Hadamard product in deep learning: introduction, advances and challenges

Journal article

While convolution and self-attention mechanisms have dominated architectural design in deep learning, this survey examines a fundamental yet understudied primitive: the Hadamard product. Despite its widespread implementation across various applications, the Hadamard product has not been systematically analyzed as a core architectural primitive. We present the first comprehensive taxonomy of its applications in deep learning, identifying four principal domains: higher-order correlation, multimodal data fusion, dynamic representation modulation, and efficient pairwise operations. The Hadamard product's ability to model nonlinear interactions with linear computational complexity makes it particularly valuable for resource-constrained deployments and edge computing scenarios. We demonstrate its natural applicability in multimodal fusion tasks, such as visual question answering, and its effectiveness in representation masking for applications including image inpainting and pruning. This systematic review not only consolidates existing knowledge about the Hadamard product's role in deep learning architectures but also establishes a foundation for future architectural innovations. Our analysis reveals the Hadamard product as a versatile primitive that offers compelling trade-offs between computational efficiency and representational power, positioning it as a crucial component in the deep learning toolkit.

Authors: Chrysos, GG; Wu, Y; Pascanu, R; Torr, P; Cevher, V

• Publication status: Published
• Peer review status: Peer reviewed
• Publisher: IEEE
• Journal: IEEE Transactions on Pattern Analysis and Machine Intelligence
• Volume: 47
• Issue: 8
• Pages: 6531 - 6549
• Publication date: 2025-04-15
• Acceptance date: 2025-04-08
• DOI: 10.1109/TPAMI.2025.3560423
• EISSN: 1939-3539
• ISSN: 0162-8828
• Language: English
• Keywords: deep learning; tensors; correlation; adaptation models; solid modeling; computational efficiency; surveys; modulation; training; computational modeling