We study the problem of finding elements in the intersection of an arbitrary conic variety in $\mathbb{F}^n$ with a given linear subspace (where $\mathbb{F}$ can be the real or complex field). This problem captures a rich family of algorithmic problems under different choices of the variety. The special case of the variety consisting of rank-1 matrices already has strong connections to central problems in different areas like quantum information theory and tensor decompositions. This problem is known to be NP-hard in the worst-case, even for the variety of rank-1 matrices. Surprisingly, despite these hardness results we give efficient algorithms that solve this problem for "typical" subspaces. Here, the subspace $U \subseteq \mathbb{F}^n$ is chosen generically of a certain dimension, potentially with some generic elements of the variety contained in it. Our main algorithmic result is a polynomial time algorithm that recovers all the elements of $U$ that lie in the variety, under some mild non-degeneracy assumptions on the variety. As corollaries, we obtain the following results: $\bullet$ Uniqueness results and polynomial time algorithms for generic instances of a broad class of low-rank decomposition problems that go beyond tensor decompositions. Here, we recover a decomposition of the form $\sum_{i=1}^R v_i \otimes w_i$, where the $v_i$ are elements of the given variety $X$. This implies new algorithmic results even in the special case of tensor decompositions. $\bullet$ Polynomial time algorithms for several entangled subspaces problems in quantum entanglement, including determining $r$-entanglement, complete entanglement, and genuine entanglement of a subspace. While all of these problems are NP-hard in the worst case, our algorithm solves them in polynomial time for generic subspaces of dimension up to a constant multiple of the maximum possible.
翻译:我们研究如何在任意的调子的交叉点中找到一个在$\mathb{F ⁇ n$和给定的线性子空间( $\mathb{F}$可以是真实或复杂的字段) 的调子。 这个问题在不同的选择下可以捕捉到丰富的算法问题。 由等级-1矩阵组成的特例已经与不同领域的中心问题有着紧密的联系, 比如量子信息理论和变异。 这个问题在最坏的情况下是NP- 硬的, 甚至对于等级-1 矩阵的种类。 令人惊讶的是,尽管这些硬性结果能够解决“ 典型” 子空间的问题。 这里, 子空间 $U\subsetebbb{F ⁇ n 美元是某些层面的。 我们的主要算法结果是一个甚至最坏的时间算法, 美元- 美元的值- 的变异性, 在较轻微的 美元- 时间 假设下, 我们的变现结果是 美元- 数字 的变化 的变数 。