Robust density estimation over star-shaped density classes
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866910788109729792 |
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| author | Liu, Xiaolong Neykov, Matey |
| author_facet | Liu, Xiaolong Neykov, Matey |
| contents | We establish a novel criterion for comparing the performance of two densities, $g_1$ and $g_2$, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, $\mathcal{F}$, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction $ε\leq \frac{1}{3}$ of the $N$ observations are arbitrarily corrupted. We obtain the minimax upper bound $\max\{ τ_{\overline{J}}^2, ε\} \wedge d^2$. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared $L_2$ loss as $$ \max\left\{ τ^{*2} \wedge d^2, ε\wedge d^2 \right\}, $$ where $τ^* := \sup\left\{ τ: Nτ^2 \leq \log \mathcal{M}_{\mathcal{F}}^{\text{loc}}(τ, c) \right\}$ for a sufficiently large constant $c$. Here, $\mathcal{M}_{\mathcal{F}}^{\text{loc}}(τ, c)$ denotes the local entropy of the set $\mathcal{F}$, and $d$ is the $L_2$ diameter of $\mathcal{F}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_10025 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Robust density estimation over star-shaped density classes Liu, Xiaolong Neykov, Matey Statistics Theory Machine Learning We establish a novel criterion for comparing the performance of two densities, $g_1$ and $g_2$, within the context of corrupted data. Utilizing this criterion, we propose an algorithm to construct a density estimator within a star-shaped density class, $\mathcal{F}$, under conditions of data corruption. We proceed to derive the minimax upper and lower bounds for density estimation across this star-shaped density class, characterized by densities that are uniformly bounded above and below (in the sup norm), in the presence of adversarially corrupted data. Specifically, we assume that a fraction $ε\leq \frac{1}{3}$ of the $N$ observations are arbitrarily corrupted. We obtain the minimax upper bound $\max\{ τ_{\overline{J}}^2, ε\} \wedge d^2$. Under certain conditions, we obtain the minimax risk, up to proportionality constants, under the squared $L_2$ loss as $$ \max\left\{ τ^{*2} \wedge d^2, ε\wedge d^2 \right\}, $$ where $τ^* := \sup\left\{ τ: Nτ^2 \leq \log \mathcal{M}_{\mathcal{F}}^{\text{loc}}(τ, c) \right\}$ for a sufficiently large constant $c$. Here, $\mathcal{M}_{\mathcal{F}}^{\text{loc}}(τ, c)$ denotes the local entropy of the set $\mathcal{F}$, and $d$ is the $L_2$ diameter of $\mathcal{F}$. |
| title | Robust density estimation over star-shaped density classes |
| topic | Statistics Theory Machine Learning |
| url | https://arxiv.org/abs/2501.10025 |