On the complexity of general matrix scaling and entropy minimization via the RAS algorithm

Abstract

Given an n × m nonnegative matrix A = (a_{ij}) and positive integral vectors r in R^n and c in R^m having a common one-norm h, the (r, c)-scaling problem is to obtain positive diagonal matrices X and Y, if they exist, such that XAY has row and column sums equal to r and c, respectively. The entropy minimization problem corresponding to A is to find an n × m matrix z = (z_{ij}) having the same zero pattern as A, the sum of whose entries is a given number h, its row and column sums are within given integral vectors of lower and upper bounds, and such that the entropy function consisting of the sum of the terms z_{ij} ln(z_{ij}/a_{ij}) is minimized. When the lower and upper bounds coincide, matrix scaling and entropy minimization are closely related. In this paper we present several complexity bounds for the \epsilon-approximate (r, c)-scaling problem, polynomial in n, m, h, 1/\epsilon, and ln V , where V and v are the largest and the smallest positive entries of A, respectively. These bounds, although not polynomial in ln(1/\epsilon), not only provide alternative complexities for the polynomial time algorithms, but could result in better overall complexities. In particular, our theoretical analysis supports the practicality of the well-known RAS algorithm

Reverse hypercontractivity using information measures

“…Indeed, it can be readily checked that (27) is a stronger constraint on (W, Z) than s * (W ; Z) ≤ (1 − 2 ) 2 that was used in [25]. Inequalities (27) and (28) can thus provide a stronger formulaic approach to establish Conjecture 1 analytically.…”

“…A computer search suggests that either of the constraints (27), (28) individually appear to suffice to establish I(W ; Z) ≤ 1 − h( ). Indeed, it can be readily checked that (27) is a stronger constraint on (W, Z) than s * (W ; Z) ≤ (1 − 2 ) 2 that was used in [25].…”

“…Computing this infimum is a convex problem and may be solved efficiently using convex solvers. A popular algorithm for this computation in practice is the alternating minimization algorithm also known as the RAS algorithm (see [27] for details about this and other algorithms). However, by choosing the specific form for the r X and r Y in (iv ), we get to bypass the computationally intensive step that calculates the infimum.…”

Certificates of optimality for minimum norm biproportional apportionments

“…A full characterization of fair share quotas can be found in [1,2,9]. Sometimes different quotas are used in real electoral systems (e.g.…”

Balancing of agricultural census data by using discrete optimization

“…the RAS algorithm [22]. In other cases, on the contrary, the variations introduced for balancing the matrix should pursue an objective that typically depends on the specific application.…”