Latent Space Interpolation in the Variational Autoencoder and the Denoising Diffusion Probabilistic Model
(2026) In Bachelor's Theses in Mathematical Sciences MASK11 20252Mathematical Statistics
- Abstract
- This work investigates image synthesis of two generative models: the Variational Autoencoder (VAE) and the Denoising Diffusion Probabilistic Model (DDPM).
One approach for synthesis is to interpolate between real samples within the models' latent spaces. To understand how we can interpolate in the models' latent space, we derive the loss functions of the models. We trained the models on MNIST and Fashion-MNIST dataset. With that knowledge, we explain why continuous linear interpolation in the latent space often finds the shortest interpolation path in the VAE: because of a the geometric structuredness and continuity of the latent space. In the DDPM, linear interpolation in the latent space tends to fail for samples with different features... (More) - This work investigates image synthesis of two generative models: the Variational Autoencoder (VAE) and the Denoising Diffusion Probabilistic Model (DDPM).
One approach for synthesis is to interpolate between real samples within the models' latent spaces. To understand how we can interpolate in the models' latent space, we derive the loss functions of the models. We trained the models on MNIST and Fashion-MNIST dataset. With that knowledge, we explain why continuous linear interpolation in the latent space often finds the shortest interpolation path in the VAE: because of a the geometric structuredness and continuity of the latent space. In the DDPM, linear interpolation in the latent space tends to fail for samples with different features because the model does not learn a smooth geometric structure for the latent representation of the data. We show how increasing the dimensionality of the latent space from two to three in the VAE results in more clarity of the synthesized samples. In the two dimensional latent space we train a degree two polynomial by the difference between each image in the pixel space and by encouraging proximity to high density regions in the latent space. This resulted in shorter interpolation path compared to linear interpolation in the latent space. (Less) - Popular Abstract
- Interpolation is a method to estimate new data points given a discrete amount of samples. In image processing, interpolation can be used to find intermediate states between two pixels in an image or between two separate images. Nowadays, image interpolation is used for image processing such as enhancing the quality of medical images or generating intermediate frames in videos to increase the frame rate. To achieve the best possible results, modern interpolation techniques increasingly rely on artificial intelligence, specifically models that are capable of image synthesis. These models are called generative models.
One of the models that we will investigate for this purpose is the variational autoencoder. The first research on... (More) - Interpolation is a method to estimate new data points given a discrete amount of samples. In image processing, interpolation can be used to find intermediate states between two pixels in an image or between two separate images. Nowadays, image interpolation is used for image processing such as enhancing the quality of medical images or generating intermediate frames in videos to increase the frame rate. To achieve the best possible results, modern interpolation techniques increasingly rely on artificial intelligence, specifically models that are capable of image synthesis. These models are called generative models.
One of the models that we will investigate for this purpose is the variational autoencoder. The first research on autoencoders began in the 1980s. The general idea was to reduce the data to its most important features in order to simplify training generative models. In practice, an encoder maps a high-dimensional data point to a point in a lower-dimensional space called the latent space. The encoded point is then decoded to reconstruct the original data point.
Based on this idea, variational autoencoders were introduced in 2013 as an extension of the traditional autoencoder. Instead of mapping each input to a single point in latent space, the encoder encodes each input to a distribution in latent space. The benefit of using the variational autoencoder is that interpolation is no longer restricted to discrete data points in the latent space. Because each encoded data point corresponds to a distribution in the latent space, it becomes possible to interpolate between two encoded data points while still obtaining meaningful reconstructions of the data.
The second model is the denoising diffusion probabilistic model. First introduced in 2020, the model consists of processes; the forward diffusion
and the reverse diffusion. In the forward process, noise is gradually added to an image
until it becomes indistinguishable from pure noise. In the reverse process, the generative model learns to predict the
noise that was added at each step in the forward process. By interpolating between diffused data points, intermediate data points can be generated. These data points are then generated through the reverse diffusion process.
We will compare these two models based on how well they produce meaningful and visually coherent image interpolations. This will be done by generating image interpolations with the two models and investigating their respective strengths and weaknesses. To further challenge the models, we will attempt to interpolate between samples with little or no structural similarity. For the variational autoencoder, we will also investigate different non-linear interpolation path in order to improve the to increase the quality of the resulting interpolation. (Less)
Please use this url to cite or link to this publication:
https://lup.lub.lu.se/student-papers/record/9236006
- author
- Hitzemann, Max LU
- supervisor
-
- Ted Kronvall LU
- organization
- course
- MASK11 20252
- year
- 2026
- type
- M2 - Bachelor Degree
- subject
- keywords
- Autoencoder, Variational Autoencoder, VAE, Denoising Diffusion Probabilistic Model, DDPM, Interpolation, Latent Space, Artificial Intelligence, AI
- publication/series
- Bachelor's Theses in Mathematical Sciences
- report number
- LUNFMS-4088-2026
- ISSN
- 1654-6229
- other publication id
- 2026:K19
- language
- English
- id
- 9236006
- date added to LUP
- 2026-06-11 17:15:20
- date last changed
- 2026-06-11 17:15:20
@misc{9236006,
abstract = {{This work investigates image synthesis of two generative models: the Variational Autoencoder (VAE) and the Denoising Diffusion Probabilistic Model (DDPM).
One approach for synthesis is to interpolate between real samples within the models' latent spaces. To understand how we can interpolate in the models' latent space, we derive the loss functions of the models. We trained the models on MNIST and Fashion-MNIST dataset. With that knowledge, we explain why continuous linear interpolation in the latent space often finds the shortest interpolation path in the VAE: because of a the geometric structuredness and continuity of the latent space. In the DDPM, linear interpolation in the latent space tends to fail for samples with different features because the model does not learn a smooth geometric structure for the latent representation of the data. We show how increasing the dimensionality of the latent space from two to three in the VAE results in more clarity of the synthesized samples. In the two dimensional latent space we train a degree two polynomial by the difference between each image in the pixel space and by encouraging proximity to high density regions in the latent space. This resulted in shorter interpolation path compared to linear interpolation in the latent space.}},
author = {{Hitzemann, Max}},
issn = {{1654-6229}},
language = {{eng}},
note = {{Student Paper}},
series = {{Bachelor's Theses in Mathematical Sciences}},
title = {{Latent Space Interpolation in the Variational Autoencoder and the Denoising Diffusion Probabilistic Model}},
year = {{2026}},
}