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Estimating the linear dimensionality of a data set in the presence of noiseis a common problem. However, data may also be corrupted by monotone nonlineardistortion that preserves the ordering of matrix entries but causes linearmethods for estimating rank to fail. In light of this, we consider the problemof computing \emph{underlying rank}, which is the lowest rank consistent withthe ordering of matrix entries, and \emph{monotone rank}, which is the lowestrank consistent with the ordering within columns. We show that each matrix ofmonotone rank $d$ corresponds to a point arrangement and a hyperplanearrangement in $\mathbb R^{d}$, and that the ordering within columns of thematrix can be used to recover information about these arrangements. UsingRadon's theorem and the related concept of the VC dimension, we can obtainlower bounds on the monotone rank of a matrix. However, we also show that themonotone rank of a matrix can exceed these bounds. In order to obtain betterbounds on monotone rank, we develop the connection between monotone rankestimation and oriented matroid theory. Using this connection, we show thatmonotone rank is difficult to compute: the problem of deciding whether a matrixhas monotone rank two is already NP-hard. However, we introduce an "orientedmatroid completion" problem as a combinatorial relaxation of the monotone rankproblem and show that checking whether a set of sign vectors has matroidcompletion rank two is easy.

Using oriented matroids to find low rank structure in presence of nonlinearity