# Probabilistic and Semantic Descriptions of Image Manifolds and Their Applications

arXiv: https://arxiv.org/abs/2307.02881 · HTML: https://mlsanity.com/paper/2307.02881/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Placed in Diffusion foundations and sampling, one of 14 directions in Diffusion models.
- Builds on 42 of its 62 references that are in the map.
- First posted on arXiv 6 Jul 2023.

## Placed in

- Diffusion models / [Diffusion foundations and sampling](https://mlsanity.com/diffusion/diffusion-foundations-and-sampling/)

## Builds on (most cited first, 30 of 42)

- [Deep Residual Learning for Image Recognition](https://mlsanity.com/paper/1512.03385/) (2015)
- [Denoising Diffusion Probabilistic Models](https://mlsanity.com/paper/2006.11239/) (2020)
- [Denoising Diffusion Implicit Models](https://mlsanity.com/paper/2010.02502/) (2020)
- [Towards Deep Learning Models Resistant to Adversarial Attacks](https://mlsanity.com/paper/1706.06083/) (2017)
- [Fashion-MNIST: a Novel Image Dataset for Benchmarking Machine Learning Algorithms](https://mlsanity.com/paper/1708.07747/) (2017)
- [Zero-Shot Text-to-Image Generation](https://mlsanity.com/paper/2102.12092/) (2021)
- [Auto-Encoding Variational Bayes](https://mlsanity.com/paper/1312.6114/) (2013)
- [Deep Learning Face Attributes in the Wild](https://mlsanity.com/paper/1411.7766/) (2014)
- [Towards Evaluating the Robustness of Neural Networks](https://mlsanity.com/paper/1608.04644/) (2016)
- [FaceNet: A Unified Embedding for Face Recognition and Clustering](https://mlsanity.com/paper/1503.03832/) (2015)
- [Variational Inference with Normalizing Flows](https://mlsanity.com/paper/1505.05770/) (2015)
- [Glow: Generative Flow with Invertible 1x1 Convolutions](https://mlsanity.com/paper/1807.03039/) (2018)
- [Normalizing Flows for Probabilistic Modeling and Inference](https://mlsanity.com/paper/1912.02762/) (2019)
- [CNN-generated images are surprisingly easy to spot... for now](https://mlsanity.com/paper/1912.11035/) (2019)
- [Ensemble Adversarial Training: Attacks and Defenses](https://mlsanity.com/paper/1705.07204/) (2017)
- [The Limitations of Deep Learning in Adversarial Settings](https://mlsanity.com/paper/1511.07528/) (2015)
- [Adversarial Patch](https://mlsanity.com/paper/1712.09665/) (2017)
- [Nesterov Accelerated Gradient and Scale Invariance for Adversarial Attacks](https://mlsanity.com/paper/1908.06281/) (2019)
- [Understanding Diffusion Models: A Unified Perspective](https://mlsanity.com/paper/2208.11970/) (2022)
- [The Intrinsic Dimension of Images and Its Impact on Learning](https://mlsanity.com/paper/2104.08894/) (2021)
- [Representation Degeneration Problem in Training Natural Language Generation Models](https://mlsanity.com/paper/1907.12009/) (2019)
- [Isolating Sources of Disentanglement in Variational Autoencoders](https://mlsanity.com/paper/1802.04942/) (2018)
- [SentiNet: Detecting Localized Universal Attacks Against Deep Learning Systems](https://mlsanity.com/paper/1812.00292/) (2018)
- [Emergence of Language with Multi-agent Games: Learning to Communicate with Sequences of Symbols](https://mlsanity.com/paper/1705.11192/) (2017)
- [Compositionality and Generalization in Emergent Languages](https://mlsanity.com/paper/2004.09124/) (2020)
- [Diffusion Normalizing Flow](https://mlsanity.com/paper/2110.07579/) (2021)
- [Hierarchical Conditional Flow: A Unified Framework for Image Super-Resolution and Image Rescaling](https://mlsanity.com/paper/2108.05301/) (2021)
- [Dual Contradistinctive Generative Autoencoder](https://mlsanity.com/paper/2011.10063/) (2020)
- [Guided Variational Autoencoder for Disentanglement Learning](https://mlsanity.com/paper/2004.01255/) (2020)
- [Compositional Generalization in Unsupervised Compositional Representation Learning: A Study on Disentanglement and Emergent Language](https://mlsanity.com/paper/2210.00482/) (2022)

## Built on by (most cited first, 3 of 3)

- [The structure of the token space for large language models](https://mlsanity.com/paper/2410.08993/) (2024)
- [From Abstraction to Reality: DARPA's Vision for Robust Sim-to-Real Autonomy](https://mlsanity.com/paper/2503.11007/) (2025)
- [DrawMotion: Generating 3D Human Motions by Freehand Drawing](https://mlsanity.com/paper/2605.20955/) (2026)

## Abstract (arXiv)

This paper begins with a description of methods for estimating image probability density functions that reflects the observation that such data is usually constrained to lie in restricted regions of the high-dimensional image space-not every pattern of pixels is an image. It is common to say that images lie on a lower-dimensional manifold in the high-dimensional space. However, it is not the case that all points on the manifold have an equal probability of being images. Images are unevenly distributed on the manifold, and our task is to devise ways to model this distribution as a probability distribution. We therefore consider popular generative models. For our purposes, generative/probabilistic models should have the properties of 1) sample generation: the possibility to sample from this distribution with the modelled density function, and 2) probability computation: given a previously unseen sample from the dataset of interest, one should be able to compute its probability, at least up to a normalising constant. To this end, we investigate the use of methods such as normalising flow and diffusion models. We then show how semantic interpretations are used to describe points on the manifold. To achieve this, we consider an emergent language framework that uses variational encoders for a disentangled representation of points that reside on a given manifold. Trajectories between points on a manifold can then be described as evolving semantic descriptions. We also show that such probabilistic descriptions (bounded) can be used to improve semantic consistency by constructing defences against adversarial attacks. We evaluate our methods with improved semantic robustness and OoD detection capability, explainable and editable semantic interpolation, and improved classification accuracy under patch attacks. We also discuss the limitation in diffusion models.