From bb3d3709344b12b3877ecae2df0fc474fd2b8222 Mon Sep 17 00:00:00 2001 From: Morten Hjorth-Jensen Date: Wed, 1 May 2024 19:53:36 +0200 Subject: [PATCH] update week 16 --- doc/pub/week16/html/week16-bs.html | 93 +++++++++++++++++-- doc/pub/week16/html/week16-reveal.html | 82 +++++++++++++++- doc/pub/week16/html/week16-solarized.html | 88 +++++++++++++++++- doc/pub/week16/html/week16.html | 88 +++++++++++++++++- doc/pub/week16/ipynb/ipynb-week16-src.tar.gz | Bin 192 -> 191 bytes doc/pub/week16/ipynb/week16.ipynb | 91 ++++++++++++++++-- doc/pub/week16/pdf/week16.pdf | Bin 180247 -> 225951 bytes doc/src/week16/week16.do.txt | 62 ++++++++++++- 8 files changed, 477 insertions(+), 27 deletions(-) diff --git a/doc/pub/week16/html/week16-bs.html b/doc/pub/week16/html/week16-bs.html index 4320ac2c..bcb42c71 100644 --- a/doc/pub/week16/html/week16-bs.html +++ b/doc/pub/week16/html/week16-bs.html @@ -40,14 +40,33 @@ 2, None, 'plans-for-the-week-of-may-6-10-2024'), - ('Diffusion models', 2, None, 'diffusion-models'), - ('Summary of course and topics covered', + ('Diffusion models, basics', 2, None, 'diffusion-models-basics'), + ('Problems with probabilistic models', 2, None, - 'summary-of-course-and-topics-covered')]} + 'problems-with-probabilistic-models'), + ('Diffusion models', 2, None, 'diffusion-models'), + ('Original idea', 2, None, 'original-idea'), + ('Diffusion learning', 2, None, 'diffusion-learning')]} end of tocinfo --> + + + + + + + diff --git a/doc/pub/week16/html/week16-reveal.html b/doc/pub/week16/html/week16-reveal.html index 9f5ffe5e..b74def96 100644 --- a/doc/pub/week16/html/week16-reveal.html +++ b/doc/pub/week16/html/week16-reveal.html @@ -149,6 +149,22 @@ + + + + + + +
@@ -192,14 +208,76 @@

Plans for the week of May 6-10, 202

+
+

Diffusion models, basics

+ +

Diffusion models are inspired by non-equilibrium thermodynamics. They +define a Markov chain of diffusion steps to slowly add random noise to +data and then learn to reverse the diffusion process to construct +desired data samples from the noise. Unlike VAE or flow models, +diffusion models are learned with a fixed procedure and the latent +variable has high dimensionality (same as the original data). +

+
+ +
+

Problems with probabilistic models

+ +

Historically, probabilistic models suffer from a tradeoff between two +conflicting objectives: \textit{tractability} and +\textit{flexibility}. Models that are \textit{tractable} can be +analytically evaluated and easily fit to data (e.g. a Gaussian or +Laplace). However, these models are unable to aptly describe structure +in rich datasets. On the other hand, models that are \textit{flexible} +can be molded to fit structure in arbitrary data. For example, we can +define models in terms of any (non-negative) function \( \phi(\boldsymbol{x}) \) +yielding the flexible distribution \( p\left(\boldsymbol{x}\right) = +\frac{\phi\left(\boldsymbol{x} \right)}{Z} \), where \( Z \) is a normalization +constant. However, computing this normalization constant is generally +intractable. Evaluating, training, or drawing samples from such +flexible models typically requires a very expensive Monte Carlo +process. +

+
+

Diffusion models

+

Diffusion models have several interesting features

+ +
-

Text to come here.

+
+

Original idea

+ +

In the original formulation, one uses a Markov chain to gradually +convert one distribution into another, an idea used in non-equilibrium +statistical physics and sequential Monte Carlo. Diffusion models build +a generative Markov chain which converts a simple known distribution +(e.g. a Gaussian) into a target (data) distribution using a diffusion +process. Rather than use this Markov chain to approximately evaluate a +model which has been otherwise defined, one can explicitly define the +probabilistic model as the endpoint of the Markov chain. Since each +step in the diffusion chain has an analytically evaluable probability, +the full chain can also be analytically evaluated. +

