Log-concave Density Estimation with Independent Components
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866912160904380416 |
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| author | Kubal, Sharvaj Campbell, Christian Robeva, Elina |
| author_facet | Kubal, Sharvaj Campbell, Christian Robeva, Elina |
| contents | We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(ε^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $ε$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_01500 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Log-concave Density Estimation with Independent Components Kubal, Sharvaj Campbell, Christian Robeva, Elina Statistics Theory Methodology We propose a method for estimating a log-concave density on $\mathbb R^d$ from samples, under the assumption that there exists an orthogonal transformation that makes the components of the random vector independent. While log-concave density estimation is hard both computationally and statistically, the independent components assumption alleviates both issues, while still maintaining a large non-parametric class. We prove that under mild conditions, at most $\tilde{\mathcal{O}}(ε^{-4})$ samples (suppressing constants and log factors) suffice for our proposed estimator to be within $ε$ of the original density in squared Hellinger distance. On the computational front, while the usual log-concave maximum likelihood estimate can be obtained via a finite-dimensional convex program, it is slow to compute -- especially in higher dimensions. We demonstrate through numerical experiments that our estimator can be computed efficiently, making it more practical to use. |
| title | Log-concave Density Estimation with Independent Components |
| topic | Statistics Theory Methodology |
| url | https://arxiv.org/abs/2401.01500 |