完全二分图嵌入、格非嵌入性与信息距离的非欧几里得幂
Complete bipartite embeddings, lattice non-embeddability, and non-Euclidean powers of information distance
- Beijing Innovation Center of Humanoid Robotics (X-Humanoid)（北京人形机器人创新中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过构造完全二分图的尺度嵌入，证明信息距离的正幂不能由实希尔伯特空间精确表示，并排除整数格嵌入，解决了Hutter的多个开放问题。
AI中文摘要：
我们将每个有限完全二分图的顶点在每个尺度上编码为二进制字符串。这些字符串之间的信息距离以与尺度无关的常数加性误差逼近缩放后的图距离。我们还为每个顶点分配一个固定的无限二进制序列；适当的前缀满足相同的距离估计，误差为对数级。这扩展了Hutter对$K_{3,3}$的构造，并解决了他关于有限完全二分图尺度嵌入的开放问题。结合他的完全二分图障碍，这些嵌入表明信息距离的任何正幂都不能由实希尔伯特空间中的距离精确表示，从而解决了他关于此类表示的问题。对于条件前缀复杂度，我们进一步证明，在任意两个字符串之间，总距离超过其互距离至多$\delta$的字符串最多有$2^{\delta+C}$个，其中$C$与端点无关。这个界限排除了整数线和每个正维整数格的尺度嵌入，解决了Hutter关于字符串和序列嵌入的格嵌入问题。限制到整数格对每个$m\ge1$的$(\mathbb R^m,\\\\|\cdot\\\\|_1)$给出相同结论。
英文摘要：
We encode the vertices of every finite complete bipartite graph as binary strings at each scale. The information distances between these strings approximate the scaled graph distances with an additive error bounded by a constant independent of the scale. We also assign one fixed infinite binary sequence to each vertex; suitable prefixes satisfy the same distance estimate with logarithmic error. This extends Hutter's construction for $K_{3,3}$ and resolves his open problem on scale-embeddings of finite complete bipartite graphs. Combined with his complete bipartite obstruction, these embeddings show that no positive power of information distance can be represented exactly by distances in a real Hilbert space, resolving his question about such representations. For conditional prefix complexity, we further prove that at most $2^{δ+C}$ strings lie between any two strings with total distance exceeding their mutual distance by at most $δ$, where $C$ is independent of the endpoints. This bound rules out scale-embeddings of the integer line and every positive-dimensional integer lattice, resolving Hutter's lattice embedding problem for both string and sequence embeddings. Restriction to the integer lattice gives the same conclusion for $(\mathbb R^m,\|\cdot\|_1)$ for every $m\ge1$.
