Broadcast Product: Redefining Shape-aligned Element-wise Multiplication and Beyond

A researcher has introduced a new mathematical operation called the broadcast product to formalize the widely used but often imprecise practice of element-wise tensor multiplication in machine learning, according to a paper posted on the arXiv preprint server [1]. The paper, authored by Yusuke Matsui and initially submitted in September 2024 before a revision in June 2026, defines the broadcast product with the symbol ⊡ [1][2]. It is designed to explicitly extend the Hadamard product through shape-aligned element duplication, addressing a common source of invalid equations in scientific computing literature where element-wise products are written despite mismatched tensor shapes [2]. The work provides a rigorous definition, analyzes the operation's algebraic properties, and demonstrates how it can be expressed using standard linear algebra [2]. Building on this framework, the paper formulates least-squares problems and sketches a proof-of-concept broadcast decomposition, showing that the formalism enables a new family of decompositions with structural properties distinct from conventional tensor decompositions [2]. The first version of the paper was 518 KB, and the revised second version is 599 KB [1]. The paper appears on arXiv, an open-access repository of electronic preprints that is not peer-reviewed but is moderated before posting [6]. Founded in 1991, arXiv passed the two-million-article milestone by the end of 2021 and, as of November 2024, receives about 24,000 submissions per month [6]. The article's abstract page features several community-developed tools under the arXivLabs framework, including the Bibliographic Explorer for navigating citation trees and the CORE Recommender for discovering related open-access papers [4][5]. arXivLabs, launched as a formalized collaboration framework in 2020, allows third-party developers to create experimental features that appear as tabs on article pages, with partners required to adhere to arXiv's values of openness, community, excellence, and user data privacy [4].