🤖 AI Summary
This work studies weighted least-squares function approximation in $L^2$ space based on random sampling: given an $m$-dimensional subspace $V_m$, how to achieve near-optimal $L^2$ approximation error with minimal sampling cost. We propose a generalized volume resampling framework that, for the first time, achieves expected near-optimal $L^2$ error—i.e., bounded by a constant multiple of the best approximation error—using only $O(m log m)$ samples. Furthermore, in embedding normed spaces, we establish almost-sure error control in the $H$-norm. Our method integrates projection determinantal point processes (DPPs), generalized volume sampling, and independent repeated DPP sampling, significantly enhancing sample diversity and feature selection efficiency. Numerical experiments demonstrate that our approach attains accuracy comparable to i.i.d. or classical volume sampling—but with substantially fewer samples.
📝 Abstract
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 $n = O(mlog(m))$, 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.