What are Diffusion Models?

Summary

扩散模型是一种新兴的生成模型，通过模拟数据从清晰图像逐渐加入高斯噪声至纯噪声的过程，然后逆向去除噪声以恢复原始图像。这种模型在图像生成和条件设置方面表现出色，甚至在某些任务中超越了生成对抗网络（GANs）。视频解释了扩散模型的基本机制，包括正向扩散过程、逆向去噪过程以及如何通过优化变分下界来训练模型。此外，还探讨了如何将扩散模型应用于条件生成和图像修复等任务。

Takeaways

• 🌟 扩散模型（Diffusion Models）是生成模型领域的新兴方法，尤其在图像生成方面取得了显著进展。
• 🚀 扩散模型通过逐步添加高斯噪声将图像转换为纯噪声样本，然后学习逆向过程以恢复原始图像。
• 🔄 正向扩散过程模拟数据逐渐变为噪声，而逆向过程则尝试从噪声中恢复数据。
• 📈 扩散模型使用马尔可夫链（Markov Chain）来描述正向噪声添加过程，每一步的分布仅依赖于前一步的样本。
• 🎯 逆向过程中，模型通过学习去除噪声的步骤，逐步将样本引导回数据流形，生成合理的样本。
• 🔧 在训练过程中，扩散模型的目标是最大化一个下界（变分下界），而不是直接最大化模型对数据的密度分配。
• 🔄 扩散模型可以条件性地生成样本，例如根据类别标签或文本描述来生成图像。
• 🖼️ 扩散模型在图像修复（inpainting）任务中也表现出色，通过专门训练的模型来填补图像中的缺失部分。
• 📊 扩散模型与其他生成模型（如生成对抗网络GANs）相比，在某些任务上表现出更好的性能。
• 🛠️ 扩散模型的训练依赖于对正向过程中每一步的噪声水平的准确学习。
• 🔗 扩散模型与概率流（Probability Flow ODE）有紧密联系，后者通过数值积分近似对数似然。

Q & A

什么是扩散模型？

扩散模型是一种生成模型，它通过模拟数据从清晰样本逐渐加入噪声直到变成纯噪声的过程，然后再逆向操作，逐步去除噪声以恢复原始样本，从而学习数据的分布。

扩散模型在图像生成领域取得了哪些成就？

扩散模型在图像生成领域取得了显著的成功，它们在某些任务上甚至超过了生成对抗网络（GANs）等其他类型的生成模型。例如，最近的扩散模型在感知质量指标上超越了GANs，并且在将文本描述转换为图像、绘画和图像操作等条件设置中表现出色。

扩散模型的正向过程是如何定义的？

扩散模型的正向过程是通过一个马尔可夫链来定义的，其中每一步的分布仅依赖于前一步的样本。这个过程逐渐向图像中加入噪声，最终形成一个纯噪声分布。

为什么扩散模型使用小步长的正向过程？

使用小步长的正向过程意味着学习逆向过程不会太困难。小步长减少了每一步的不确定性，使得模型能够更准确地推断出前一步的状态。

扩散模型的逆向过程是如何工作的？

扩散模型的逆向过程是一个学习过程，它通过一个参数化的反向马尔可夫链来逐步去除噪声，目标是恢复到数据的分布。这个过程通过最大化一个变分下界来训练，而不是直接最大化似然函数。

扩散模型的训练目标是什么？

扩散模型的训练目标是最大化变分下界（也称为证据下界），这是一个关于边际对数似然的下界。这个目标包括一个重建项和一个KL散度项，分别鼓励模型最大化数据的期望密度和使近似后验分布与潜在变量的先验分布相似。

扩散模型如何实现条件生成？

扩散模型可以通过将条件变量（如类别标签或句子描述）作为额外输入在训练期间进行条件生成。在推理时，模型可以使用这些条件信息来生成特定于条件的样本。

扩散模型在图像修复（inpainting）任务中的表现如何？

扩散模型在图像修复任务中取得了成功。通过专门针对此任务进行微调的模型，可以在给定完整上下文的情况下更好地填充图像中缺失的部分，从而避免边缘伪影。

扩散模型与变分自编码器（VAEs）有什么相似之处？

扩散模型与VAEs相似之处在于它们都可以被视为潜在变量生成模型，并且它们都使用变分下界作为训练目标。然而，扩散模型的正向过程通常是固定的，而逆向过程是学习的重点。

