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

TMLR Paper7612 Authors

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 propose a broadcast decomposition. Preliminary experiments on synthetic data suggest that the proposed decomposition captures structures that are challenging for conventional tensor decompositions. This work establishes a mathematical foundation for broadcast-aware tensor operations, connecting practical implementations with rigorous tensor analysis.

Submission Type: Regular submission (no more than 12 pages of main content)

Assigned Action Editor: ~Stephen_Becker1

Submission Number: 7612