# Approximation of Grassmanian cubatures through random noise

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 ?