Semi-Tensor-Product Based Convolutional Neural Networks
Daizhan Cheng, Xiao Zhang
TL;DR
The paper tackles boundary artifacts and data irregularities in CNNs by introducing a padding-free convolution built on the semi-tensor product (STP) that operates across cross-dimensional domains $\mathbb{R}^\infty$. A unified linear-algebraic formulation uses a receptive-field matrix (RFM) and a block Hadamard product to realize STP-based convolution (STP-CP) without padding, enabling domain-aware processing of irregular and multi-scale inputs. The authors demonstrate STP-CP on irregular, partially damaged, and 3D volumetric data, showing robustness to missing data and scale mismatches and extending to higher-order signals. They provide a theoretical forward-propagation foundation for STP-CNNs and outline directions for training analysis, large-scale validation, and library integration.
Abstract
The semi-tensor product of vectors generalizes the conventional inner product, enabling algebraic operations between vectors of different dimensions. Building upon this foundation, we introduce a domain-based convolutional product and integrate it with the STP to formulate a padding-free convolutional operation. This new operation inherently avoids zero or other artificial padding, thereby eliminating redundant information and boundary artifacts commonly present in conventional convolutional neural networks. Based on this operation, we further develop an STP-based CNN framework that extends convolutional computation to irregular and cross-dimensional data domains. Applications to image processing and third-order signal identification demonstrate the proposed method's effectiveness in handling irregular, incomplete, and high-dimensional data without the distortions caused by padding.