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Adaptive exact recovery in sparse nonparametric models
Statistics Theory · Submitted 6 Nov 2024
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim 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.
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The abstract, sentence by sentence
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 $\beta\in(0,1)$. We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | 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 $\varepsilon>0$. | background 0.94 | 0.38 | 0.02 | asserted 0.98 |
| s1 | We assume that the function $f$ is regular and that it is a sum of $k$-variate functions, where $k$ varies from $1$ to $s$ ($1\leq s\leq d$). | background 0.79 | 0.20 | 0.02 | asserted 1.00 |
| s2 | These functions are unknown to us and only few of them are nonzero. | background 0.99 | 0.10 | 0.02 | asserted 1.00 |
| s3 | 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 $\varepsilon\to \infty$. | contribution 0.53 | 0.22 | 0.07 | asserted 0.81 |
| s4 | This may be viewed as a variable selection problem. | interpretation 0.81 | 0.14 | 0.03 | hedged possibility 0.94 |
| s5 | 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. | contribution 0.90 | 0.97 | 0.65 | asserted 1.00 |
| s6 | The procedure is adaptive to a degree of model sparsity described by the sparsity parameter $\beta\in(0,1)$. | contribution 0.48 | 0.86 | 0.11 | asserted 1.00 |
| s7 | We also derive conditions that make the exact variable selection impossible. | result 0.91 | 0.94 | 0.10 | asserted 1.00 |
| s8 | Our results augment previous work in this area. | contribution 0.69 | 0.35 | 0.04 | asserted 0.99 |
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Original abstract
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 $\beta\in(0,1)$. We also derive conditions that make the exact variable selection impossible. Our results augment previous work in this area.
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