The Hadamard Product in Deep Learning: Foundations, Advances, and Challenges

Abstract

The Hadamard product, or element-wise multiplication, has emerged as a fundamental operation in modern deep learning architectures, enabling efficient and flexible modeling of feature interactions. Despite its apparent simplicity, the Hadamard product underpins critical mechanisms such as gating in recurrent neural networks, multiplicative feature fusion in multimodal learning, and lightweight modulation in efficient models. This survey provides a comprehensive overview of the role of the Hadamard product in deep learning, detailing its mathematical foundations, historical development, and key applications across diverse architectures. We highlight the advantages conferred by its computational efficiency and structural regularity, while also analyzing the major challenges it introduces, including expressiveness limitations, optimization instability, and difficulties in capturing global dependencies. By systematically categorizing existing works, we expose both the strengths and the limitations of Hadamard-based designs. Finally, we outline promising future research directions aimed at enhancing the expressive capacity, improving the stability, and broadening the applicability of Hadamard interactions in emerging paradigms such as foundation models and neuralsymbolic systems. Our goal is to provide both practitioners and researchers with a clear understanding of the current landscape and the open opportunities surrounding the Hadamard product in deep learning.

Keywords: Hadamard Product, Element-wise Multiplication, Deep Learning, Gating Mechanisms, Attention Mechanisms, Multimodal Fusion, Lightweight Neural Networks, Feature Interaction, Model Efficiency, Neural Network Optimization