Adaptive exact recovery in sparse nonparametric models
Fuente:
arXiv
Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
|
| Schlagworte: | |
| Online-Zugang: | |
| Tags: |
Tag hinzufügen
Keine Tags, Fügen Sie den ersten Tag hinzu!
|
| _version_ | 1866915369222930432 |
|---|---|
| author | Stepanova, Natalia Turcicova, Marie |
| author_facet | Stepanova, Natalia Turcicova, Marie |
| contents | We observe an unknown regression function of $d$ variables $f(\boldsymbol{t})$, $\boldsymbol{t} \in[0,1]^d$, in the Gaussian white noise model of intensity $\varepsilon>0$. We assume that the function $f$ is regular and that it is a sum of $k$-variate functions, where $k$ varies from $1$ to $s$ ($1\leq s\leq d$). These functions are unknown to us and only few of them are nonzero. In this article, we address the problem of identifying the nonzero components of $f$ in the case when $d=d_\varepsilon\to \infty$ as $\varepsilon\to 0$ and $s$ is either fixed or $s=s_\varepsilon\to \infty$, $s=o(d)$ as $\varepsilon\to \infty$. This may be viewed as a variable selection problem. We derive the conditions when exact variable selection in the model at hand is possible and provide a selection procedure that achieves this type of selection. The procedure is adaptive to a degree of model sparsity described by the sparsity parameter $β\in(0,1)$. We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2411_04320 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Adaptive exact recovery in sparse nonparametric models Stepanova, Natalia Turcicova, Marie Statistics Theory 62G08 (Primary), 62H12, 62G20 (Secondary) We observe an unknown regression function of $d$ variables $f(\boldsymbol{t})$, $\boldsymbol{t} \in[0,1]^d$, in the Gaussian white noise model of intensity $\varepsilon>0$. We assume that the function $f$ is regular and that it is a sum of $k$-variate functions, where $k$ varies from $1$ to $s$ ($1\leq s\leq d$). These functions are unknown to us and only few of them are nonzero. In this article, we address the problem of identifying the nonzero components of $f$ in the case when $d=d_\varepsilon\to \infty$ as $\varepsilon\to 0$ and $s$ is either fixed or $s=s_\varepsilon\to \infty$, $s=o(d)$ as $\varepsilon\to \infty$. This may be viewed as a variable selection problem. We derive the conditions when exact variable selection in the model at hand is possible and provide a selection procedure that achieves this type of selection. The procedure is adaptive to a degree of model sparsity described by the sparsity parameter $β\in(0,1)$. We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area. |
| title | Adaptive exact recovery in sparse nonparametric models |
| topic | Statistics Theory 62G08 (Primary), 62H12, 62G20 (Secondary) |
| url | https://arxiv.org/abs/2411.04320 |