Optimality and Constructions of Spanning Bipartite Block Designs

TL;DR

This paper introduces Spanning Bipartite Block Designs (SBBD) for efficient estimation of effects in bipartite graph-based experiments, proving their optimality and providing construction methods, with potential applications in deep learning.

Contribution

The paper proposes SBBDs for bipartite graph experiments, proves their variance balance and A-optimality under certain conditions, and offers new construction methods including BIBDs.

Findings

SBBDs achieve variance balance in estimators.

SBBDs with semi-regular or regular blocks are A-optimal.

Construction methods include using ($r, heta$)-designs and BIBDs.

Abstract

We consider a statistical problem to estimate variables (effects) that are associated with the edges of a complete bipartite graph $K_{v_1, v_2}=(V_1, V_2 \, ; E)$. Each data is obtained as a sum of selected effects, a subset of $E$. In order to estimate efficiently, we propose a design called Spanning Bipartite Block Design (SBBD). For SBBDs such that the effects are estimable, we proved that the estimators have the same variance (variance balanced). If each block (a subgraph of $K_{v_1, v_2}$) of SBBD is a semi-regular or a regular bipartite graph, we show that the design is A-optimum. We also show a construction of SBBD using an ($r,\lambda$)-design and an ordered design. A BIBD with prime power blocks gives an A-optimum semi-regular or regular SBBD. At last, we mention that this SBBD is able to use for deep learning.