-

Summary of course and topics covered

+

Diffusion learning

+ +

Learning in this framework involves estimating small perturbations to +a diffusion process. Estimating small, analytically tractable, +perturbations is more tractable than explicitly describing the full +distribution with a single, non-analytically-normalizable, potential +function. Furthermore, since a diffusion process exists for any +smooth target distribution, this method can capture data distributions +of arbitrary form. +

diff --git a/doc/pub/week16/html/week16-solarized.html b/doc/pub/week16/html/week16-solarized.html index 5327a9cb..a64b028f 100644 --- a/doc/pub/week16/html/week16-solarized.html +++ b/doc/pub/week16/html/week16-solarized.html @@ -67,14 +67,33 @@ 2, None, 'plans-for-the-week-of-may-6-10-2024'), - ('Diffusion models', 2, None, 'diffusion-models'), - ('Summary of course and topics covered', + ('Diffusion models, basics', 2, None, 'diffusion-models-basics'), + ('Problems with probabilistic models', 2, None, - 'summary-of-course-and-topics-covered')]} + 'problems-with-probabilistic-models'), + ('Diffusion models', 2, None, 'diffusion-models'), + ('Original idea', 2, None, 'original-idea'), + ('Diffusion learning', 2, None, 'diffusion-learning')]} end of tocinfo --> + + + + + + +

Advanced machine learning and data analysis for the physical sciences

@@ -111,13 +130,72 @@

Plans for the week of May 6-10, 202 +









+

Diffusion models, basics

+ +

Diffusion models are inspired by non-equilibrium thermodynamics. They +define a Markov chain of diffusion steps to slowly add random noise to +data and then learn to reverse the diffusion process to construct +desired data samples from the noise. Unlike VAE or flow models, +diffusion models are learned with a fixed procedure and the latent +variable has high dimensionality (same as the original data). +

+ +









+

Problems with probabilistic models

+ +

Historically, probabilistic models suffer from a tradeoff between two +conflicting objectives: \textit{tractability} and +\textit{flexibility}. Models that are \textit{tractable} can be +analytically evaluated and easily fit to data (e.g. a Gaussian or +Laplace). However, these models are unable to aptly describe structure +in rich datasets. On the other hand, models that are \textit{flexible} +can be molded to fit structure in arbitrary data. For example, we can +define models in terms of any (non-negative) function \( \phi(\boldsymbol{x}) \) +yielding the flexible distribution \( p\left(\boldsymbol{x}\right) = +\frac{\phi\left(\boldsymbol{x} \right)}{Z} \), where \( Z \) is a normalization +constant. However, computing this normalization constant is generally +intractable. Evaluating, training, or drawing samples from such +flexible models typically requires a very expensive Monte Carlo +process. +

+









Diffusion models

+

Diffusion models have several interesting features

+ +









+

Original idea

-

Text to come here.

+

In the original formulation, one uses a Markov chain to gradually +convert one distribution into another, an idea used in non-equilibrium +statistical physics and sequential Monte Carlo. Diffusion models build +a generative Markov chain which converts a simple known distribution +(e.g. a Gaussian) into a target (data) distribution using a diffusion +process. Rather than use this Markov chain to approximately evaluate a +model which has been otherwise defined, one can explicitly define the +probabilistic model as the endpoint of the Markov chain. Since each +step in the diffusion chain has an analytically evaluable probability, +the full chain can also be analytically evaluated. +











-

Summary of course and topics covered

+

Diffusion learning

+ +

Learning in this framework involves estimating small perturbations to +a diffusion process. Estimating small, analytically tractable, +perturbations is more tractable than explicitly describing the full +distribution with a single, non-analytically-normalizable, potential +function. Furthermore, since a diffusion process exists for any +smooth target distribution, this method can capture data distributions +of arbitrary form. +

+
© 1999-2024, Morten Hjorth-Jensen. Released under CC Attribution-NonCommercial 4.0 license diff --git a/doc/pub/week16/html/week16.html b/doc/pub/week16/html/week16.html index 963be699..602fc475 100644 --- a/doc/pub/week16/html/week16.html +++ b/doc/pub/week16/html/week16.html @@ -144,14 +144,33 @@ 2, None, 'plans-for-the-week-of-may-6-10-2024'), - ('Diffusion models', 2, None, 'diffusion-models'), - ('Summary of course and topics covered', + ('Diffusion models, basics', 2, None, 'diffusion-models-basics'), + ('Problems with probabilistic models', 2, None, - 'summary-of-course-and-topics-covered')]} + 'problems-with-probabilistic-models'), + ('Diffusion models', 2, None, 'diffusion-models'), + ('Original idea', 2, None, 'original-idea'), + ('Diffusion learning', 2, None, 'diffusion-learning')]} end of tocinfo --> + + + + + + +

Advanced machine learning and data analysis for the physical sciences

@@ -188,13 +207,72 @@

Plans for the week of May 6-10, 202 +









+

Diffusion models, basics

+ +

Diffusion models are inspired by non-equilibrium thermodynamics. They +define a Markov chain of diffusion steps to slowly add random noise to +data and then learn to reverse the diffusion process to construct +desired data samples from the noise. Unlike VAE or flow models, +diffusion models are learned with a fixed procedure and the latent +variable has high dimensionality (same as the original data). +

