Fermionic neural network (FermiNet) is a recently proposed wavefunction
Ansatz, which is used in variational Monte Carlo (VMC) methods to solve the
many-electron Schr\"{o}dinger equation. FermiNet proposes
permutation-equivariant architectures, on which a Slater determinant is applied
to induce antisymmetry. FermiNet is proved to have universal approximation
capability with a single determinant, namely, it suffices to represent any
antisymmetric function given sufficient parameters. However, the asymptotic
computational bottleneck comes from the Slater determinant, which scales with
$O(N^3)$ for $N$ electrons. In this paper, we substitute the Slater determinant
with a pairwise antisymmetry construction, which is easy to implement and can
reduce the computational cost to $O(N^2)$. We formally prove that the pairwise
construction built upon permutation-equivariant architectures can universally
represent any antisymmetric function. Besides, this universality can be
achieved via continuous approximators when we aim to represent ground-state
wavefunctions.

本文提出了一种基于 Fermionic 神经网络和排列等变结构的方法用于解决多电子 Schrödinger 方程，使用分块反对称构建代替 Slater 行列式以减小计算成本。该方法通过连续逼近器实现波函数的基态表示。

$O (N^2)$ 费米子神经网络中的全局反对称性

$O(N^2)$ Universal Antisymmetry in Fermionic Neural Networks

Representation learning of knowledge graphs encodes entities and relation
types into a continuous low-dimensional vector space, learns embeddings of
entities and relation types. Most existing methods only concentrate on
knowledge triples, ignoring logic rules which contain rich background
knowledge. Although there has been some work aiming at leveraging both
knowledge triples and logic rules, they ignore the transitivity and
antisymmetry of logic rules. In this paper, we propose a novel approach to
learn knowledge representations with entities and ordered relations in
knowledges and logic rules. The key idea is to integrate knowledge triples and
logic rules, and approximately order the relation types in logic rules to
utilize the transitivity and antisymmetry of logic rules. All entries of the
embeddings of relation types are constrained to be non-negative. We translate
the general constrained optimization problem into an unconstrained optimization
problem to solve the non-negative matrix factorization. Experimental results
show that our model significantly outperforms other baselines on knowledge
graph completion task. It indicates that our model is capable of capturing the
transitivity and antisymmetry information, which is significant when learning
embeddings of knowledge graphs.

本文提出了一种新颖的方法，将知识三元组和逻辑规则整合起来，通过有序关系在连续的低维向量空间中对实体和关系类型进行编码，同时利用逻辑规则的传递性和反对称性，来实现知识表示的学习。通过实验证明，该模型在知识图谱完成任务方面显著优于其他基线模型，表明该模型可以捕获知识图谱中的传递性和反对称性信息。