Compressive Hermite interpolation: sparse, high-dimensional approximation from gradient-augmented measurements
For high-dimensional function approximation, this work shows that incorporating gradient information improves approximation quality without increasing sample complexity, offering a practical advantage over standard methods.
The paper extends sparse polynomial approximation to use both function and gradient samples, achieving error bounds in a stronger Sobolev norm with the same asymptotic sample complexity as function-only methods, which is algebraic in sparsity and at most logarithmic in dimension.
We consider the sparse polynomial approximation of a multivariate function on a tensor product domain from samples of both the function and its gradient. When only function samples are prescribed, weighted $\ell^1$ minimization has recently been shown to be an effective procedure for computing such approximations. We extend this work to the gradient-augmented case. Our main results show that for the same asymptotic sample complexity, gradient-augmented measurements achieve an approximation error bound in a stronger Sobolev norm, as opposed to the $L^2$-norm in the unaugmented case. For Chebyshev and Legendre polynomial approximations, this sample complexity estimate is algebraic in the sparsity $s$ and at most logarithmic in the dimension $d$, thus mitigating the curse of dimensionality to a substantial extent. We also present several experiments numerically illustrating the benefits of gradient information over an equivalent number of function samples only.