Tempered Calculus for ML: Application to Hyperbolic Model Embedding
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917788878110720 |
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| author | Nock, Richard Amid, Ehsan Nielsen, Frank Soen, Alexander Warmuth, Manfred K. |
| author_facet | Nock, Richard Amid, Ehsan Nielsen, Frank Soen, Alexander Warmuth, Manfred K. |
| contents | Most mathematical distortions used in ML are fundamentally integral in nature: $f$-divergences, Bregman divergences, (regularized) optimal transport distances, integral probability metrics, geodesic distances, etc. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We start with a generalization of Riemann integration that also encapsulates functions that are not strictly additive but are, more generally, $t$-additive, as in nonextensive statistical mechanics. Notably, this recovers Volterra's product integral as a special case. We then generalize the Fundamental Theorem of calculus using an extension of the (Euclidean) derivative. This, along with a series of more specific Theorems, serves as a basis for results showing how one can specifically design, alter, or change fundamental properties of distortion measures in a simple way, with a special emphasis on geometric- and ML-related properties that are the metricity, hyperbolicity, and encoding. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vs Euclidean scale. We unveil a new application for which the Poincaré disk model has very appealing features, and our theory comes in handy: \textit{model} embeddings for boosted combinations of decision trees, trained using the log-loss (trees) and logistic loss (combinations). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_04163 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Tempered Calculus for ML: Application to Hyperbolic Model Embedding Nock, Richard Amid, Ehsan Nielsen, Frank Soen, Alexander Warmuth, Manfred K. Machine Learning I.2.6 Most mathematical distortions used in ML are fundamentally integral in nature: $f$-divergences, Bregman divergences, (regularized) optimal transport distances, integral probability metrics, geodesic distances, etc. In this paper, we unveil a grounded theory and tools which can help improve these distortions to better cope with ML requirements. We start with a generalization of Riemann integration that also encapsulates functions that are not strictly additive but are, more generally, $t$-additive, as in nonextensive statistical mechanics. Notably, this recovers Volterra's product integral as a special case. We then generalize the Fundamental Theorem of calculus using an extension of the (Euclidean) derivative. This, along with a series of more specific Theorems, serves as a basis for results showing how one can specifically design, alter, or change fundamental properties of distortion measures in a simple way, with a special emphasis on geometric- and ML-related properties that are the metricity, hyperbolicity, and encoding. We show how to apply it to a problem that has recently gained traction in ML: hyperbolic embeddings with a "cheap" and accurate encoding along the hyperbolic vs Euclidean scale. We unveil a new application for which the Poincaré disk model has very appealing features, and our theory comes in handy: \textit{model} embeddings for boosted combinations of decision trees, trained using the log-loss (trees) and logistic loss (combinations). |
| title | Tempered Calculus for ML: Application to Hyperbolic Model Embedding |
| topic | Machine Learning I.2.6 |
| url | https://arxiv.org/abs/2402.04163 |