The paper proposes an artificial neural network (ANN) being a global
approximator for a special class of functions, which are known as generalized
homogeneous. The homogeneity means a symmetry of a function with respect to a
group of transformations having topological characterization of a dilation. In
this paper, a class of the so-called linear dilations is considered. A
homogeneous universal approximation theorem is proven. Procedures for an
upgrade of an existing ANN to a homogeneous one are developed. Theoretical
results are supported by examples from the various domains (computer science,
systems theory and automatic control).

本文提出了一种人工神经网络（ANN），作为一类特殊函数（称为广义齐次函数）的全局逼近器。证明了一种齐次通用逼近定理，并提出了将现有 ANN 升级为齐次 ANN 的方法。理论结果通过计算机科学、系统理论和自动控制领域的示例进行支持。

同构人工神经网络

Homogeneous Artificial Neural Network

Information retrieval involves selecting artifacts from a corpus that are
most relevant to a given search query. The flavor of retrieval typically used
in classical applications can be termed as homogeneous and relaxed, where
queries and corpus elements are both natural language (NL) utterances
(homogeneous) and the goal is to pick most relevant elements from the corpus in
the Top-K, where K is large, such as 10, 25, 50 or even 100 (relaxed).
Recently, retrieval is being used extensively in preparing prompts for large
language models (LLMs) to enable LLMs to perform targeted tasks. These new
applications of retrieval are often heterogeneous and strict -- the queries and
the corpus contain different kinds of entities, such as NL and code, and there
is a need for improving retrieval at Top-K for small values of K, such as K=1
or 3 or 5. Current dense retrieval techniques based on pretrained embeddings
provide a general-purpose and powerful approach for retrieval, but they are
oblivious to task-specific notions of similarity of heterogeneous artifacts. We
introduce Adapted Dense Retrieval, a mechanism to transform embeddings to
enable improved task-specific, heterogeneous and strict retrieval. Adapted
Dense Retrieval works by learning a low-rank residual adaptation of the
pretrained black-box embedding. We empirically validate our approach by showing
improvements over the state-of-the-art general-purpose embeddings-based
baseline.

信息检索中的全面检索方法被应用于大型语言模型的前提学习中，这些检索方法早期用于经典应用，但近期多用于异构且严格的应用，需要改进小 K 值的检索。我们提出了一种改进的密集检索方法，通过学习预训练嵌入的低秩残差适应来实现任务特定、异构且严格的检索，并在实验证明我们的方法相比于基于通用嵌入的基线有所改进。

增强嵌入用于定制检索

Augmented Embeddings for Custom Retrievals

Strongly Rayleigh distributions are natural generalizations of product and
determinantal probability distributions and satisfy strongest form of negative
dependence properties. We show that the "natural" Monte Carlo Markov Chain
(MCMC) is rapidly mixing in the support of a {\em homogeneous} strongly
Rayleigh distribution. As a byproduct, our proof implies Markov chains can be
used to efficiently generate approximate samples of a $k$-determinantal point
process. This answers an open question raised by Deshpande and Rademacher.

强瑞利分布是概率分布的自然推广，在其均匀形式的支持下，“自然的” Monte Carlo Markov Chain 快速混合；此外，该证明还暗示 Markov Chains 可以有效地生成 $k$ 确定性点过程的近似样本，这回答了 Deshpande 和 Rademacher 提出的一个开放性问题。