提出广播乘积新符号，解决张量运算中形状不匹配的数学难题
Broadcast Product: Redefining Shape-aligned Element-wise Multiplication and Beyond
- 引入符号$oxdot$显式定义广播乘积，统一张量对齐规则
- 证明其可由标准线性代数表达，支持最小二乘建模
- 为张量分解提供新结构，适合需要精确数学描述的场景
广播操作广泛用于科学计算库，但其数学表述常隐含且在机器学习文献中不一致，导致元素乘积因张量形状不匹配而产生无效方程。本文通过引入广播乘积符号$oxdot$，显式扩展了哈达玛积，通过形状对齐的元素复制实现。我们给出了广播乘积的严格定义，分析其代数性质，并展示其可表示为标准线性代数形式。基于该框架，我们构建了最小二乘问题并给出广播分解的初步证明。作为初步示例，该形式化方法可导出与传统张量分解不同的新型分解家族。本工作为广播感知的张量操作建立了数学基础，连接实际实现与严谨张量分析。
原文摘要 · Abstract (English)
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.
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