Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.

Consider $n+1$ samples $X_0, \ldots, X_n \sim D$. Is it possible to find an upper bound of

$$E_D(min_{v_1,\ldots, v_n} || X_0 - \sum_{i} v_i X_i||^2)$$

as a function of statistics of $D$? Otherwise stated, is it possible to bound the expected distance of a point sampled from $D$ and the smallest subspace containing $n$ points drawn from the same distribution.

Let $X$ be a $d$-dimensional random vector distributed according to probability measure $D$. At least the second moment of the coordinates of $X$ is finite.

Consider $n+1$ samples $X_0, \ldots, X_n \sim D$. Is it possible to find an upper bound of

$$E_D(\min_{v_1,\ldots, v_n} || X_0 - \sum_{i} v_i X_i||^2)$$

as a function of statistics of $D$? Otherwise stated, is it possible to bound the expected distance of a point sampled from $D$ and the smallest subspace containing $n$ points drawn from the same distribution.