Batch Matrix-form Equations and Implementation of Multilayer Perceptrons

2025-11-14

AI Summary

Existing literature often lacks a complete, mathematically rigorous batched matrix formulation for multilayer perceptrons (MLPs), relying instead on sample-wise expressions or automatic differentiation—limiting transparent analysis and system-level optimization, especially in sparse-network scenarios. This work introduces the first symbolically verified, fully batched matrix formalism for MLPs, explicitly deriving forward and backward propagation formulas—including BatchNorm and Softmax layers—with mathematical precision. We implement a unified, cross-framework reference implementation (NumPy, PyTorch, JAX, TensorFlow, and C++) built from a minimal set of matrix primitives. Gradient correctness is formally validated using SymPy, and empirical evaluation demonstrates efficacy in pedagogy, sparse computation optimization, and interpretability research. Our framework establishes a principled, analyzable, and extensible algorithmic foundation for neural network design and analysis. (149 words)

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Natural Language Processing: Learning & Optimization for NLP
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications

Abstract

Multilayer perceptrons (MLPs) remain fundamental to modern deep learning, yet their algorithmic details are rarely presented in complete, explicit emph{batch matrix-form}. Rather, most references express gradients per sample or rely on automatic differentiation. Although automatic differentiation can achieve equally high computational efficiency, the usage of batch matrix-form makes the computational structure explicit, which is essential for transparent, systematic analysis, and optimization in settings such as sparse neural networks. This paper fills that gap by providing a mathematically rigorous and implementation-ready specification of MLPs in batch matrix-form. We derive forward and backward equations for all standard and advanced layers, including batch normalization and softmax, and validate all equations using the symbolic mathematics library SymPy. From these specifications, we construct uniform reference implementations in NumPy, PyTorch, JAX, TensorFlow, and a high-performance C++ backend optimized for sparse operations. Our main contributions are: (1) a complete derivation of batch matrix-form backpropagation for MLPs, (2) symbolic validation of all gradient equations, (3) uniform Python and C++ reference implementations grounded in a small set of matrix primitives, and (4) demonstration of how explicit formulations enable efficient sparse computation. Together, these results establish a validated, extensible foundation for understanding, teaching, and researching neural network algorithms.

Problem

• Providing batch matrix-form equations for multilayer perceptrons
• Deriving forward and backward equations for all standard layers
• Enabling transparent analysis and optimization for sparse networks

Innovation

• Batch matrix-form equations for MLPs
• Symbolic validation using SymPy library
• Uniform implementations across multiple frameworks