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What are the typical case we need to use Non-negative least squares NNLS

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

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)

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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.

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

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