发表机构
Department of Physics, University of Maryland（马里兰大学物理系）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究提出了基于SU(d)对称空间的广义霍普菲尔德网络，其临界容量比传统向量网络提升近一个数量级，通过李代数方法规避几何约束，还引入RGB图像编解码协议验证SU(3)的记忆检索，并将其量子化为Sachdev-Ye玻璃态模型。
AI 中文摘要
本文引入了广义霍普菲尔德网络，其中记忆和神经元是位于黎曼流形上的连续变量。我们明确关注与特殊幺正群SU(d)相关的对称空间，并使用数值和解析（复制）技术证明，从d=3开始，其临界容量比向量网络提高了近一个数量级，且随d进一步快速增长。为规避非线性几何约束，我们采用李代数方法（遵循V. Galitski, Phys. Rev. A 84, 012118 (2011)），在辅助希尔伯特空间中用线性代数精确描述经典神经网络。研究表明，与传统霍普菲尔德网络不同，SU(d)霍普菲尔德的记忆检索对应神经元沿带刺矩阵的顶部本征向量对齐，相比其他具有连续神经元变量的模型，其受随机矩阵串扰的影响更小。本文简要讨论了实现SU(d)霍普菲尔德的物理平台，并展示了物理（除算法外）检索机制，其中记忆通过广义朗道-利夫希茨-吉尔伯特动力学自然恢复。为说明SU(3)的记忆检索，我们引入了颜色（RGB）图像编解码协议，并对受损线索上的图像恢复进行了明确运行。最后，我们对广义霍普菲尔德进行了量子化，结果表明其可简化为Sachdev-Ye玻璃态类型模型。其多体谱通常具有暗带和记忆带两种类型，其中记忆带表现出混沌维格纳-戴森能级统计，隐藏了赫布数据。
英文摘要
Generalized Hopfield networks are introduced where memories and neurons are continuous variables that lie on a Riemannian manifold. We explicitly focus on symmetric spaces associated with the special unitary groups SU(d), and use both numerical and analytical (replica) techniques to demonstrate an almost order of magnitude enhancement in critical capacity over the vector networks starting with d=3 and further rapidly growing with d. To circumvent the non-linear geometric constraints, we use a Lie algebraic method [following V. Galitski, Phys. Rev. A 84, 012118 (2011)] to exactly describe the classical neural network in terms of linear algebra in an auxiliary Hilbert space. It is shown that in contrast to the traditional Hopfield networks, memory recall in SU(d) Hopfields corresponds to neuron alignment along a top eigenvector of a spiked matrix, which is less susceptible to random matrix crosstalk than other models with continuous neuron variables. Physical platforms to realize SU(d) Hopfields are briefly discussed and physical (in addition to algorithmic) recall mechanism is demonstrated, where memory recovery occurs naturally through generalized Landau-Lifshitz-Gilbert dynamics. To illustrate SU(3) memory recall, we introduce a color (RGB) image encoding/decoding protocol and explicitly run image recovery on corrupted cues. Finally, we quantize the generalized Hopfields which are shown to reduce to Sachdev-Ye glassy type of models. Their many-body spectra generally feature two types of dark and memory bands, where the latter exhibits chaotic Wigner-Dyson level statistics that hides Hebbian data.
Comments14 pages, 9 figures
