Suppose there are two vectors: $\mathbf{x} \in \mathbb{R}^N$ and $\mathbf{y} \in \mathbb{Z}^N_+$. If I know that $\mathbf{x} = \alpha \cdot \mathbf{y}$, where $\alpha \in \mathbb{R}$, is there a way to estimate $\alpha$ given only $\mathbf{x}$? Put another way, if I know that a vector of floats was created by multiplying a vector of positive integers by a floating point scalar, can I estimate that scalar accurately?
Edit:
The problem can be framed another way: Given a vector of real numbers, find the (smallest/largest) scalar that converts them all to integers.