Projecting onto rectangular matrices with prescribed row and column sums

Heinz H Bauschke,Shambhavi Singh,Xianfu Wang

doi:10.1186/s13663-021-00708-1

Heinz H Bauschke, Shambhavi Singh + Show 1 more

Open Access

https://doi.org/10.1186/s13663-021-00708-1

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Abstract

In 1990, Romero presented a beautiful formula for the projection onto the set of rectangular matrices with prescribed row and column sums. Variants of Romero’s formula were rediscovered by Khoury and by Glunt, Hayden, and Reams for bistochastic (square) matrices in 1998. These results have found various generalizations and applications.In this paper, we provide a formula for the more general problem of finding the projection onto the set of rectangular matrices with prescribed scaled row and column sums. Our approach is based on computing the Moore–Penrose inverse of a certain linear operator associated with the problem. In fact, our analysis holds even for Hilbert–Schmidt operators, and we do not have to assume consistency. We also perform numerical experiments featuring the new projection operator.

Highlights

The goal of this paper is to provide explicit projection operators in more general settings
Remark 3.10 A reviewer pointed out that projection algorithms can be employed to solve linear programming problems provided a strict complementary condition holds. This does suggest a possibly interesting future project: explore whether the projections in this paper are useful in solving some linear programming problems on rectangular matrices with prescribed row and column sums
The second column reports what percent of the first feasible matrices obtained were closest to the starting point T0 in the given order. This is done by measuring T0 – T, where · is the operator norm, and Tk is the first feasible matrices obtained using a given algorithm (Dyk, DR, or method of alternating projections (MAP))

Summary

Note that

T∗f = ζ (y ⊗ x)∗f = ζ y, f x = x; (y, x) = (Te, T∗f ) = A(T) ∈ ran A. (y, x) = (Te, T∗f ) = A(T) ∈ ran A. We have all the results together to start tackling the Moore–Penrose inverse of A. Theorem 2.5 (Moore–Penrose inverse of A part 2) Let (y, x) ∈ Y × X. (i): In this case, A(T) = (0, T∗f ) and A∗(y, x) = f ⊗ x. (iii): In this case, A is the zero operator and the Desoer–Whalen conditions (see [6, page 51]) make it obvious that A† is the zero operator as well (ii): This can be proved similar to (i). (iii): In this case, A is the zero operator and the Desoer–Whalen conditions (see [6, page 51]) make it obvious that A† is the zero operator as well

Let us define the auxiliary function

PRe and

MAP updates via

Using iterations for feasibility

Solutions found