论文标题
在域适应性中使用嘈杂样本的最小二乘近似值保证了错误
Error Guarantees for Least Squares Approximation with Noisy Samples in Domain Adaptation
论文作者
论文摘要
给定的$ n $样本$ f \ colon d \ to \ mathbb c $随机点相对于$ \ varrho_s $，我们开发了$ l_2（d，\ varrho_t）$ - 近似错误的理论分析。对于$ \ varrho_s $的派iclar选择，取决于$ \ varrho_t $，众所周知，从有限尺寸函数空间$ v_m $，$ \ dim（v_m）= m <\ infty $具有与$ v_m $ up lo的最佳近距离相同的差异时，加权最小二乘法$ v_m $ $ v_m $ $ v_m（v_m）= m <\ infty $相同。如果源度量$ \ varrho_s $和目标度量$ \ varrho_t $不同，我们在域的适应设置中是转移学习的子场。我们对界限中误差的结果恶化建模。此外，对于嘈杂的样本，我们的界限描述了偏见变化的权衡，具体取决于近似空间的尺寸$ m $ $ v_m $。所有结果都具有很高的可能性。为了进行演示，我们考虑在Unifom随机样品中给出的$ d $二维多维数据集中定义的功能。我们分析了多项式，半周期余弦和非周期性sobolev space $ h _ {\ mathrm {mix}}}^2 $的有界正顺序基础。克服此$ h _ {\ text {mix}}^2 $基础的数值问题，这给出了一种具有二次错误衰减的新颖稳定近似方法。数值实验表明我们的结果的适用性。
Given $n$ samples of a function $f\colon D\to\mathbb C$ in random points drawn with respect to a measure $\varrho_S$ we develop theoretical analysis of the $L_2(D, \varrho_T)$-approximation error. For a parituclar choice of $\varrho_S$ depending on $\varrho_T$, it is known that the weighted least squares method from finite dimensional function spaces $V_m$, $\dim(V_m) = m < \infty$ has the same error as the best approximation in $V_m$ up to a multiplicative constant when given exact samples with logarithmic oversampling. If the source measure $\varrho_S$ and the target measure $\varrho_T$ differ we are in the domain adaptation setting, a subfield of transfer learning. We model the resulting deterioration of the error in our bounds. Further, for noisy samples, our bounds describe the bias-variance trade off depending on the dimension $m$ of the approximation space $V_m$. All results hold with high probability. For demonstration, we consider functions defined on the $d$-dimensional cube given in unifom random samples. We analyze polynomials, the half-period cosine, and a bounded orthonormal basis of the non-periodic Sobolev space $H_{\mathrm{mix}}^2$. Overcoming numerical issues of this $H_{\text{mix}}^2$ basis, this gives a novel stable approximation method with quadratic error decay. Numerical experiments indicate the applicability of our results.
