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This paper studies the "explanation problem" for tree- and linearly-orderedarray data, a problem motivated by database applications and recently solvedfor the one-dimensional tree-ordered case. In this paper, one is given a matrixA whose rows and columns have semantics: special subsets of the rows andspecial subsets of the columns are meaningful, others are not. A submatrix in Ais said to be meaningful if and only if it is the cross product of a meaningfulrow subset and a meaningful column subset, in which case we call it an "allowedrectangle." The goal is to "explain" A as a sparse sum of weighted allowedrectangles. Specifically, we wish to find as few weighted allowed rectangles aspossible such that, for all i,j, a_{ij} equals the sum of the weights of allrectangles which include cell (i,j).  In this paper we consider the natural cases in which the matrix dimensionsare tree-ordered or linearly-ordered. In the tree-ordered case, we are given arooted tree T1 whose leaves are the rows of A and another, T2, whose leaves arethe columns. Nodes of the trees correspond in an obvious way to the sets oftheir leaf descendants. In the linearly-ordered case, a set of rows or columnsis meaningful if and only if it is contiguous.  For tree-ordered data, we prove the explanation problem NP-Hard and give arandomized 2-approximation algorithm for it. For linearly-ordered data, weprove the explanation problem NP-Hard and give a 2.56-approximation algorithm.To our knowledge, these are the first results for the problem of sparsely andexactly representing matrices by weighted rectangles.

On Parsimonious Explanations for 2-D Tree- and Linearly-Ordered Data