AI 中文总结
研究单层ReLU网络验证问题，通过输入共享图表示未确定ReLU的共享模式，在图为匹配且边满足正则秩一条件时，导出二阶块稀疏SOS松弛紧性的充分条件，扩展了无边情况的紧稀疏松弛结果。
AI 中文摘要
Azuma、Kim和Yamashita将单层ReLU网络的验证问题表述为二次约束二次规划，并为无边情况和单单元设置建立了紧半定松弛。在这项工作中，我们通过输入共享图表示盒输入集上未确定ReLU的共享模式，并关注该图为匹配的情况。然后，我们为与该图的连通分量分解相关的二阶块稀疏SOS松弛的紧性导出了一个明确、可检验的充分条件。在匹配假设下，全局问题分解为孤立顶点块和单边块。关键难点在于为两单元边块建立紧性。对于正则秩一边，我们表明每个两单元局部集的凸包可以由通过公共共享标量耦合的两个简化单单元包精确描述。结合Azuma等人的单单元紧性结果与Farkas引理和仿射消除，我们为每个边块获得了一个局部二阶证书。孤立顶点块简化为盒输入集上的单单元问题，因此以相同阶处理。我们证明，当输入共享图为匹配且每条边满足正则秩一条件时，二阶块稀疏SOS松弛是紧的。这将无边情况的紧稀疏松弛结果扩展到了具有非平凡两单元相互作用的第一个稀疏设置。
英文摘要
Azuma, Kim, and Yamashita formulated the verification problem for one-layer ReLU networks as a quadratically constrained quadratic program and established tight semidefinite relaxations for the edgeless case and for one-unit settings. In this work, we represent the sharing pattern of undecided ReLUs over a box input set through an input-sharing graph and focus on the case where this graph is a matching. We then derive an explicit, checkable sufficient condition for the tightness of the order-$2$ block-sparse SOS relaxation associated with the connected-component decomposition of this graph. Under the matching assumption, the global problem decomposes into isolated-vertex blocks and single-edge blocks. The key difficulty, which is absent from the edgeless case, is establishing tightness for a two-unit edge block. For regular rank-one edges, we show that the convex hull of each two-unit local set can be described exactly by two reduced one-unit hulls coupled through a common shared scalar. Combining the one-unit tightness result of Azuma et al. with Farkas' lemma and affine elimination, we obtain a local order-$2$ certificate for each edge block. Isolated-vertex blocks reduce to one-unit problems over box input sets and are therefore handled at the same order. We prove that, when the input-sharing graph is a matching and every edge satisfies the regular rank-one condition, the order-$2$ block-sparse SOS relaxation is tight. This extends the tight sparse relaxation result for the edgeless case to the first sparse setting with a nontrivial two-unit interaction.
