Find vectors from inner products The problem is given all the inner products of N N-dimensional vector, can we work out the vector set?
For example given a N-by-N matrix $K$
$\{K\}_{ij} = v_i^Tv_j$ for all $i,j \in \{1,...,N\}$
Is it possible to find all $v_i$?
And will it be easier if we add norm constrain to the problem such that $||v_i||_2=1$?

No. You are essentially asking for a kind of matrix "square root": given a matrix $K$, find a matrix $V$ (whose columns are your $v_i$) such that $V^TV=K$. This $V$ is generally not unique; multiplying $V$ on the left by any orthogonal matrix is still a solution. In fact, the only matrix $K$ with a unique $V$ is the zero matrix.
Requiring $v_i$ to be unit does not help; for $K=I$, any orthogonal matrix $V$ has this property.
If you only care about finding a $V$, even if it isn't unique, then a solution exists if and only if $K$ is positive semi-definite, in which case one solution is given by the Cholesky decomposition.
