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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.01303 |
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| _version_ | 1866913490441076736 |
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| author | Ravelonanosy, Mahefa Ratsisetraina Menkovski, Vlado Portegies, Jacobus W. |
| author_facet | Ravelonanosy, Mahefa Ratsisetraina Menkovski, Vlado Portegies, Jacobus W. |
| contents | We investigate the ability of Diffusion Variational Autoencoder ($Δ$VAE) with unit sphere $\mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $Δ$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_01303 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE Ravelonanosy, Mahefa Ratsisetraina Menkovski, Vlado Portegies, Jacobus W. Machine Learning Artificial Intelligence Algebraic Topology 51H20 55N35 68T09 68T07 We investigate the ability of Diffusion Variational Autoencoder ($Δ$VAE) with unit sphere $\mathcal{S}^2$ as latent space to capture topological and geometrical structure and disentangle latent factors in datasets. For this, we introduce a new diagnostic of disentanglement: namely the topological degree of the encoder, which is a map from the data manifold to the latent space. By using tools from homology theory, we derive and implement an algorithm that computes this degree. We use the algorithm to compute the degree of the encoder of models that result from the training procedure. Our experimental results show that the $Δ$VAE achieves relatively small LSBD scores, and that regardless of the degree after initialization, the degree of the encoder after training becomes $-1$ or $+1$, which implies that the resulting encoder is at least homotopic to a homeomorphism. |
| title | Topological degree as a discrete diagnostic for disentanglement, with applications to the $Δ$VAE |
| topic | Machine Learning Artificial Intelligence Algebraic Topology 51H20 55N35 68T09 68T07 |
| url | https://arxiv.org/abs/2409.01303 |