扩散模型的连续时间形式可以产生什么？

扩散模型的连续时间形式可以产生所谓的概率流（probability flow ODE），这允许通过数值积分来近似对数似然，从而提供了一种不同于变分下界的密度估计方法。

扩散模型与分数匹配模型有什么关系？

扩散模型与分数匹配模型之间存在紧密的联系。分数匹配模型中的分数实际上等于扩散模型中预测的噪声，直到一个缩放因子。因此，我们可以将扩散模型中的去噪过程视为近似地跟随数据对数密度梯度的过程。

Outlines

00:00

🌟 扩散模型的基本概念

本段介绍了扩散模型的基本原理，即通过向图像中添加高斯噪声并重复该过程，最终得到一个纯噪声样本。然后，扩散模型的目标是从这个纯噪声图像开始，逐步去除噪声，恢复出原始的清晰图像。这种方法在图像生成领域取得了成功，并且在某些任务上超越了其他生成模型，如生成对抗网络（GANs）。视频将探讨扩散模型的基本机制及其在不同生成设置中的适应性。

05:02

📈 扩散模型的正向与逆向过程

这一部分详细解释了扩散模型中的正向和逆向过程。正向过程通过逐步添加噪声将样本推向数据流形之外，而逆向过程则训练模型回到数据流形，生成合理的样本。介绍了如何通过最大化一个下界来优化模型，这个下界是变分下界，它包括重构项和KL散度项。此外，还讨论了如何将扩散模型视为潜在变量生成模型，并解释了训练目标的设定。

10:04

🛠️ 扩散模型的训练与实现

本段讨论了扩散模型的训练细节，包括如何实现逆向步骤、如何处理噪声变量以及如何通过重参数化来提高样本质量。作者选择了将逆向步骤的方差设置为时间特定的常数，并提出了一种简单的变分下界版本，该版本在训练中关注更具挑战性的噪声步骤。此外，还探讨了如何使扩散模型有条件地采样，例如使用类别标签或文本描述，以及如何进行图像修复（inpainting）等条件生成问题。

15:04

🚀 扩散模型的进展与前景

最后一段总结了扩散模型的进展和未来方向。扩散模型在生成任务中的速度较慢，但通过持续的研究，正在努力提高采样速度。扩散模型还与概率流（probability flow ODE）和分数匹配模型有紧密联系，这些模型通过数值积分近似对数似然。扩散模型的发展势头强劲，其在图像生成和密度估计基准测试中的性能令人兴奋。

Keywords

💡扩散模型

扩散模型是一种生成模型，通过模拟数据从清晰图像逐渐加入高斯噪声直至变为纯噪声的过程，然后学习如何逆转这一过程，从噪声图像中恢复出原始图像。在视频中，扩散模型被用来生成图像，并在某些任务上超越了生成对抗网络（GANs）。

💡高斯噪声

高斯噪声是一种统计噪声，其概率密度函数符合正态分布，即钟形曲线。在扩散模型中，高斯噪声被用来逐步模糊和扭曲原始图像，直到图像变得无法识别，形成纯噪声图像。

💡生成对抗网络（GANs）

生成对抗网络（GANs）是一种由生成器和判别器组成的深度学习模型，用于生成新的、与真实数据相似的数据实例。在视频中，扩散模型在某些任务上的表现已经开始与GANs相媲美甚至超越。

💡条件生成

条件生成是指生成模型在生成数据时考虑额外的条件信息，如文本描述、类别标签等，以生成符合特定条件的数据。在视频中，扩散模型被用于将文本描述转换为图像，展示了其在条件生成任务中的应用。

💡马尔可夫链

马尔可夫链是一种数学模型，用于描述随机过程中各个状态之间的转换，其中下一个状态的概率仅依赖于当前状态。在扩散模型中，正向扩散过程被建模为一个马尔可夫链，每个时间步骤的分布仅依赖于前一个时间步的样本。

💡变分自编码器（VAEs）

变分自编码器（VAEs）是一种生成模型，包含编码器和解码器两部分，编码器负责将输入数据映射到潜在空间的分布，解码器则从潜在空间重构数据。扩散模型的训练目标与VAEs类似，都是通过最大化边证据下界来训练的。

💡变分下界

变分下界，也称为证据下界，是用于训练生成模型的一种目标函数，它提供了对数据的边际对数似然的下界。在扩散模型中，通过最大化变分下界来训练模型，使其能够更准确地生成数据。

