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This paper addresses the mathematical ambiguity surrounding broadcast operations in machine learning by formally defining a "broadcast product" to resolve inconsistencies between implicit library implementations and theoretical literature. By establishing rigorous algebraic properties for these shape-aligned element-wise multiplications, it lays groundwork for more robust tensor decompositions and least-squares formulations essential for scalable model training.
Broadcast Product: Redefining Shape-aligned Element-wise Multiplication and Beyond
arXiv:2409.17502v2 Announce Type: replace Abstract: 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.
Abstract: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.
TMLR2026. OpenReview: this https URL
Machine Learning (cs.LG)
arXiv:2409.17502 [cs.LG]
(or arXiv:2409.17502v2 [cs.LG] for this version)
https://doi.org/10.48550/arXiv.2409.17502 Focus to learn more arXiv-issued DOI via DataCite
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Full text under the publisher's open licence · 237 words