This article is aimed at those who want to understand exactly how Diffusion Models<\/a> work, with no prior knowledge expected. I\u2019ve tried to use illustrations wherever possible to provide visual intuitions on each part of these models. I\u2019ve kept mathematical notation and equations to a minimum, and where they are necessary I\u2019ve tried to define and explain them as they occur.<\/p>\n I\u2019ve framed this article around three main questions:<\/p>\n The examples will be based on the glyffuser<\/a>, a minimal text-to-image diffusion model that I previously implemented and wrote about<\/a>. The architecture of this model is a standard text-to-image denoising diffusion model without any bells or whistles. It was trained to generate pictures of new \u201cChinese\u201d glyphs from English definitions. Have a look at the picture below\u200a\u2014\u200aeven if you\u2019re not familiar with Chinese writing, I hope you\u2019ll agree that the generated glyphs look pretty similar to the real ones!<\/p>\n Generative Ai<\/a> models are often said to take a big pile of data and \u201clearn\u201d it. For text-to-image diffusion models, the data takes the form of pairs of images and descriptive text. But what exactly is it that we want the model to learn? First, let\u2019s forget about the text for a moment and concentrate on what we are trying to generate: the images.<\/p>\n Broadly, we can say that we want a generative AI model to learn the underlying probability distribution<\/em> of the data. What does this mean? Consider the one-dimensional normal (Gaussian) distribution below, commonly written \ud835\udca9<\/strong>(\u03bc<\/em>,\u03c3<\/em>\u00b2<\/strong>) and parameterized<\/em> with mean \u03bc <\/em>= 0 and variance \u03c3<\/em>\u00b2<\/strong> = 1. The black curve below shows the probability density function. We can sample<\/em> from it: drawing values such that over a large number of samples, the set of values reflects the underlying distribution. These days, we can simply write something like We could think of a set of numbers sampled from the above normal distribution as a simple dataset, like that shown as the orange histogram above. In this particular case, we can calculate the parameters of the underlying distribution using maximum likelihood estimation<\/em>, i.e. by working out the mean and variance. The normal distribution estimated from the samples is shown by the dotted line above. To take some liberties with terminology, you might consider this as a simple example of \u201clearning\u201d an underlying probability distribution. We can also say that here we explicitly<\/em> learnt the distribution, in contrast with the implicit<\/em> methods that diffusion models use.<\/p>\n Conceptually, this is all that generative AI is doing\u200a\u2014\u200alearning a distribution, then sampling from that distribution!<\/p>\n What, then, does the underlying probability distribution of a more complex dataset look like, such as that of the image dataset we want to use to train our diffusion model?<\/p>\n First, we need to know what the representation<\/em> of the data is. Generally, a machine learning (ML) model requires data inputs with a consistent representation, i.e. format. For the example above, it was simply numbers (scalars). For images, this representation is commonly a fixed-length vector.<\/p>\n The image dataset used for the glyffuser model is ~21,000 pictures of Chinese glyphs. The images are all the same size, 128 \u00d7 128 = 16384 pixels, and greyscale (single-channel color). Thus an obvious choice for the representation is a vector x<\/strong> of length 16384, where each element corresponds to the color of one pixel: x <\/strong>= (x<\/em>\u2081<\/strong>,x<\/em>\u2082,\u2026,x<\/em>\u2081\u2086\u2083\u2088\u2084<\/strong>). We can call the domain of all possible images for our dataset \u201cpixel space\u201d.<\/p>\n We make the assumption that our individual data samples, x<\/em>, are actually sampled from an underlying probability distribution, q<\/em>(x<\/em>), in pixel space, much as the samples from our first example were sampled from an underlying normal distribution in 1-dimensional space. Note: the notation x <\/em>\u223c q<\/em>(x<\/em>) is commonly used to mean: \u201cthe random variable x<\/em> sampled from the probability distribution q<\/em>(x<\/em>).\u201d<\/p>\n This distribution is clearly much more complex than a Gaussian and cannot be easily parameterized\u200a\u2014\u200awe need to learn it with a ML model, which we\u2019ll discuss later. First, let\u2019s try to visualize the distribution to gain a better intution.