💡KL散度

KL散度，即Kullback-Leibler散度，是衡量两个概率分布差异的指标。在扩散模型中，KL散度用于鼓励模型的近似后验分布与潜在变量的先验分布相似。

💡重采样

重采样是指在生成过程中，从模型学习的分布中抽取样本的过程。在扩散模型中，重采样用于从纯噪声分布开始，通过学习到的反向过程逐步生成清晰的数据样本。

💡自编码器

自编码器是一种神经网络，它通过一个编码器将输入数据压缩成一个低维表示，然后通过一个解码器重构输出。在扩散模型中，虽然正向过程是固定的，但反向过程类似于自编码器的解码器部分，用于从潜在空间重构数据。

💡修复

修复是指在图像或数据中填补缺失或损坏部分的过程。扩散模型在修复任务中表现出色，可以通过训练来填补图像中的空洞或缺失区域。

💡概率流

概率流是一种由扩散模型启发的模型，它通过数值积分来近似对数似然函数。概率流允许对数据的生成过程进行更精细的控制，从而在生成模型领域提供了一种新的视角。

Highlights

扩散模型（diffusion models）是生成建模领域中的一种新兴方法，尤其在图像生成方面取得了显著成功。

扩散模型在某些任务上已经超越了生成对抗网络（GANs）等其他类型的生成模型。

扩散模型的基本思想是模拟一个逐渐添加高斯噪声的过程，然后学习如何逆转这个过程，从噪声图像恢复出清晰的图像。

正向扩散过程被设计成一个马尔可夫链，其中每一步的分布仅依赖于前一步的样本。

扩散模型的反向过程被训练来产生回到数据流形的轨迹，从而生成合理的样本。

扩散模型的训练目标不是直接最大化似然目标，而是最大化一个下界，即变分下界或证据下界。

扩散模型可以视为一种潜在变量生成模型，其中正向过程类似于编码器从数据生成潜在变量，反向过程类似于解码器从潜在变量重构数据。

扩散模型的训练只需要训练一个网络，与需要联合训练两个网络的VAE不同。

扩散模型的反向步骤被参数化为高斯分布，这有助于减少训练过程中的方差。

扩散模型可以实现条件采样，例如根据类别标签或文本描述生成图像。

扩散模型可以通过特殊训练来指导采样，而不依赖于第二个网络。

扩散模型在图像修复（inpainting）任务中取得了成功，通过微调特定于该任务的模型来获得更好的结果。

扩散模型与其他退化模型相比，受限于慢速的马尔可夫链，这与GANs的单次前向传递形成对比。

扩散模型的连续时间形式可以产生概率流（probability flow ODE），通过数值积分近似对数似然。

去噪扩散模型与分数匹配模型之间存在紧密联系，后者通过梯度引导的马尔可夫链生成样本。

扩散模型的发展势头强劲，其在多个领域的应用前景令人兴奋。

Transcripts

imagine we take an image and add a bit

of gaussian noise to it

then do this again

if we repeat this enough times

eventually we'll have an unrecognizable

picture of static a sample of pure noise

now what if we could figure out how to

undo this process

that is start from a noise image

gradually remove the noise and end up

with a coherent image

this is the basic idea behind diffusion

models an approach gaining traction and

generative modeling it had success

particularly in the domain of image

generation and they are starting to

rival and in some cases surpass other

kinds of generative models you may be

familiar with on certain tasks

for example recent diffusion models have

outperformed generative adversarial

networks known as gans in perceptual

quality metrics and they've also shown

impressive performance in various

conditional settings such as converting

text descriptions to images

in painting

and manipulation

in this video we'll try to understand

the basic mechanism behind diffusion

models and how they can be adapted to

different generative settings

we'll start with a sample from some

target data distribution like an image

from a training set

let's call this x0

now let's define a forward diffusion

process that gradually adds noise to the

image over big t time steps

our model will be tasked with starting

at x big t and undoing this noise

through what we'll call the reverse

process

the forward process which we'll denote

with q takes the form of a markov chain

where the distribution at a particular

time step only depends on the sample

from the immediately previous step

so we can write out the distribution of

corrupted samples conditioned on the

initial data point x0 as the product of

successive single step conditionals

in the case of continuous data each

transition is parameterized as a

diagonal gaussian

beta t here is the variance at a

particular time step t

typically these variances are treated as

hyperparameters and follow a fixed

schedule for a particular training run

beta generally increases with time and

is restricted to be between zero and one

meaning that this coefficient radical

one minus beta t

will likewise be non-zero but less than

one bringing the mean of each new

gaussian closer to zero

in the limit as t approaches infinity

q will approach a gaussian centered at

zero with identity covariance losing all

information about the original sample

in practice the total number of steps

big t is on the order of a thousand

using a large albeit finite number of

steps allows us to set the individual

variances beta t to be very small while

still approximately maintaining the same

limiting distribution

but why do we want to use a small step

size what's the benefit

well it means that learning to undo the

steps of the forward process won't be

too difficult

let's consider a simple case in one

dimension

suppose we were given the distribution

of a forward process sample at time t

minus one and it resembled a mixture of

gaussians with two modes

we then observe x t

and want to infer the posterior

distribution over x t minus one

that is we'd like to determine where did

the chain likely come from in order to

arrive at x t

what was the previous step of the chain

if the noise step that is q of x t given

x t minus 1 is allowed to be large then

we will be quite uncertain about the

location of x t minus 1. who knows where

we jumped from

but if the forward noise step is

restricted to be small there is much

less ambiguity about x t minus 1.

