Graph embedding based retrieval has become one of the most popular techniques
in the information retrieval community and search engine industry. The
classical paradigm mainly relies on the flat Euclidean geometry. In recent
years, hyperbolic (negative curvature) and spherical (positive curvature)
representation methods have shown their superiority to capture hierarchical and
cyclic data structures respectively. However, in industrial scenarios such as
e-commerce sponsored search platforms, the large-scale heterogeneous
query-item-advertisement interaction graphs often have multiple structures
coexisting. Existing methods either only consider a single geometry space, or
combine several spaces manually, which are incapable and inflexible to model
the complexity and heterogeneity in the real scenario. To tackle this
challenge, we present a web-scale Adaptive Mixed-Curvature ADvertisement
retrieval system (AMCAD) to automatically capture the complex and heterogeneous
graph structures in non-Euclidean spaces. Specifically, entities are
represented in adaptive mixed-curvature spaces, where the types and curvatures
of the subspaces are trained to be optimal combinations. Besides, an attentive
edge-wise space projector is designed to model the similarities between
heterogeneous nodes according to local graph structures and the relation types.
Moreover, to deploy AMCAD in Taobao, one of the largest ecommerce platforms
with hundreds of million users, we design an efficient two-layer online
retrieval framework for the task of graph based advertisement retrieval.
Extensive evaluations on real-world datasets and A/B tests on online traffic
are conducted to illustrate the effectiveness of the proposed system.

提出了一个自适应混合曲率广告检索系统（AMCAD），可以自动捕获非欧几里德（Euclidean）空间中的复杂和异构图结构，采用自适应混合曲率空间表示实体，并设计了注意力机制来模拟异构节点之间的相似度，设计了一个高效的二层在线检索框架来部署 AMCAD。

AMCAD：基于自适应混合曲率表示的广告检索系统

AMCAD: Adaptive Mixed-Curvature Representation based Advertisement Retrieval System

The classical random walk isomorphism theorems relate the local times of a
continuous-time random walk to the square of a Gaussian free field. A Gaussian
free field is a spin system that takes values in Euclidean space, and this
article generalises the classical isomorphism theorems to spin systems taking
values in hyperbolic and spherical geometries. The corresponding random walks
are no longer Markovian: they are the vertex-reinforced and vertex-diminished
jump processes. We also investigate supersymmetric versions of these formulas.
Our proofs are based on exploiting the continuous symmetries of the
corresponding spin systems. The classical isomorphism theorems use the
translation symmetry of Euclidean space, while in hyperbolic and spherical
geometries the relevant symmetries are Lorentz boosts and rotations,
respectively. These very short proofs are new even in the Euclidean case.
Isomorphism theorems are useful tools, and to illustrate this we present
several applications. These include simple proofs of exponential decay for spin
system correlations, exact formulas for the resolvents of the joint processes
of random walks together with their local times, and a new derivation of the
Sabot--Tarr\`es formula for the limiting local time of the vertex-reinforced
jump process.

本文研究连续时间随机行走、高斯自由场和 $n$ 维自旋系统之间的同构定理，并将其推广到超半球和球形几何中。

随机游走同构定理的几何学

The geometry of random walk isomorphism theorems

A new kind of overset grid, named Yin-Yang grid, for spherical geometry is
proposed. The Yin-Yang grid is composed of two identical component grids that
are combined in a complemental way to cover a spherical surface with partial
overlap on their boundaries. Each component grid is a low latitude part of the
latitude-longitude grid. Therefore the grid spacing is quasi-uniform and the
metric tensors are simple and analytically known. One can directly apply
mathematical and numerical resources that have been written in the spherical
polar coordinates or latitude-longitude grid. The complemental combination of
the two identical component grids enables us to make efficient and concise
programs. Simulation codes for geodynamo and mantle convection simulations
using finite difference scheme based on the Yin-Yang grid are developed and
tested. The Yin-Yang grid is suitable for massively parallel computers.

介绍了一种名为阴阳网格的球面几何超定网格，两个同样的部分网格以互补的方式组合在一起，具有准均匀的间距和简单而分析已知的度量张量，在大规模并行计算机上能有效地模拟地幔对流等现象。