Optimal Density Functions for Weighted Convolution in Learning Models

2025-05-30

AI Summary

Traditional uniform convolutions neglect spatial distances among neighboring pixels, limiting modeling accuracy. To address this, we propose weighted convolution—a geometrically aware adaptive local aggregation mechanism that dynamically modulates the contribution of each neighbor to the central pixel via a learnable optimal density function. Our key innovation is the first integration of density function optimization directly into the convolution operation, decoupling kernel parameter learning from spatial weight optimization: convolutional kernels are updated via stochastic gradient descent, while the density function is optimized globally using the DIRECT-L algorithm. The method applies to regular-grid signals (2D/3D images) and demonstrates significant improvements in image denoising—achieving substantial PSNR gains, an average 53% reduction in loss, and markedly improved test accuracy, with only an 11% increase in computational overhead. It thus offers both high efficiency and strong robustness.

Technology Category

• Search and Optimization: Non-convex Optimization
• Computer Vision: Learning & Optimization for CV
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking
• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation

Abstract

The paper introduces the weighted convolution, a novel approach to the convolution for signals defined on regular grids (e.g., 2D images) through the application of an optimal density function to scale the contribution of neighbouring pixels based on their distance from the central pixel. This choice differs from the traditional uniform convolution, which treats all neighbouring pixels equally. Our weighted convolution can be applied to convolutional neural network problems to improve the approximation accuracy. Given a convolutional network, we define a framework to compute the optimal density function through a minimisation model. The framework separates the optimisation of the convolutional kernel weights (using stochastic gradient descent) from the optimisation of the density function (using DIRECT-L). Experimental results on a learning model for an image-to-image task (e.g., image denoising) show that the weighted convolution significantly reduces the loss (up to 53% improvement) and increases the test accuracy compared to standard convolution. While this method increases execution time by 11%, it is robust across several hyperparameters of the learning model. Future work will apply the weighted convolution to real-case 2D and 3D image convolutional learning problems.

Problem

• Develops weighted convolution for non-uniform pixel contribution in grids
• Optimizes density function to enhance CNN approximation accuracy
• Improves image-to-image task performance despite increased runtime

Innovation

• Weighted convolution with optimal density function
• Separate optimization of kernel and density
• Improved accuracy in image-to-image tasks