Clustering, Coding, and the Concept of Similarity
Fuente:
arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2014
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| _version_ | 1866911873075511296 |
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| author | McCarty, L. Thorne |
| author_facet | McCarty, L. Thorne |
| contents | This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, ${g}_{ij}({\bf x})$, which we interpret as a measure of dissimilarity. The probabilistic model consists of a stochastic process with an invariant probability measure which matches the density of the sample input data. The link between the two models is a potential function, $U({\bf x})$, and its gradient, $\nabla U({\bf x})$. We use the gradient to define the dissimilarity metric, which guarantees that our measure of dissimilarity will depend on the probability measure. Finally, we use the dissimilarity metric to define a coordinate system on the embedded Riemannian manifold, which gives us a low-dimensional encoding of our original data. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1401_2411 |
| institution | arXiv |
| publishDate | 2014 |
| record_format | arxiv |
| spellingShingle | Clustering, Coding, and the Concept of Similarity McCarty, L. Thorne Machine Learning This paper develops a theory of clustering and coding which combines a geometric model with a probabilistic model in a principled way. The geometric model is a Riemannian manifold with a Riemannian metric, ${g}_{ij}({\bf x})$, which we interpret as a measure of dissimilarity. The probabilistic model consists of a stochastic process with an invariant probability measure which matches the density of the sample input data. The link between the two models is a potential function, $U({\bf x})$, and its gradient, $\nabla U({\bf x})$. We use the gradient to define the dissimilarity metric, which guarantees that our measure of dissimilarity will depend on the probability measure. Finally, we use the dissimilarity metric to define a coordinate system on the embedded Riemannian manifold, which gives us a low-dimensional encoding of our original data. |
| title | Clustering, Coding, and the Concept of Similarity |
| topic | Machine Learning |
| url | https://arxiv.org/abs/1401.2411 |