<\/p>\n As humans find it difficult to see in more than 3 dimensions, we need to reduce the dimensionality of our data. A small digression on why this works: the manifold hypothesis<\/a> posits that natural datasets lie on lower dimensional manifolds embedded in a higher dimensional space\u200a\u2014\u200athink of a line embedded in a 2-D plane, or a plane embedded in 3-D space. We can use a dimensionality reduction technique such as UMAP<\/a> to project our dataset from 16384 to 2 dimensions. The 2-D projection retains a lot of structure, consistent with the idea that our data lie on a lower dimensional manifold embedded in pixel space. In our UMAP, we see two large clusters corresponding to characters in which the components are arranged either horizontally (e.g. \u660e) or vertically (e.g. \u8349). An interactive version of the plot below with popups on each datapoint is linked here<\/a>.<\/p>\n Let\u2019s now use this low-dimensional UMAP dataset as a visual shorthand for our high-dimensional dataset. Remember, we assume that these individual points have been sampled from a continuous underlying probability distribution q<\/em>(x<\/em>). To get a sense of what this distribution might look like, we can apply a KDE (kernel density estimation) over the UMAP dataset. (Note: this is just an approximation for visualization purposes.)<\/p>\n This gives a sense of what q<\/em>(x<\/em>) should look like: clusters of glyphs correspond to high-probability regions of the distribution. The true q<\/em>(x<\/em>) lies in 16384 dimensions\u200a\u2014\u200athis is the distribution we want to learn with our diffusion model.<\/p>\n We showed that for a simple distribution such as the 1-D Gaussian, we could calculate the parameters (mean and variance) from our data. However, for complex distributions such as images, we need to call on ML methods. Moreover, what we will find is that for diffusion models in practice, rather than parameterizing the distribution directly, they learn it implicitly<\/em> through the process of learning how to transform noise into data over many steps.<\/p>\n The aim of generative AI such as diffusion models is to learn the complex probability distributions underlying their training data and then sample from these distributions.<\/p>\n Diffusion models have recently come into the spotlight as a particularly effective method for learning these probability distributions. They generate convincing images by starting from pure noise and gradually refining it. To whet your interest, have a look at the animation below that shows the denoising process generating 16 samples.<\/p>\n In this section we\u2019ll only talk about the mechanics of how these models work but if you\u2019re interested in how they arose from the broader context of generative models, have a look at the further reading<\/a> section below.<\/p>\n Let\u2019s first precisely define noise, since the term is thrown around a lot in the context of diffusion. In particular, we are talking about Gaussian noise: consider the samples we talked about in the section about probability distributions<\/a>. You could think of each sample as an image of a single pixel of noise. An image that is \u201cpure Gaussian noise\u201d, then, is one in which each pixel value is sampled from an independent standard Gaussian distribution, \ud835\udca9<\/strong>(0,1). For a pure noise image in the domain of our glyph dataset, this would be noise drawn from 16384 separate Gaussian distributions. You can see this in the previous animation. One thing to keep in mind is that we can choose the means of these noise distributions, i.e. center<\/em> them, on specific values\u200a\u2014\u200athe pixel values of an image, for instance.<\/p>\n For convenience, you\u2019ll often find the noise distributions for image datasets written as a single multivariate distribution \ud835\udca9<\/strong>(0,I<\/em><\/strong>) where I<\/em><\/strong> is the identity matrix, a covariance matrix with all diagonal entries equal to 1 and zeroes elsewhere. This is simply a compact notation for a set of multiple independent Gaussians\u200a\u2014\u200ai.e. there are no correlations between the noise on different pixels. In the basic implementations of diffusion models, only uncorrelated (a.k.a. \u201cisotropic\u201d) noise is used. This article<\/a> contains an excellent interactive introduction on multivariate Gaussians.<\/p>\n Below is an adaptation of the somewhat-famous diagram from Ho et al<\/em>.