Theorems on Positive Data: On the Uniqueness of NMF
Figure 1
A three-dimensional space is scaled such that the
vectors are in the hyper plane: .
By the mapping to the hyper plane, a plane in is mapped to a line and a simplicial cone is mapped to an area. In the
figure, it can be observed that the dashed triangle (desired solution) is the
only triangle (third-order simplicial cone) that
contains the shaded area (positive span of ) while being within the solid border (the
dual of ). The NMF can be concluded to be unique by
Theorem 1.
