用球面赫林格坎托罗维奇流实现带权重点云的高效记忆检索
Sinkhorn Based Associative Memory Retrieval Using Spherical Hellinger Kantorovich Dynamics
- 基于去偏Sinkhorn散度构建能量函数，通过梯度流更新点位置与权重
- 在局部分离条件下收敛到局部最小值，且最小值贴近原始存储模式
- 对高维随机模式模型，记忆容量呈指数级增长，适合大规模数据
我们提出一种用于经验测度（加权点云）的密集关联记忆。存储模式和查询均为有限支撑的概率测度，检索定义为最小化基于去偏Sinkhorn散度的霍普菲尔德型对数求和指数能量。我们推导出检索动力学为球面赫林格坎托罗维奇（SHK）梯度流，可同时更新支撑点位置与权重。离散化该流得到确定性算法，利用Sinkhorn势能计算重心传输步骤并进行乘法单纯形重加权。在局部分离和PL型条件下，证明了吸引域不变性、几何收敛至局部极小值，以及极小值与对应存储模式保持接近的界。在随机模式模型下，进一步表明这些Sinkhorn吸引域以高概率互不相交，意味着在环境维度上具有指数级容量。合成高斯点云记忆实验显示，相比欧氏霍普菲尔德基线，本方法能从扰动查询中鲁棒恢复。
原文摘要 · Abstract (English)
We propose a dense associative memory for empirical measures (weighted point clouds). Stored patterns and queries are finitely supported probability measures, and retrieval is defined by minimizing a Hopfield-style log-sum-exp energy built from the debiased Sinkhorn divergence. We derive retrieval dynamics as a spherical Hellinger Kantorovich (SHK) gradient flow, which updates both support locations and weights. Discretizing the flow yields a deterministic algorithm that uses Sinkhorn potentials to compute barycentric transport steps and a multiplicative simplex reweighting. Under local separation and PL-type conditions we prove basin invariance, geometric convergence to a local minimizer, and a bound showing the minimizer remains close to the corresponding stored pattern. Under a random pattern model, we further show that these Sinkhorn basins are disjoint with high probability, implying exponential capacity in the ambient dimension. Experiments on synthetic Gaussian point-cloud memories demonstrate robust recovery from perturbed queries versus a Euclidean Hopfield-type baseline.
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