in [this repository](https://github.com/dfalbel/denoising-diffusion) and we expect that you +can now visualize how the components connect to each other. + +The code repository also contains experiment results conducted using the GuildAI for +R integration. Please feel free to contact us on [GitHub](https://github.com/dfalbel/denoising-diffusion) +if you have questions. + +Thank you for reading! \ No newline at end of file diff --git a/_posts/2023-04-21-denoising-diffusion-impl/images/book.jpg b/_posts/2023-04-21-denoising-diffusion-impl/images/book.jpg new file mode 100644 index 0000000000000000000000000000000000000000..0b3d970e8a03ee89d6a2b7ce3b274dca2bfdcbbd GIT binary patch literal 122813 zcmeFZbwC_Vvp2d}aMuJ4?(PyKxCVC!&IT5jT|5cy?h-tBaEA~qcz__mU4y&dC6DDf z=l#BWzx(ICr`g{AO?OpwRdvry&ukYD(+^7kEO}`;X#fl~F~cwe01wM_Yf@mKB>*5N z#|S_I004*p92htN3=}=)XD|rAN>eB%{u94}V){R6V4;`;1{Q!0EuEm56N>Smr5`i{ zgW!Mh#zS!dG&4g5KFa*7l#x?J!%AH~A_iF=^cuu$#bpd}Hs9_~*~pFoeF{D$%W#E*0^uz$!Q zgVMwPiQ8zQ4bX)2>+j)r-#wZErF@tLyaXUUd4l)^0SOTi5g8c?1q}}!4HXsbDGn|s z9_cf3GSX)xBox%FbQF}#R3s$yd<@L&9NgU8X|=lnlT51#?pNPr#~sIim)SZo+LY?y~G06A1ogvXJFfhzk)f`NsDM|gsWgp7g; zC8))E^b{N{JRAZ7JUo=z8`=(l$40<;&LRE;SJecO(h-mIZA>;2l|*F+zS`&^HJ7Q= 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