CW Complex Hypothesis for Image Data

Yi Wang, Zhiren Wang
Proceedings of the 41st International Conference on Machine Learning, PMLR 235:51596-51613, 2024.

Abstract

We examine both the manifold hypothesis (Bengio et al., 2013) and the union of manifold hypothesis (Brown et al., 2023), and argue that, in contrast to these hypotheses, the local intrinsic dimension varies from point to point even in the same connected component. We propose an alternative CW complex hypothesis that image data is distributed in “manifolds with skeletons". We support the hypothesis through visualization of distributions of image data of random geometric objects, as well as by introducing and testing a criterion on natural image datasets. One motivation of our work is to explain why diffusion models have difficulty generating accurate higher dimensional details such as human hands. Under the CW complex hypothesis and with both theoretical and empirical evidences, we provide an interpretation that the mixture of higher and lower dimensional components in data obstructs diffusion models from efficient learning.

Cite this Paper

BibTeX
@InProceedings{pmlr-v235-wang24bs, title = {{CW} Complex Hypothesis for Image Data}, author = {Wang, Yi and Wang, Zhiren}, booktitle = {Proceedings of the 41st International Conference on Machine Learning}, pages = {51596--51613}, year = {2024}, editor = {Salakhutdinov, Ruslan and Kolter, Zico and Heller, Katherine and Weller, Adrian and Oliver, Nuria and Scarlett, Jonathan and Berkenkamp, Felix}, volume = {235}, series = {Proceedings of Machine Learning Research}, month = {21--27 Jul}, publisher = {PMLR}, pdf = {https://raw.githubusercontent.com/mlresearch/v235/main/assets/wang24bs/wang24bs.pdf}, url = {https://proceedings.mlr.press/v235/wang24bs.html}, abstract = {We examine both the manifold hypothesis (Bengio et al., 2013) and the union of manifold hypothesis (Brown et al., 2023), and argue that, in contrast to these hypotheses, the local intrinsic dimension varies from point to point even in the same connected component. We propose an alternative CW complex hypothesis that image data is distributed in “manifolds with skeletons". We support the hypothesis through visualization of distributions of image data of random geometric objects, as well as by introducing and testing a criterion on natural image datasets. One motivation of our work is to explain why diffusion models have difficulty generating accurate higher dimensional details such as human hands. Under the CW complex hypothesis and with both theoretical and empirical evidences, we provide an interpretation that the mixture of higher and lower dimensional components in data obstructs diffusion models from efficient learning.} }
Endnote
%0 Conference Paper %T CW Complex Hypothesis for Image Data %A Yi Wang %A Zhiren Wang %B Proceedings of the 41st International Conference on Machine Learning %C Proceedings of Machine Learning Research %D 2024 %E Ruslan Salakhutdinov %E Zico Kolter %E Katherine Heller %E Adrian Weller %E Nuria Oliver %E Jonathan Scarlett %E Felix Berkenkamp %F pmlr-v235-wang24bs %I PMLR %P 51596--51613 %U https://proceedings.mlr.press/v235/wang24bs.html %V 235 %X We examine both the manifold hypothesis (Bengio et al., 2013) and the union of manifold hypothesis (Brown et al., 2023), and argue that, in contrast to these hypotheses, the local intrinsic dimension varies from point to point even in the same connected component. We propose an alternative CW complex hypothesis that image data is distributed in “manifolds with skeletons". We support the hypothesis through visualization of distributions of image data of random geometric objects, as well as by introducing and testing a criterion on natural image datasets. One motivation of our work is to explain why diffusion models have difficulty generating accurate higher dimensional details such as human hands. Under the CW complex hypothesis and with both theoretical and empirical evidences, we provide an interpretation that the mixture of higher and lower dimensional components in data obstructs diffusion models from efficient learning.
APA
Wang, Y. & Wang, Z.. (2024). CW Complex Hypothesis for Image Data. Proceedings of the 41st International Conference on Machine Learning, in Proceedings of Machine Learning Research 235:51596-51613 Available from https://proceedings.mlr.press/v235/wang24bs.html.

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