Finding a linear combination with constraints on coefficients Let there be $n$ unit vectors $\{\boldsymbol{u}_i\}_{1\leq i\leq n }$ in an $m$ dimensional space. The vectors are not necessarily a basis of the space. Let $\boldsymbol{v}$ be a unit vector in the $m$ dim. space. I suspect that there is a linear combination such that
$$\boldsymbol{v} \approx\sum_{i=1}^n a_i\boldsymbol{u}_i,$$
where $a_i\geq 0$, and $a_i$ are relatievly small. What do I mean by this? the actual problem is the following: The $\boldsymbol{u}_i$ vectors represent a text of a single topic. For instance $\boldsymbol{u}_1$ could represent texts on politics, and $\boldsymbol{u}_3$ could represent texts on football, etc. If I have a text represented by a vector $\boldsymbol{v}$, and the text is composed of politics and football, then I expect to find numerically that $a_1$ and $a_3$ are significantly big (so that $\boldsymbol{v}$ remains of norm 1), while $a_2, a_4, ..., a_n \approx 0$. Also, I expect $\{\boldsymbol{u}_i\}$ to be a set of linearly independent vectors since they describe different topics (this could be helpful so that the linear combination has sense).
I need to know how to include the constraints on the coefficients in order to solve this problem numerically. Any references or help on how to implement this idea would be of great benefit. Thank you.
 A: If we have a linear system $A x = b$ with infinitely many solutions, then finding the solution with the minimum $2$-norm is the least-norm problem
$$\begin{array}{ll} \text{minimize} & \|x\|_2^2\\ \text{subject to} & A x = b\end{array}$$
Assuming that $A$ has full row rank, the least-norm solution is
$$x_{\ln} := A^T (A A^T)^{-1} b$$
Including nonnegativity constraints, $x \geq 0_n$, if $x_{\ln} \geq 0_n$, then we are done. If not, then we solve the following quadratic program (QP)
$$\begin{array}{ll} \text{minimize} & \|x\|_2^2\\ \text{subject to} & A x = b\\ & x \geq 0_n\end{array}$$
A: You should consider carefully whether you need to satisfy 
$\sum a_{i}u_{i}=Ua=v$
exactly, or whether you can live with a solution where $\| Ua-v \|_{2}$ is small enough (and then of course you have to decide what's a small enough error.)  
You should also consider carefully what property of the vector $a$ you want to optimize.  If (as another answer has suggested) you want to minimize $\| a \|_{2}$, then Tikhonov regularization is the way to go.  However, if you want to minimize the number of nonzero $a_{i}$ coefficients, then minimizing $\| a \|_{1}$ (know as LASSO) is likely to be the way you want to go.  
