Border complexity captures functions that can be approximated by low-complexity ones. Debordering is the task of proving an upper bound on some non-border complexity measure in terms of a border complexity measure, thus getting rid of limits. Debordering lies at the heart of foundational complexity theory questions relating Valiant's determinant versus permanent conjecture (1979) and its geometric complexity theory (GCT) variant proposed by Mulmuley and Sohoni (2001). The debordering of matrix multiplication tensors by Bini (1980) played a pivotal role in the development of efficient matrix multiplication algorithms. Consequently, debordering finds applications in both establishing computational complexity lower bounds and facilitating algorithm design. Recent years have seen notable progress in debordering various restricted border complexity measures. In this survey, we highlight these advances and discuss techniques underlying them.

翻译：边界复杂度刻画了可由低复杂度函数逼近的函数。去边界化任务旨在依据边界复杂度度量证明某些非边界复杂度度量的上界，从而消除极限过程。去边界化处于基础计算复杂性理论的核心问题之中，这些问题关联着Valiant的行列式与永久式猜想（1979）及其由Mulmuley和Sohoni（2001）提出的几何复杂性理论变体。Bini（1980）对矩阵乘法张量的去边界化在高效矩阵乘法算法的发展中发挥了关键作用。因此，去边界化在建立计算复杂性下界和促进算法设计两方面都具有应用价值。近年来，针对各类受限边界复杂度度量的去边界化研究取得了显著进展。本综述重点阐述这些进展，并探讨其背后的技术原理。