+ +









+

Problems with probabilistic models

+ +

Historically, probabilistic models suffer from a tradeoff between two +conflicting objectives: \textit{tractability} and +\textit{flexibility}. Models that are \textit{tractable} can be +analytically evaluated and easily fit to data (e.g. a Gaussian or +Laplace). However, these models are unable to aptly describe structure +in rich datasets. On the other hand, models that are \textit{flexible} +can be molded to fit structure in arbitrary data. For example, we can +define models in terms of any (non-negative) function \( \phi(\boldsymbol{x}) \) +yielding the flexible distribution \( p\left(\boldsymbol{x}\right) = +\frac{\phi\left(\boldsymbol{x} \right)}{Z} \), where \( Z \) is a normalization +constant. However, computing this normalization constant is generally +intractable. Evaluating, training, or drawing samples from such +flexible models typically requires a very expensive Monte Carlo +process. +

+









Diffusion models

+

Diffusion models have several interesting features

+
    +
  • extreme flexibility in model structure,
  • +
  • exact sampling,
  • +
  • easy multiplication with other distributions, e.g. in order to compute a posterior, and
  • +
  • the model log likelihood, and the probability of individual states, to be cheaply evaluated.
  • +
+









+

Original idea

-

Text to come here.

+

In the original formulation, one uses a Markov chain to gradually +convert one distribution into another, an idea used in non-equilibrium +statistical physics and sequential Monte Carlo. Diffusion models build +a generative Markov chain which converts a simple known distribution +(e.g. a Gaussian) into a target (data) distribution using a diffusion +process. Rather than use this Markov chain to approximately evaluate a +model which has been otherwise defined, one can explicitly define the +probabilistic model as the endpoint of the Markov chain. Since each +step in the diffusion chain has an analytically evaluable probability, +the full chain can also be analytically evaluated. +











-

Summary of course and topics covered

+

Diffusion learning

+ +

Learning in this framework involves estimating small perturbations to +a diffusion process. Estimating small, analytically tractable, +perturbations is more tractable than explicitly describing the full +distribution with a single, non-analytically-normalizable, potential +function. Furthermore, since a diffusion process exists for any +smooth target distribution, this method can capture data distributions +of arbitrary form. +

+
© 1999-2024, Morten Hjorth-Jensen. 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They +define a Markov chain of diffusion steps to slowly add random noise to +data and then learn to reverse the diffusion process to construct +desired data samples from the noise. Unlike VAE or flow models, +diffusion models are learned with a fixed procedure and the latent +variable has high dimensionality (same as the original data). + + +!split +===== Problems with probabilistic models ===== + +Historically, probabilistic models suffer from a tradeoff between two +conflicting objectives: \textit{tractability} and +\textit{flexibility}. Models that are \textit{tractable} can be +analytically evaluated and easily fit to data (e.g. a Gaussian or +Laplace). However, these models are unable to aptly describe structure +in rich datasets. On the other hand, models that are \textit{flexible} +can be molded to fit structure in arbitrary data. For example, we can +define models in terms of any (non-negative) function $\phi(\bm{x})$ +yielding the flexible distribution $p\left(\bm{x}\right) = +\frac{\phi\left(\bm{x} \right)}{Z}$, where $Z$ is a normalization +constant. However, computing this normalization constant is generally +intractable. Evaluating, training, or drawing samples from such +flexible models typically requires a very expensive Monte Carlo +process. + !split ===== Diffusion models ===== +Diffusion models have several interesting features +* extreme flexibility in model structure, +* exact sampling, +* easy multiplication with other distributions, e.g. in order to compute a posterior, and +* the model log likelihood, and the probability of individual states, to be cheaply evaluated. + +!split +===== Original idea ===== -Text to come here. +In the original formulation, one uses a Markov chain to gradually +convert one distribution into another, an idea used in non-equilibrium +statistical physics and sequential Monte Carlo. Diffusion models build +a generative Markov chain which converts a simple known distribution +(e.g. a Gaussian) into a target (data) distribution using a diffusion +process. Rather than use this Markov chain to approximately evaluate a +model which has been otherwise defined, one can explicitly define the +probabilistic model as the endpoint of the Markov chain. Since each +step in the diffusion chain has an analytically evaluable probability, +the full chain can also be analytically evaluated. !split -===== Summary of course and topics covered ===== +===== Diffusion learning ===== + +Learning in this framework involves estimating small perturbations to +a diffusion process. Estimating small, analytically tractable, +perturbations is more tractable than explicitly describing the full +distribution with a single, non-analytically-normalizable, potential +function. Furthermore, since a diffusion process exists for any +smooth target distribution, this method can capture data distributions +of arbitrary form. + + +