On estimation and detection of a function from tensor product spaces

We observe an unknown $d$-variables function $f(t)$, $ t\in[0,1]^d$ in the white Gaussian noise of a level $\varepsilon>0$. We suppose that $f\in\mathcal{F}$, where $\mathcal{F}$ is a ball in Hilbert space $\mathcal{L}^d\subset L_2([0,1]^d)$ of tensor product structure. Under minimax setup, we consider two problems: to estimate $f$ (for quadratic losses) and to detect $f$, i.e., to test the null hypothesis $H_0:f=0$ against alternatives $H_1: f\in\mathcal{F}_d$, $\|f\|_2\ge r_\varepsilon$. We are interesting in the case $d=d_\varepsilon\to\infty$. We study sharp, rate and log-asymptotics (as $\varepsilon\to 0$, $d\to\infty$) in the problems. In particular, we show that log-asymptotics are different essentially for $d\ll\log\varepsilon^{-1}$ and for $d\gg\log\varepsilon^{-1}$.