Training 3D Spatially Embedded Neural Networks for Regression via Gradient Descent

This study examines a feed-forward neural network embedded in Euclidean R3 space. Nuclei coordinates condition network parameters. Backpropagating from loss-parameter to loss-coordinate derivatives enables spatially-embedded gradient descent. A dense multi-layer perceptron (MLP) learns price-prediction on the California Housing dataset. The model demonstrates performance comparable to conventional, non-spatial MLP predictions. Robustness examinations via re-initialisation sensitivity tests, and spatial analysis via node ablation and activation imaging, reveal complexity and interpretability characteristics unique to the spatially embedded model.

本研究针对嵌入于欧几里得R³空间（Euclidean R³ space）的前馈神经网络（feed-forward neural network）展开分析。该网络以核坐标调控网络参数，通过从损失-参数导数到损失-坐标导数的反向传播，可实现空间嵌入型梯度下降。一个稠密多层感知器（Multi-Layer Perceptron, MLP）基于加州住房数据集（California Housing dataset）开展房价预测任务，其预测性能可与传统非空间型MLP的预测效果相媲美。通过重初始化敏感性测试开展鲁棒性检验，结合节点消融与激活成像进行空间分析，结果揭示了该空间嵌入型模型所独有的复杂性与可解释性特征。