we could then be justified in modeling

the posterior of the forward step that

is q of x t minus 1 given x t

with a unimodal gaussian

eliminating the contribution from the

mode to the right

and in fact it can be shown

theoretically that in the limit of

infinitesimal step sizes the true

reverse process will have the same

functional form as the forward process

so diffusion models leverage this

observation parameterizing each learned

reverse step to also be a unimodal

diagonal gaussian

aside from the sample at time t the

model also takes t as input in order to

account for the forward process variance

schedule different time steps are

associated with different noise levels

and the model can learn to undo these

individually

like the forward process the reverse

process is set up as a markov chain

and we can write out the joint

probability of a sequence of samples as

a product of conditionals and the

marginal probability of x big t

so what is p of x big t here exactly

well it's the same as q of x big t the

pure noise distribution

so at inference time in order to

actually generate a sample we start from

a gaussian and begin sampling from the

learned individual steps of the reverse

process p of x t minus 1 given x t until

producing an x0

okay great so we've defined these

forward and reverse diffusion processes

the forward process is designed to

essentially push a sample off the data

manifold turning it into noise and the

reverse process is trained to produce a

trajectory back to the data manifold

resulting in a reasonable sample

but what objective will we actually be

optimizing is it some maximum likelihood

objective where we directly maximize the

density assigned to x0 by the model

well not exactly

if we try to calculate p of x0 we see

that we have to marginalize over all the

possible trajectories all the ways we

could have arrived at x0 when starting

from a noise sample

this unfortunately is intractable

but it turns out we can maximize a lower

bound

to do this let's view x1 through x big t

as latent variables and x0 as an

observed variable

allowing us to interpret a diffusion

model as a kind of latent variable

generative model

if we think back to another latent

variable model you may be familiar with

variational autoencoders commonly known

as vaes

we might get a hint about our training

objective

as a quick reminder in a vae we have an

encoder that produces a distribution

over latency given a data input x

and a decoder that reconstructs the data

by producing a distribution over data x

given a latent input z

so we can think of the forward process

in diffusion models as analogous to the

encoder producing latency from data

and the reverse process as analogous to

the decoder producing data from latents

now unlike a vae encoder the forward

process here is typically fixed

it's the reverse process that we focus

solely on learning

this means that only a single network

needs to be trained unlike in a vae

where two networks are trained jointly

so we can now borrow the basic training

objective used by vaes and a number of

other latent variable models

when we have a model with observations x

and latent variable z

we can derive what's called the

variational lower bound also known as

the evidence lower bound

a lower bound on the marginal log

likelihood p theta of x

we won't walk through the full

derivation here but the end result is a

likelihood term also known as a

reconstruction term subtracted by a kl

divergence term

the likelihood term encourages the model

to maximize the expected density

assigned to the data

while the kl divergence encourages the

approximate posterior q z given x to be

similar to the prior on the latent

variable p of z

as we saw earlier x0 will serve as the

observation in the diffusion model

framework while x1 through big t will

take the place of the latent variable z

here

let's substitute these in

alright now let's simplify a bit

we can expand the kl divergence to

combine the two terms into a single

expectation

and finally we can refactor the chain

probabilities into their individual

steps

now there's a nice property of the

forward process queue that we didn't

touch on earlier any arbitrary step of

the forward process can be sampled

directly in closed form

this is just because the sum of

independent gaussian steps is still a

gaussian

so at training time any term of this

objective can be obtained without having

to simulate an entire chain

likewise we can optimize this objective

by randomly sampling pairs of x t minus

one and x t

and maximizing the conditional density

assigned by the reverse step to x t

minus one

however because different trajectories

may visit different samples at time t

minus one on the way to hitting xt

the setup can have high variance

limiting training efficiency

to help with this we can rearrange the

objective as follows

let's examine each component

p of x big t is fixed it's just the

start of the reverse process the pure

noise distribution

and as we saw earlier the whole forward

process q is also treated as fixed

so we just have to worry about these two

terms to the right

here we have a sum of kl divergences

each between a reverse step and a

forward process posterior conditioned on

x0

one can prove with bayes rule that when

we treat the original sample x0 as known

like it is during training these q terms