<\/a>\u2019s seminal paper \u201cDenoising Diffusion Probabilistic Models<\/em>\u201d which gives an overview of the whole diffusion process:<\/p>\n I found that there was a lot to unpack in this diagram and simply understanding what each component meant was very helpful, so let\u2019s go through it and define everything step by step.<\/p>\n We previously used x <\/em>\u223c q<\/em>(x<\/em>) to refer to our data. Here, we\u2019ve added a subscript, x<\/em>\u209c, to denote timestep t<\/em> indicating how many steps of \u201cnoising\u201d have taken place. We refer to the samples noised a given timestep as x <\/em>\u223c q<\/em>(x<\/em>\u209c). x<\/em>\u2080\u200b is clean data and x<\/em>\u209c (t<\/em> = T<\/em>) \u223c \ud835\udca9<\/strong>(0,1) is pure noise.<\/p>\n We define a forward diffusion<\/em> process whereby we corrupt samples with noise. This process is described by the distribution q<\/em>(x<\/em>\u209c|x<\/em>\u209c\u208b\u2081). If we could access the hypothetical reverse process q<\/em>(x<\/em>\u209c\u208b\u2081|x<\/em>\u209c), we could generate samples from noise. As we cannot access it directly because we would need to know x<\/em>\u2080\u200b, we use ML to learn the parameters, \u03b8<\/em>, of a model of this process, \ud835\udc5d\u03b8<\/em>(\ud835\udc65\u209c\u208b\u2081\u2223\ud835\udc65\u209c). (That should be p<\/em> subscript \u03b8 <\/em>but medium cannot render it.)<\/p>\n In the following sections we go into detail on how the forward and reverse diffusion processes work.<\/p>\n Used as a verb, \u201cnoising\u201d an image refers to applying a transformation that moves it towards pure noise by scaling down its pixel values toward 0 while adding proportional Gaussian noise. Mathematically, this transformation is a multivariate Gaussian distribution centered on the pixel values of the preceding image.<\/p>\n In the forward diffusion process, this noising distribution is written as q<\/em>(x<\/em>\u209c|x<\/em>\u209c\u208b\u2081) where the vertical bar symbol \u201c|\u201d is read as \u201cgiven\u201d or \u201cconditional on\u201d, to indicate the pixel means are passed forward from q<\/em>(x<\/em>\u209c\u208b\u2081) At t<\/em> = T<\/em> where T<\/em> is a large number (commonly 1000) we aim to end up with images of pure noise (which, somewhat confusingly, is also a Gaussian distribution, as discussed previously<\/a>).<\/p>\n The marginal<\/em> distributions q<\/em>(x<\/em>\u209c) represent the distributions that have accumulated the effects of all the previous noising steps (marginalization<\/em> refers to integration over all possible conditions, which recovers the unconditioned distribution).<\/p>\n Since the conditional distributions are Gaussian, what about their variances? They are determined by a variance schedule<\/em> that maps timesteps to variance values. Initially, an empirically determined schedule of linearly increasing values from 0.0001 to 0.02 over 1000 steps was presented in Ho et al<\/em>.<\/a> Later research by Nichol & Dhariwal<\/a> suggested an improved cosine schedule. They state that a schedule is most effective when the rate of information destruction through noising is relatively even per step throughout the whole noising process.<\/p>\n As we encounter Gaussian distributions both as pure noise q<\/em>(x<\/em>\u209c, t<\/em> = T<\/em>) and as the noising distribution q<\/em>(x<\/em>\u209c|x<\/em>\u209c\u208b\u2081), I\u2019ll try to draw the distinction by giving a visual intuition of the distribution for a single noising step, q<\/em>(x<\/em>\u2081\u2223x<\/em>\u2080), for some arbitrary, structured 2-dimensional data:<\/p>\n The distribution q<\/em>(x<\/em>\u2081\u2223x<\/em>\u2080) is Gaussian, centered around each point in x<\/em>\u2080, shown in blue. Several example points x<\/em>\u2080\u207d\u2071\u207e are picked to illustrate this, with q<\/em>(x<\/em>\u2081\u2223x<\/em>\u2080 = x<\/em>\u2080\u207d\u2071\u207e) shown in orange.<\/p>\n In practice, the main usage of these distributions is to generate specific instances of noised samples for training (discussed further below). We can calculate the parameters of the noising distributions at any timestep t<\/em> directly from the variance schedule, as the chain of Gaussians is itself also Gaussian. This is very convenient, as we don\u2019t need to perform noising sequentially\u2014for any given starting data x<\/em>\u2080\u207d\u2071\u207e, we can calculate the noised sample x<\/em>\u209c\u207d\u2071\u207e by sampling from q<\/em>(x<\/em>\u209c\u2223x<\/em>\u2080 = x<\/em>\u2080\u207d\u2071\u207e) directly.