神经编码的开、闭与非退化嵌入维数
Open, Closed, and Non-Degenerate Embedding Dimensions of Neural Codes
AI总结：
本研究聚焦神经编码的开、闭与非退化嵌入维数，结合凸开集向日葵与刚性结构工具，构造出满足特定条件的编码，首次给出闭嵌入维数大于开嵌入维数且有限的编码实例。
摘要：
我们研究神经编码的开、闭与非退化嵌入维数，它们分别是能找到由开凸集、闭凸集或非退化凸集构成的编码实现的最小维度，其中非退化的定义由克鲁兹、朱斯蒂、伊茨科夫和克龙霍尔姆提出。对于给定的编码$\mathcal C$，我们将嵌入维数向量定义为由这些嵌入维数组成的三元组$(a,b,c)$。已有结果保证$\max\{a,b\} \le c$，而我们证明当这些维数中任意一个至少为2时，这是此类向量的唯一限制。具体而言，对于满足$2\le \min \{a,b\}$和$\max\{a,b\}\le c$的每一个三元组$(a,b,c)$，我们构造出一个编码$\mathcal C_{(a,b,c)}$，其嵌入维数向量恰好为$(a,b,c)$。我们的构造结合了凸神经编码领域的两种现有工具：凸开集向日葵，以及刚性结构——后者由陈、约翰斯顿、伦特、鲁伊斯·德·佩雷斯和萧在近期工作中定义。我们的构造首次给出了闭嵌入维数大于开嵌入维数但仍为有限值的编码实例。
英文摘要：
We study the open, closed, and non-degenerate embedding dimensions of neural codes, which are the smallest respective dimensions in which one can find a realization of a code consisting of convex sets that are open, closed, or non-degenerate in a sense defined by Cruz, Giusti, Itskov, and Kronholm. For a given code $\mathcal C$ we define the embedding dimension vector to be the triple $(a,b,c)$ consisting of these embedding dimensions. Existing results guarantee that $\max\{a,b\} \le c$, and we show that when any of these dimensions is at least 2 this is the only restriction on such vectors. Specifically, for every triple $(a,b,c)$ with $2\le \min \{a,b\}$ and $\max\{a,b\}\le c$ we construct a code $\mathcal C_{(a,b,c)}$ whose embedding dimension vector is exactly $(a,b,c)$. Our constructions combine two existing tools in the convex neural codes literature: sunflowers of convex open sets, and rigid structures, the latter of which was recently defined in work of Chan, Johnston, Lent, Ruys de Perez, and Shiu. Our constructions provide the first examples of codes whose closed embedding dimension is larger than their open embedding dimension, but still finite.
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