Detection of a Sparse Variable Function
Detection of a Sparse Variable Function
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稀疏变量函数的检测
DOI：
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发表时间：
2015
期刊：
影响因子：
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通讯作者：
I. Suslina
中科院分区：
文献类型：
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作者：
Yu. I. Ingster;I. Suslina
We observe an unknown d-variable function f = f(t), t = (t1, . . . , td) ∈ [0, 1]d, f ∈ L2([0, 1]d), in Gaussian white noise of level ε > 0. We test the null hypothesis H0 : f = 0 against an alternative H1. Under the alternative, we assume that the unknown function is separated from zero:
f≥rε$$ \left\Vert f\right\Vert \ge {r}_{\varepsilon } $$
for some positive family rε→ε→00$$ {r}_{\varepsilon}\underset{\varepsilon \to 0}{\to }0 $$. Moreover, we assume that the unknown d-variable function f is a function of a smaller number of variables s (“sparse variable” function) that satisfies some regularity constraints. We also consider the problem of adaptation in k = 1, . . . , s. We assume that d = dε → ∞. The integer s ∈ ℕ is either fixed or s = sε → ∞, s = o(d). We study minimax error probabilities and obtain minimax separation rates that provide distinguishability in the problems. Then we apply the results obtained in the case of alternatives from Sobolev balls with a deleted L2-ball.
We observe an unknown d-variable function f = f(t), t = (t1, . . . , td) ∈ [0, 1]d, f ∈ L2([0, 1]d), in Gaussian white noise of level ε > 0. We test the null hypothesis H0 : f = 0 against an alternative H1. Under the alternative, we assume that the unknown function is separated from zero:
f≥rε$$ \left\Vert f\right\Vert \ge {r}_{\varepsilon } $$
for some positive family rε→ε→00$$ {r}_{\varepsilon}\underset{\varepsilon \to 0}{\to }0 $$. Moreover, we assume that the unknown d-variable function f is a function of a smaller number of variables s (“sparse variable” function) that satisfies some regularity constraints. We also consider the problem of adaptation in k = 1, . . . , s. We assume that d = dε → ∞. The integer s ∈ ℕ is either fixed or s = sε → ∞, s = o(d). We study minimax error probabilities and obtain minimax separation rates that provide distinguishability in the problems. Then we apply the results obtained in the case of alternatives from Sobolev balls with a deleted L2-ball.