Nonnegative Rank

Formal definition

There are several equivalent definitions, all modifying the definition of the linear rank slightly. Apart from the definition given above, there is the following: The nonnegative rank of a nonnegative m×n-matrix A is equal to the smallest number q such there exists a nonnegative m×q-matrix B and a nonnegative q×n-matrix C such that A = BC (the usual matrix product). To obtain the linear rank, drop the condition that B and C must be nonnegative.

Further, the nonnegative rank is the smallest number of nonnegative rank-one matrices into which the matrix can be decomposed additively:

${\mbox{rank}}_{+}(A)=\min\{q\mid \sum _{{j=1}}^{q}R_{j}=A,\;{\mbox{rank}}\,R_{1}=\dots ={\mbox{rank}}\,R_{q}=1,\;R_{1},\dots ,R_{q}\geq 0\},$

where R_j ≥ 0 stands for "R_j is nonnegative".^[1] (To obtain the usual linear rank, drop the condition that the R_j have to be nonnegative.)

Given a nonnegative $m \times n$ ${\mbox{rank}}\,(A)\leq {\mbox{rank}}_{+}(A)\leq \min(m,n),$ where ${\mbox{rank}}\,(A)$

A Fallacy

The rank of the matrix A is the largest number of columns which are linearly independent, i.e., none of the selected columns can be written as a linear combination of the other selected columns. It is not true that adding nonnegativity to this characterization gives the nonnegative rank: The nonnegative rank is in general less than or equal to the largest number of columns such that no selected column can be written as a nonnegative linear combination of the other selected columns.

Connection with the linear rank

It is always true that rank(A) ≤ rank₊(A). In fact rank₊(A) = rank(A) holds whenever rank(A) ≤ 2 [2].

In the case rank(A) ≥ 3, however, rank(A) < rank₊(A) is possible. For example, the matrix ${\mathbf {A}}={\begin{bmatrix}1&1&0&0\\1&0&1&0\\0&1&0&1\\0&0&1&1\end{bmatrix}},$

satisfies rank(A) = 3 < 4 = rank₊(A) [2].

Computing the nonnegative rank

The nonnegative rank of a matrix can be determined algorithmically.^[2]

It has been proved that determining whether ${{rank}_{+}}(A)=rank(A)$

Obvious questions concerning the complexity of nonnegative rank computation remain unanswered to date. For example, the complexity of determining the nonnegative rank of matrices of fixed rank k is unknown for k > 2.

Ancillary facts

Nonnegative rank has important applications in Combinatorial optimization:^[4] The minimum number of facets of an extension of a polyhedron P is equal to the nonnegative rank of its so-called slack matrix.^[5]

References

^ Abraham Berman and Robert J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, SIAM
^ J. Cohen and U. Rothblum. "Nonnegative ranks, decompositions and factorizations of nonnegative matrices". Linear Algebra and its Applications, 190:149–168, 1993.
^ Stephen Vavasis, On the complexity of nonnegative matrix factorization, SIAM Journal on Optimization 20 (3) 1364-1377, 2009.
^ Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441–466, 1991.
^ See this blog post Archived 2014-10-07 at the Wayback Machine

[1] Abraham Berman and Robert J. Plemmons. Nonnegative Matrices in the Mathematical Sciences, SIAM

[2] J. Cohen and U. Rothblum. "Nonnegative ranks, decompositions and factorizations of nonnegative matrices". Linear Algebra and its Applications, 190:149–168, 1993.

[3] Stephen Vavasis, On the complexity of nonnegative matrix factorization, SIAM Journal on Optimization 20 (3) 1364-1377, 2009.

[4] Mihalis Yannakakis. Expressing combinatorial optimization problems by linear programs. J. Comput. Syst. Sci., 43(3):441–466, 1991.

[5] See this blog post Archived 2014-10-07 at the Wayback Machine

[1]