An approximate Spielman-Teng theorem for the least singular value s_n(M_n) of a random n × n square matrix M_n is a statement of the following form: there exist constants C, c > 0 such that for all η > 0, Pr(s_n(M_n) ≤ η) ≲ n^Cη + exp(−n^c). The goal of this paper is to develop a simple and novel framework for proving such results for discrete random matrices. As an application, we prove an approximate Spielman-Teng theorem for {0, 1}-valued matrices, each of whose rows is an independent vector with exactly n/2 zero components. This improves on previous work of Nguyen and Vu, and is the first such result in a ‘truly combinatorial’ setting.

随机n × n方阵M _n的最小奇异值s _n（M _n）的近似Spielman-Teng定理是以下形式的陈述：存在常数C，c > 0使得对于所有η > 0 ，PR（小号_ñ（中号_ñ）≤ η）≲ ñ ^ç η + EXP（ - ñ ^ç）。本文的目的是开发一个简单新颖的框架来证明离散随机矩阵的这种结果。作为一个应用程序，我们证明了{0，1}值矩阵的近似Spielman-Teng定理，其每一行都是具有正好为n / 2个零分量的独立向量。这是对Nguyen和Vu以前工作的改进，并且是“真正组合”设置中的第一个这样的结果。