论文标题
概率误差界限，用于重现内核希尔伯特空间中功能的近似值
Probability error bounds for approximation of functions in reproducing kernel Hilbert spaces
论文作者
论文摘要
我们发现概率错误的范围限制了函数的近似值$ f $在可分离的再现核Hilbert Space $ \ MATHCAL {H} $中，基本空间$ x $上的bernel $ k $首先，首先是在有限的函数的有限线性组合方面$ \ mathrm {span} \ {k_ {x_i} \}^n_ {i = 1} $，对于点的随机序列$ x =（x_i）_i $ in $ x $。给定概率度量$ p $，让$ p_k $是$ \ mathrm {d} p_k（x）= k（x，x，x）\ mathrm {d} p（x）$，$ x \ in x $定义的度量l_ {p，k}λ：= \int_xλ（x）k_x \ mathrm {d} p（x）\ in \ mathcal {h}，\]，其中积分存在于Bochner Sense中。然后，使用此操作员，我们定义了一个新的复制内核Hilbert Space，该空间由$ \ Mathcal {h} _p $表示，即$ L_ {p，k} $的运算符范围。我们的主要结果是根据操作员$ l_ {p，k} $建立界限，这是因为Hilbert的空间距离任意函数$ f \ in \ mathcal {h} $与类型$ k_ {x_i} $的函数的线性组合，对于$（x_i）$（x_i）__i $ sampled $ phlift a phl then a thriplass flat flats flat flats flat flats flat flats flat flats flats flat flats，对于点$（x_i）_ {i = 1}^\ infty $构成所谓的唯一性集的序列，正交投影$π^n_x $ to $ \ \ \ mathrm {span} \ {k_ {k_ {k_i} {x_i} \}^n__ = 1} $ conterator the Storge to the Storge to the Storge to the Storge to the Storge to the Storge conterator to the Storge to the Storge to the storte to the storte the poterator。我们证明，在假设$ \ MATHCAL {H} _p $在$ \ Mathcal {H} $中密集的$ \ Mathcal {H} $，$ P $的任何iID样本序列都会产生概率$ 1 $的独特设置。此结果在较弱的规范中（例如统一或$ l^p $ norms）的先前误差范围有所改善，这些规范仅产生概率而不是A.C的收敛。收敛。两个示例表明该结果适用于紧凑的间隔和hardy空间$ h^2（\ mathbb {d}）$的适用性。
We find probability error bounds for approximations of functions $f$ in a separable reproducing kernel Hilbert space $\mathcal{H}$ with reproducing kernel $K$ on a base space $X$, firstly in terms of finite linear combinations of functions of type $K_{x_i}$ and then in terms of the projection $π^n_x$ on $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$, for random sequences of points $x=(x_i)_i$ in $X$. Given a probability measure $P$, letting $P_K$ be the measure defined by $\mathrm{d} P_K(x)=K(x,x)\mathrm{d} P(x)$, $x\in X$, our approach is based on the nonexpansive operator \[L^2(X;P_K)\niλ\mapsto L_{P,K}λ:=\int_X λ(x)K_x\mathrm{d} P(x)\in \mathcal{H},\] where the integral exists in the Bochner sense. Using this operator, we then define a new reproducing kernel Hilbert space, denoted by $\mathcal{H}_P$, that is the operator range of $L_{P,K}$. Our main result establishes bounds, in terms of the operator $L_{P,K}$, on the probability that the Hilbert space distance between an arbitrary function $f\in\mathcal{H}$ and linear combinations of functions of type $K_{x_i}$, for $(x_i)_i$ sampled independently from $P$, falls below a given threshold. For sequences of points $(x_i)_{i=1}^\infty$ constituting a so-called uniqueness set, the orthogonal projections $π^n_x$ to $\mathrm{Span}\{K_{x_i}\}^n_{i=1}$ converge in the strong operator topology to the identity operator. We prove that, under the assumption that $\mathcal{H}_P$ is dense in $\mathcal{H}$, any sequence of iid samples from $P$ yields a uniqueness set with probability $1$. This result improves on previous error bounds in weaker norms, such as uniform or $L^p$ norms, which yield only convergence in probability and not a.c. convergence. Two examples that show the applicability of this result to a uniform distribution on a compact interval and to the Hardy space $H^2(\mathbb{D})$ are presented as well.
