论文标题
定向的矩形和组合神经代码
Oriented Matroids and Combinatorial Neural Codes
论文作者
论文摘要
组合神经代码$ \ mathscr c \ subseteq 2^{[n]} $是凸的，如果它作为$ \ mathbb r^d $的凸面开放子集的相交模式出现。我们将凸神经代码的新兴理论与确定的矩形理论相关联，无论是在几何和计算复杂性方面。在分类方面，我们表明，将面向无环的矩阵的地图符合其山顶的积极部分的代码是忠实的函数。我们将Novik，Postnikov和Sturmfels引入的定向的矩阵理想转化为从定向的Matroid类别中的函子到戒指的类别。然后，我们表明所产生的环映射自然地映射到曲霉神经代码的神经环。 对于几何和计算复杂性，我们表明，当且仅当它位于Jeffs引入的代码的部分代码中，代码与凸多属的实现具有实现。我们表明，以前发表的非凸代码的示例不在任何方向的矩阵下，并且我们构建了位于非代表性的矩阵下方的非convex代码的示例。通过这种构建，我们可以应用MNëv-Sturmfels普遍性，以表明确定组合代码是否为凸是否是NP-HARD。
A combinatorial neural code $\mathscr C\subseteq 2^{[n]}$ is convex if it arises as the intersection pattern of convex open subsets of $\mathbb R^d$. We relate the emerging theory of convex neural codes to the established theory of oriented matroids, both categorically and with respect to geometry and computational complexity. On the categorical side, we show that the map taking an acyclic oriented matroid to the code of positive parts of its topes is a faithful functor. We adapt the oriented matroid ideal introduced by Novik, Postnikov, and Sturmfels into a functor from the category of oriented matroids to the category of rings; then, we show that the resulting ring maps naturally to the neural ring of the matroid's neural code. For geometry and computational complexity, we show that a code has a realization with convex polytopes if and only if it lies below the code of a representable oriented matroid in the partial order of codes introduced by Jeffs. We show that previously published examples of non-convex codes do not lie below any oriented matroids, and we construct examples of non-convex codes lying below non-representable oriented matroids. By way of this construction, we can apply Mnëv-Sturmfels universality to show that deciding whether a combinatorial code is convex is NP-hard.
