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Let $G_{1,d}$ be the $1$-grassmanian in $d$ dimensions, that is the set of linear projections from $\mathbb R^d$ to $\mathbb R$. We can see it as $\mathbb S(\mathbb R^d)$, as any projection can be defined by its (one-dimensional) direction in the $\mathbb R^d$ space.

To a direction $c \in \mathbb S(\mathbb R^d)$ and and integer $n \in \mathbb N$, we associate the homogeneous polynomial $P_c^{n}(x) = \langle c, x\rangle^n$.

Denote $\text{Hom}_n$ the set of homogeneous polynomials of order $n$ in $d$ variables. I know that there exist bases of $\text{Hom}_n$ that consists of such polynomials $P_c^{n}(x)$. Let $D$ be the dimension of $\text{Hom}_n$ (as far as i remember, $D = \binom{n+d-1}{d-1}$).

I have two questions:

  1. If I take $c_1,....c_D$ independently (say from the $\mathcal U\left(\mathbb S(\mathbb R^d)\right)$ distribution, or even the $\mathcal U\left(\mathbb S_+(\mathbb R^d)\right)$ distribution), will the set of polynomials $\left\{P_{c_i}^{n}, i \in 1,...,D\right\}$ be a basis of $\text{Hom}_n$ ?

  2. If I take $c_1,...c_E$, $E < D$, still independent and uniform, and say $P \in \text{Hom}_n$, is it possible to compute (or at least bound, e.g. in expectation) the projection loss of $P$ into this basis ?

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