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：我们研究在 $\mathbb{F}^n$ 中任意二次型簇与给定线性子空间交集内元素的求解问题（其中 $\mathbb{F}$ 可为实数域或复数域）。该问题通过选择不同的代数簇，可涵盖一系列丰富的算法问题。当代数簇为秩1矩阵构成的特殊情形时，该问题已与量子信息论、张量分解等不同领域的核心问题产生深刻关联。已知即使对于秩1矩阵簇，该问题在最坏情况下是NP难的。令人意外的是，尽管存在这些困难结论，我们提出了针对"典型"子空间有效求解该问题的算法。此处子空间 $U \subseteq \mathbb{F}^n$ 以特定维数被一般性选取，且可能包含该簇中的若干一般性元素。我们的主要算法结果是在对簇施加温和的非退化假设下，给出一个多项式时间算法，可恢复 $U$ 中所有位于该簇内的元素。作为推论，我们获得以下结果：$\bullet$ 对于超越张量分解的广义低秩分解问题类的一般实例，给出唯一性结论与多项式时间算法。此处我们恢复形如 $\sum_{i=1}^R v_i \otimes w_i$ 的分解，其中 $v_i$ 属于给定簇 $X$。即使对于张量分解这一特例，这也蕴含了新的算法结论。$\bullet$ 针对量子纠缠中多个纠缠子空间问题（包括判断子空间的 $r$-纠缠性、完全纠缠性及真纠缠性）的多项式时间算法。尽管这些问题在最坏情况下均为NP难的，我们的算法可在多项式时间内解决维数不超过最大可能常数倍的一般子空间的此类问题。