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We consider the problem of approximating a function from $L^2$ by an elementof 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 usingindependent and identically distributed points, we consider weightedleast-squares using projection determinantal point processes (DPP) or volumesampling. These distributions introduce dependence between the points thatpromotes diversity in the selected features $\varphi(x_i)$. We first provide ageneralized version of volume-rescaled sampling yielding quasi-optimalityresults in expectation with a number of samples $n = O(m\log(m))$, that meansthat the expected $L^2$ error is bounded by a constant times the bestapproximation error in $L^2$. Also, further assuming that the function is insome normed vector space $H$ continuously embedded in $L^2$, we further provethat the approximation is almost surely bounded by the best approximation errormeasured in the $H$-norm. This includes the cases of functions from $L^\infty$or reproducing kernel Hilbert spaces. Finally, we present an alternativestrategy consisting in using independent repetitions of projection DPP (orvolume sampling), yielding similar error bounds as with i.i.d. or volumesampling, but in practice with a much lower number of samples. Numericalexperiments illustrate the performance of the different strategies.

Weighted least-squares approximation with determinantal point processes and generalized volume sampling