Minimal length power-sum representation for polynomials

Question

I am interested in the polynomial basis in power of linear forms. For example, for any multivariate polynomial form $p(x)$ of variable $x\in \mathbb{R}^n$ and of degree $d$. We have a polynomial basis $\mathcal{B}=\{\langle \alpha, x \rangle^d : \alpha \in I_{n,d} \}$ where $I_{n,d}$ is the set of all weak compositions of d with n parts. Hence, $p(x)$ can be uniquely written as a power-sum as $$p(x) = \sum_{\alpha \in I_{n,d}} \lambda_{\alpha}\langle \alpha, x \rangle^d.$$ My question is : how to find a polynomial basis different to $\mathcal{B}$ but still in power of linear forms such that the number of non-zero coefficient $\lambda$ is minimized? In other words, I want to represent $p(x)$ as sparse as possible in a basis of power of linear forms. Thanks a lot for any comments.