A recent breakthrough in Edmonds' problem showed that the noncommutative rank can be computed in deterministic polynomial time, and various algorithms for it were devised. However, only quite complicated algorithms are known for finding a so-called shrunk subspace, which acts as a dual certificate for the value of the noncommutative rank. In particular, the operator Sinkhorn algorithm, perhaps the simplest algorithm to compute the noncommutative rank with operator scaling, does not find a shrunk subspace. Finding a shrunk subspace plays a key role in applications, such as separation in the Brascamp-Lieb polytope, one-parameter subgroups in the null-cone membership problem, and primal-dual algorithms for matroid intersection and fractional matroid matching. In this paper, we provide a simple Sinkhorn-style algorithm to find the smallest shrunk subspace over the complex field in deterministic polynomial time. To this end, we introduce a generalization of the operator scaling problem, where the spectra of the marginals must be majorized by specified vectors. Then we design an efficient Sinkhorn-style algorithm for the generalized operator scaling problem. Applying this to the shrunk subspace problem, we show that a sufficiently long run of the algorithm also finds an approximate shrunk subspace close to the minimum exact shrunk subspace. Finally, we show that the approximate shrunk subspace can be rounded if it is sufficiently close. Along the way, we also provide a simple randomized algorithm to find the smallest shrunk subspace. As applications, we design a faster algorithm for fractional linear matroid matching and efficient weak membership and optimization algorithms for the rank-2 Brascamp-Lieb polytope.

翻译：Edmonds 问题最近的突破表明, 非通缩级别可以在确定性多元时间里计算, 并且为此设计了各种算法。 但是, 找到所谓的缩缩子空间, 只有相当复杂的算法才为人所知, 也就是找到所谓的缩缩子空间, 这是用于计算非混合级别价值的双重证书。 特别是, 操作员 Sinkhorn 算法, 可能是计算操作员缩放非混合级别的最简单的算法, 找不到一个缩水的子空间。 找到一个缩水的子空间在应用程序中扮演着关键角色, 例如在 Brascamp- Lieb 多边算法中分离, 在无纸色成员问题中只知道一个一米的亚类算法, 而在本文中, 操作员 Sinkhorn 算法的最小缩水下算法中, 我们也可以找到一个更近的平流的平流层- 平流流的递增缩问题, 然后我们用一个最短的平流的平流层算法 来显示一个最短的直流的直径直流的递化的递化的亚级算。