Broadcast Product: Shape-aligned Element-wise Multiplication and Beyond
This work provides a new mathematical tool for tensor manipulation, potentially benefiting researchers and practitioners in machine learning and numerical computing, though it appears incremental as it builds on existing element-wise operations.
The paper introduces the broadcast product operator for tensors, which performs element-wise multiplication after shape alignment, and demonstrates its use in simplifying complex tensor operations and proposing a new tensor decomposition for dimensionality reduction.
We propose a new operator defined between two tensors, the broadcast product. The broadcast product calculates the Hadamard product after duplicating elements to align the shapes of the two tensors. Complex tensor operations in libraries like \texttt{numpy} can be succinctly represented as mathematical expressions using the broadcast product. Finally, we propose a novel tensor decomposition using the broadcast product, highlighting its potential applications in dimensionality reduction.