Deep Spectral Embeddings: Unveiling Intrinsic Manifolds via Eigenfunctions
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2025
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| _version_ | 1866901997305724928 |
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| author | SÉRGIO DE ANDRADE, PAULO |
| author_facet | SÉRGIO DE ANDRADE, PAULO |
| contents | High-dimensional data frequently resides on or near low-dimensional non-linear manifolds, whose intrinsic geometry is often obscured by noise and irrelevant features. Traditional manifold learning techniques, such as Laplacian Eigenmaps and Isomap, leverage spectral decomposition to reveal these underlying structures, but their reliance on explicitly constructed similarity graphs and linear eigenvalue problems limits their scalability and ability to capture highly complex non-linearities. This paper introduces Deep Spectral Embeddings (DSE), a novel framework that synergistically combines the power of deep neural networks with the robust theoretical foundations of spectral graph theory. DSE aims to learn data representations where the embedding coordinates directly correspond to the eigenfunctions of an implicitly defined manifold Laplacian, thereby unveiling the intrinsic geometric structure in an end-to-end differentiable manner. We propose an architecture that leverages deep learning to construct an adaptive similarity graph and simultaneously learn a non-linear mapping whose output dimensions approximate the leading eigenfunctions. Through this approach, DSE overcomes the limitations of fixed graph constructions and linear approximations, offering a more flexible and powerful tool for non-linear dimensionality reduction and manifold learning. Our findings demonstrate that DSE can effectively uncover complex intrinsic manifolds, leading to superior performance in various downstream tasks such as clustering, visualization, and anomaly detection. |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_17689871 |
| institution | Zenodo |
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| publishDate | 2025 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | Deep Spectral Embeddings: Unveiling Intrinsic Manifolds via Eigenfunctions SÉRGIO DE ANDRADE, PAULO High-dimensional data frequently resides on or near low-dimensional non-linear manifolds, whose intrinsic geometry is often obscured by noise and irrelevant features. Traditional manifold learning techniques, such as Laplacian Eigenmaps and Isomap, leverage spectral decomposition to reveal these underlying structures, but their reliance on explicitly constructed similarity graphs and linear eigenvalue problems limits their scalability and ability to capture highly complex non-linearities. This paper introduces Deep Spectral Embeddings (DSE), a novel framework that synergistically combines the power of deep neural networks with the robust theoretical foundations of spectral graph theory. DSE aims to learn data representations where the embedding coordinates directly correspond to the eigenfunctions of an implicitly defined manifold Laplacian, thereby unveiling the intrinsic geometric structure in an end-to-end differentiable manner. We propose an architecture that leverages deep learning to construct an adaptive similarity graph and simultaneously learn a non-linear mapping whose output dimensions approximate the leading eigenfunctions. Through this approach, DSE overcomes the limitations of fixed graph constructions and linear approximations, offering a more flexible and powerful tool for non-linear dimensionality reduction and manifold learning. Our findings demonstrate that DSE can effectively uncover complex intrinsic manifolds, leading to superior performance in various downstream tasks such as clustering, visualization, and anomaly detection. |
| title | Deep Spectral Embeddings: Unveiling Intrinsic Manifolds via Eigenfunctions |
| url | https://doi.org/10.5281/zenodo.17689871 |