High-capacity associative memories based on Kernel Logistic Regression (KLR) are known for their exceptional performance but are hindered by high computational costs. This paper investigates the compressibility of KLR-trained Hopfield networks to understand the geometric principles of its robust encoding. We provide a comprehensive geometric theory based on spontaneous symmetry breaking and Walsh analysis, and validate it with compression experiments (quantization and pruning). Our experiments reveal a striking contrast: the network is extremely robust to low-precision quantization but highly sensitive to pruning. Our theory explains this via a ``sparse function, dense representation'' principle, where a sparse input mapping is implemented with a dense, bimodal parameterization. Our findings not only provide a practical path to hardware-efficient kernel memories but also offer new insights into the geometric principles of robust representation in neural systems.

翻译：基于核逻辑回归的高容量关联记忆虽以卓越性能著称，但其高昂计算成本制约了实际应用。本文通过研究核逻辑回归训练的Hopfield网络的可压缩性，探究其鲁棒编码的几何原理。我们基于自发对称性破缺与沃尔什分析提出完备几何理论，并通过压缩实验（量化与剪枝）加以验证。实验揭示显著对比：网络对低精度量化具有极高鲁棒性，但对剪枝却异常敏感。理论通过"稀疏函数-稠密表示"原理解释该现象——稀疏输入映射由稠密双峰参数化实现。本研究不仅为硬件高效核记忆提供实用路径，更揭示了神经系统中鲁棒表示的几何原理新见解。