利用吸引子神经网络优化学习多个低维流形的容量-分辨率权衡
Capacity-resolution trade-off in the optimal learning of multiple low-dimensional manifolds by attractor neural networks
摘要：
递归神经网络（RNN）是解释吸引子如何从嘈杂的高维动力学中出现的强大工具。我们在这里研究如何学习具有N个神经元的RNN中的~N^（2）对相互作用，以嵌入D << N维的L流形。我们表明，容量，即最大比率L/N，随着|对数|^（-D），其中λ是由沿着每个流形的神经活动编码的位置上的误差。因此，RNN是灵活的存储设备，能够以高空间分辨率存储大量流形。我们的研究结果依赖于从统计力学和随机矩阵理论的分析工具的组合，扩展加德纳的经典学习理论的情况下，具有较强的空间相关性的模式。
英文摘要：
Recurrent neural networks (RNN) are powerful tools to explain how attractors may emerge from noisy, high-dimensional dynamics. We study here how to learn the ~N^(2) pairwise interactions in a RNN with N neurons to embed L manifolds of dimension D << N. We show that the capacity, i.e. the maximal ratio L/N, decreases as |log(epsilon)|^(-D), where epsilon is the error on the position encoded by the neural activity along each manifold. Hence, RNN are flexible memory devices capable of storing a large number of manifolds at high spatial resolution. Our results rely on a combination of analytical tools from statistical mechanics and random matrix theory, extending Gardner's classical theory of learning to the case of patterns with strong spatial correlations.
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