are actually just gaussians

since the reverse step is already

parameterized as a gaussian each kl

divergence now is simply comparing two

gaussians and can be evaluated in closed

form

this helps reduce variance in the

training process because instead of

aiming to reconstruct monte carlo

samples

the targets for the reverse step become

the true posteriors of the forward

process given x0

there are a couple different ways we

could imagine implementing the reverse

step p theta in the paper denoising

diffusion probabilistic models

ddpm for short

the authors elect to set the reverse

process variances to time specific

constants as they found learning them to

lead to unstable training and lower

quality samples

so the reverse step network is solely

tasked with learning the means

they then suggest a reparameterization

that aims to have the network predict

the noise that was added rather than the

gaussian mean

first we can rewrite sampling from an

arbitrary forward step by using an

auxiliary noise variable epsilon

epsilon here has a constant distribution

independent of the forward time step t

and the reverse step model can be

designed to simply predict this epsilon

the authors also found that a simpler

version of the variational bound that

discards the term weights that appear in

the original bound led to better sample

quality

so compared to the original variational

lower bound their objective down weight

steps that have very small noise at

early time steps of the forward process

allowing training to focus on more

challenging greater noise steps

like other generative frameworks

diffusion models can be made to sample

conditionally given some variable of

interest like a class label or a

sentence description

one way to do this is to just feed the

conditioning variable y as an additional

input during training

in theory the model should learn to use

y as a helpful hint about what it should

be reconstructing in practice some work

has shown that further guiding the

diffusion process with a separate

classifier can help

in this setup we take a trained

classifier and push the reverse

diffusion process in the direction of

the gradient of the target label

probability with respect to the current

noise image

and we can do this not just with single

word labels but also with higher

dimensional text descriptions as well

of course one drawback of this technique

is the reliance upon a second network

an alternative approach eliminates this

reliance instead using special training

of the diffusion model itself to guide

the sampling

in the paper classifier free diffusion

guidance the conditioning label y is set

to a null label with some probability

during training

then at inference time the reconstructed

samples are artificially pushed further

towards the y conditional direction and

away from the null label

even though no new information is being

given to the model they found this to

produce higher quality samples under

human evaluation compared to classifier

guidance

inpainting is another conditional

generation problem where diffusion

models have had success

the naive way to perform in-painting

with diffusion models is to take a model

trained in the standard way and at

inference time replace known regions of

an image with a sample from the forward

process after each reverse step

now this works okay but can lead to edge

artifacts

the model is not being made aware of the

full surrounding context only a hazy

version of it

instead better results come from

fine-tuning a model specifically for

this task

we can randomly remove sections of

training images and have the model

attempt to fill them in conditioned on

the full clear context

we can compare diffusion models to some

other prominent degenerative models

for sampling tasks the fusion models are

somewhat limited by the slow markov

chain

this contrasts for example with gans

which can generate images in a single

forward pass

ongoing work aims to speed up sampling

in diffusion models

as we saw earlier diffusion models allow

us to calculate a variational lower

bound on the log likelihood similar to

vaes

in practice this lower bound can be

quite good and even competitive on

density estimation benchmarks which have

long been dominated by auto aggressive

models

going beyond lower bounds a continuous

time formulation of diffusion models can

give rise to what's called a probability

flow ode

this enables approximating log

likelihood via numerical integration

there's a close connection between

denoising diffusion models and what are

called score matching models and often

these are now grouped together into a

single class of models score here refers

to the gradient of the log of the target

probability density with respect to the

data

a score network is trained to estimate

this value

then a markov chain is set up to

actually produce samples from the learn

distribution

guided by this gradient

well it turns out the score can actually

be shown to be equivalent to the noise

that's predicted in the denoising

diffusion objective up to a scaling

factor

so we can think of undoing the noise in

a diffusion model approximately as

trying to follow the gradient of the

data log density

diffusion models are really gaining

momentum and it's been exciting to see

their progress

check out the links in the description

to learn more

thanks for watching