What are the typical case we need to use [Non-negative least squares NNLS](https://en.wikipedia.org/wiki/Non-negative_least_squares)


$$
||Ax - B||^2
$$


instead of least-square $$ Ax-B$$ (or vice versa)?


And is there any drawback in applying them on large $A$ matrix (e.g. dimension 1000 row x 10 columns)



When to use non-negative-least square and least-square
You would want to constrain $x$ to non-negative values whenever a negative $x$ makes no physical sense, say because $x$ represents the intensity of a pixel, or the price of an object, or a frequency count, or a chemical concentration, etc. An overview of typical applications is given in [Nonnegativity Constraints in Numerical Analysis.](http://users.wfu.edu/plemmons/papers/nonneg.pdf)


A large dimension $n$ of the $n\times p$ matrix $A$ is not problematic, a large dimension $p$ (larger than $n$) is a complication, see [Non-negative least squares for high-dimensional linear models: consistency and sparse recovery without regularization](http://arxiv.org/abs/1205.0953v2)


