We consider the problem of approximating a function from $L^2$ by an element
of a given $m$-dimensional space $V_m$, associated with some feature map
$\varphi$, using evaluations of the function at random points $x_1,\dots,x_n$.
After recalling some results on optimal weighted least-squares using
independent and identically distributed points, we consider weighted
least-squares using projection determinantal point processes (DPP) or volume
sampling. These distributions introduce dependence between the points that
promotes diversity in the selected features $\varphi(x_i)$. We first provide a
generalized version of volume-rescaled sampling yielding quasi-optimality
results in expectation with a number of samples $n = O(m\log(m))$, that means
that the expected $L^2$ error is bounded by a constant times the best
approximation error in $L^2$. Also, further assuming that the function is in
some normed vector space $H$ continuously embedded in $L^2$, we further prove
that the approximation is almost surely bounded by the best approximation error
measured in the $H$-norm. This includes the cases of functions from $L^\infty$
or reproducing kernel Hilbert spaces. Finally, we present an alternative
strategy consisting in using independent repetitions of projection DPP (or
volume sampling), yielding similar error bounds as with i.i.d. or volume
sampling, but in practice with a much lower number of samples. Numerical
experiments illustrate the performance of the different strategies.

通过使用随机点值函数的加权最小二乘逼近方法，该研究论文提供了一种依赖于投影行列式点过程（DPP）或体积采样的加权最小二乘泛化版本，证明了在期望意义下使用 O (mlog (m)) 个样本时预期的 L^2 误差受到常数倍的 L^2 最佳逼近误差的限制，并证明了在函数属于连续嵌入 L^2 的某个范数向量空间 H 的情况下，逼近几乎一定受到 H 范数下最佳逼近误差的限制，最后通过数值实验展示了不同策略的性能。