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The projection of a vector on a unit ball has a closed form solution, given by $$\frac{x}{max(1,\|x\|)}$$.
Is there a similar formula for the projection of a matrix over a unit ball of matrices (matrix of values containing between -1 and 1) ?. Is there a closed form solution ?
Based on the comments from OP, the problem is to find the closest projection of a given matrix $A$ to the set of the matrices $\mathcal{C}$. Set of matrices $\mathcal{C}$ is defined as the set of all matrices whose entries lie in the interval $[-1,1]$. Thus OP's problem can be written as $$X^{*}=\arg\min_{X\in\mathcal{C}}||X-A||^2_F$$ First observation is that this is a convex problem. In fact, this is a straightforward problem to solve. Note that the objective can be written as $$||X-A||^2_F=\sum_{i,j}(X_{ij}-A_{ij})^2$$ It is easy to see that solution for one entry is independent of other. Thus, it is enough to find solutions for $X_{ij}$ individually. Since $X_{ij}$ is bound to lie in $[-1,1]$, its enough to find closest point of $A_{ij}$ in $[-1,1]$. This can be easily seen to be $$X_{ij}=\begin{cases}A_{ij}, & \mbox{if}~-1<A_{ij}<1\\-1, & \mbox{if}~-\infty<A_{ij}<-1 \\1, & \mbox{if}~~1<A_{ij}<\infty\end{cases}$$