<\/p>\n Let\u2019s now return to our glyph dataset (once again using the UMAP visualization as a visual shorthand). The top row of the figure below shows our dataset sampled from distributions noised to various timesteps: x<\/em>\u209c \u223c q<\/em>(x<\/em>\u209c). As we increase the number of noising steps, you can see that the dataset begins to resemble pure Gaussian noise. The bottom row visualizes the underlying probability distribution q<\/em>(x<\/em>\u209c).<\/p>\n It follows that if we knew the reverse distributions q<\/em>(x<\/em>\u209c\u208b\u2081\u2223x<\/em>\u209c), we could repeatedly subtract a small amount of noise, starting from a pure noise sample x<\/em>\u209c at t <\/em>= T<\/em> to arrive at a data sample x<\/em>\u2080 \u223c q<\/em>(x<\/em>\u2080). In practice, however, we cannot access these distributions without knowing x<\/em>\u2080 beforehand. Intuitively, it\u2019s easy to make a known image much noisier, but given a very noisy image, it\u2019s much harder to guess what the original image was.<\/p>\n So what are we to do? Since we have a large amount of data, we can train an ML model to accurately guess the original image that any given noisy image came from. Specifically, we learn the parameters \u03b8<\/em> of an ML model that approximates the reverse noising distributions, p\u03b8<\/em>(x<\/em>\u209c\u208b\u2081 \u2223 x<\/em>\u209c) for t<\/em> = 0,\u00a0\u2026, T<\/em>. In practice, this is embodied in a single noise prediction model<\/em> trained over many different samples and timesteps. This allows it to denoise any given input, as shown in the figure below.<\/p>\n Next, let\u2019s go over how this noise prediction model is implemented and trained in practice.<\/p>\n First, we define the ML model\u200a\u2014\u200agenerally a deep neural network of some sort\u200a\u2014\u200athat will act as our noise prediction model. This is what does the heavy lifting! In practice, any ML model that inputs and outputs data of the correct size can be used; the U-net<\/a>, an architecture particularly suited to learning images, is what we use here and frequently chosen in practice. More recent models also use vision transformers<\/em><\/a>.<\/p>\n Then we run the training loop depicted in the figure above:<\/p>\n Note: A neural network is essentially a function with a huge number of parameters (on the order of 10\u2076 <\/strong>for the glyffuser). Neural network ML models are trained by iteratively updating their parameters using backpropagation<\/em> to minimize a given loss function over many training data examples. This<\/a> is an excellent introduction. These parameters effectively store the network\u2019s \u201cknowledge\u201d.<\/p>\n A noise prediction model trained in this way eventually sees many different combinations of timesteps and data examples. The glyffuser, for example, was trained over 100 epochs<\/em> (runs through the whole data set), so it saw around 2 million data samples. Through this process, the model implicity learns the reverse diffusion distributions over the entire dataset at all different timesteps. This allows the model to sample the underlying distribution q<\/em>(x<\/em>\u2080) by stepwise denoising starting from pure noise. Put another way, given an image noised to any given level, the model can predict how to reduce the noise based on its guess of what the original image. By doing this repeatedly, updating its guess of the original image each time, the model can transform any noise to a sample that lies in a high-probability region of the underlying data distribution.<\/p>\n We can now revisit this video of the glyffuser denoising process. Recall a large number of steps from sample to noise e.g. T<\/em> = 1000 is used during training to make the noise-to-sample trajectory very easy for the model to learn, as changes between steps will be small. Does that mean we need to run 1000 denoising steps every time we want to generate a sample?<\/p>\n Luckily, this is not the case. Essentially, we can run the single-step noise prediction but then rescale it to any given step, although it might not be very good if the gap is too large! This allows us to approximate the full sampling trajectory with fewer steps. The video above uses 120 steps, for instance (most implementations will allow the user to set the number of sampling steps).<\/p>\n Recall that predicting the noise at a given step is equivalent to predicting the original image x<\/em>\u2080, and that we can access the equation for any noised image deterministically using only the variance schedule and x<\/em>\u2080. Thus, we can calculate x<\/em>\u209c\u208b\u2096 based on any denoising step. The closer the steps are, the better the approximation will be.