Broadcast operations are widely used in scientific computing libraries, yet their mathematical formulation is often implicit and inconsistently represented in machine learning literature. This problem frequently leads to invalid equations when element-wise products are written despite mismatched tensor shapes. In this paper, we formalize such operations by introducing the broadcast product $\boxdot$, which explicitly extends the Hadamard product through shape-aligned element duplication. We provide a rigorous definition of the broadcast product, analyze its algebraic properties, and show how it can be expressed using standard linear algebra. Building on this framework, we formulate least-squares problems and sketch a proof-of-concept broadcast decomposition. As a preliminary illustration, we show that the formalism enables a new family of decompositions with distinct structural properties from conventional tensor decompositions. This work establishes a mathematical foundation for broadcast-aware tensor operations, connecting practical implementations with rigorous tensor analysis.

翻译：广播操作在科学计算库中被广泛使用，但其数学形式化在机器学习文献中常常隐式表达且表述不一致。当逐元素乘积在张量形状不匹配的情况下被书写时，这一问题常导致无效方程。本文通过引入广播积 $\boxdot$ 对这一操作进行形式化，该操作通过形状对齐的元素复制显式扩展了阿达玛积。我们给出了广播积的严格定义，分析了其代数性质，并展示了如何用标准线性代数表达该操作。基于这一框架，我们构建了最小二乘问题，并概述了一种概念验证的广播分解。作为初步例证，我们展示了该形式化方法能够催生一类具有与传统张量分解不同结构特性的新型分解。本研究为支持广播的张量运算建立了数学基础，连接了实际实现与严谨的张量分析。