Detection of a sparse-variable function

We observe an unknown $d$-variable function $f=f(t)$, $t=(t_1,\dots,t_d)\in[0,\,1]^d,$ $f\in L_2([0,\,1]^d)$ in Gaussian white noise of level $\varepsilon>0$. We test the null hypothesis $H_0\colon f=0$ against the alternative $H_1$. Under the alternative, we suppose that unknown function is bounded away from zero:
$$ \|f\|\ge r_\varepsilon,$$
for some positive family $\underset{\varepsilon\to0}{r_\varepsilon\to0}$. Moreover, we assume that unknown $d$-variable $f$ is a function of a smaller number of variables $s$ (“sparse variable” function), and this function satisfies some regularity constraints. We also consider the problem of adaptation in $k=1,\dots,s$. We assume that $d=d_\varepsilon\to\infty$. The integer $s\in\mathbb N$ could be fixed or $s=s_\varepsilon\to\infty$, $s=o(d)$. We study the minimax error probabilities and obtain the minimax separation rates that provide distinguishability in the problems. Then, we apply the results obtained for the case of the alternatives from the Sobolev balls with the remote $L_2$-ball.