<\/p>\n Too few steps, however, and the results become worse as the steps become too large for the model to effectively approximate the denoising trajectory. If we only use 5 sampling steps, for example, the sampled characters don\u2019t look very convincing at all:<\/p>\n There is then a whole literature on more advanced sampling methods beyond what we\u2019ve discussed so far, allowing effective sampling with much fewer steps. These often reframe the sampling as a differential equation to be solved deterministically, giving an eerie quality to the sampling videos\u200a\u2014\u200aI\u2019ve included one at the end<\/a> if you\u2019re interested. In production-level models, these are usually preferred over the simple method discussed here, but the basic principle of deducing the noise-to-sample trajectory is the same. A full discussion is beyond the scope of this article but see e.g. this paper<\/a> and its corresponding implementation<\/a> in the Hugging Face To me, it was still not 100% clear why training the model on noise prediction generalises so well. I found that an alternative interpretation of diffusion models known as \u201cscore-based modeling\u201d filled some of the gaps in intuition (for more information, refer to Yang Song\u2019s definitive article<\/a> on the topic.)<\/p>\n I try to give a visual intuition in the bottom row of the figure above: essentially, learning the noise in our diffusion model is equivalent<\/a> (to a constant factor) to learning the score function<\/em>, which is the gradient of the log of the probability distribution: \u2207\u2093 log q<\/em>(x<\/em>). As a gradient, the score function represents a vector field with vectors pointing towards the regions of highest probability density. Subtracting the noise at each step is then equivalent to moving following the directions in this vector field towards regions of high probability density.<\/p>\nIntro<\/h3>\n
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![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_vJlBMS83Bb0b3JeEbKLo3A-1024x557.png?resize=1024%2C557&ssl=1\")
What exactly is it that diffusion models\u00a0learn?<\/h3>\n
Probability distributions<\/h4>\n
x = random.gauss(0, 1)<\/code> in Python to sample from the standard normal distribution, although the computational sampling process itself is non-trivial!<\/p>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_xmmfBzgyuFwo2y9Wz1tn0Q.png?resize=583%2C217&ssl=1\")
Data representations<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_OUpCAiPGlP0ZPSD4SEHAlg-1024x1024.png?resize=1024%2C1024&ssl=1\")
Dataset visualization<\/h3>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_oWvKkHUWoy1alT50ugt8Lg.png?resize=720%2C450&ssl=1\")
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_dYvWE6ExaKSeo_YWU-v97w-1024x429.png?resize=1024%2C429&ssl=1\")
<\/figure>\nTakeaway<\/h4>\n
How and why do diffusion models\u00a0work?<\/h3>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_V07XdbZNdDA0IL-IfGHgPQ.gif?resize=522%2C552&ssl=1\")
<\/figure>\nWhat is\u00a0\u201cnoise\u201d?<\/h4>\n
Diffusion process\u00a0overview<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_ACv-2kbERa3pZHO1p8OXVQ-1024x381.png?resize=1024%2C381&ssl=1\")
Forward diffusion, or \u201cnoising\u201d<\/h4>\n
Forward diffusion intuition<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_PcgCmi3HU44EC_osMkEuMA-1024x508.png?resize=1024%2C508&ssl=1\")
Forward diffusion visualization<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_Hx4xD8VSNvfKx6nkr-xxtQ-1024x394.png?resize=1024%2C394&ssl=1\")
Reverse diffusion overview<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_5jbkLcVGGY-GcBN44WRlYQ-1024x221.png?resize=1024%2C221&ssl=1\")
How the model is implemented<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_PP6y1_sfgU-r8h4UdfHnOA-1024x312.png?resize=1024%2C312&ssl=1\")
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Reverse diffusion in\u00a0practice<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_V07XdbZNdDA0IL-IfGHgPQ-1.gif?resize=522%2C552&ssl=1\")
<\/figure>\n![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_TmF7SZlxFhNliilElvqOtw.gif?resize=522%2C552&ssl=1\")
<\/figure>\ndiffusers<\/code> library for more information.<\/p>\nAlternative intuition from score\u00a0function<\/h4>\n
![\"\"](\"https:\/\/i0.wp.com\/towardsdatascience.com\/wp-content\/uploads\/2025\/02\/1_958_vRbvMlDvEHgMo4jGfA-1024x558.png?resize=1024%2C558&ssl=1\")