This paper introduces assignment flows for density matrices as state spaces for representing and analyzing data associated with vertices of an underlying weighted graph. Determining an assignment flow by geometric integration of the defining dynamical system causes an interaction of the non-commuting states across the graph, and the assignment of a pure (rank-one) state to each vertex after convergence. Adopting the Riemannian Bogoliubov-Kubo-Mori metric from information geometry leads to closed-form local expressions which can be computed efficiently and implemented in a fine-grained parallel manner. Restriction to the submanifold of commuting density matrices recovers the assignment flows for categorial probability distributions, which merely assign labels from a finite set to each data point. As shown for these flows in our prior work, the novel class of quantum state assignment flows can also be characterized as Riemannian gradient flows with respect to a non-local non-convex potential, after proper reparametrization and under mild conditions on the underlying weight function. This weight function generates the parameters of the layers of a neural network, corresponding to and generated by each step of the geometric integration scheme. Numerical results indicates and illustrate the potential of the novel approach for data representation and analysis, including the representation of correlations of data across the graph by entanglement and tensorization.

翻译：本文引入密度矩阵上的分配流作为状态空间，用于表示和分析与底层加权图顶点相关联的数据。通过几何积分确定定义动力系统会导致非交换态在图上的相互作用，并在收敛后为每个顶点分配一个纯态（秩为一的态）。采用信息几何中的黎曼Bogoliubov-Kubo-Mori度量，可得到能高效计算并以细粒度并行方式实现的闭式局部表达式。限制到交换密度矩阵的子流形后，该方法回归到分类概率分布的分配流，后者仅为每个数据点分配有限集合中的标签。如我们先前工作中对这些流所展示的，经过适当重新参数化并在底层权重函数满足温和条件时，新型量子态分配流也可表征为关于非局部非凸势的黎曼梯度流。该权重函数生成神经网络各层的参数，这些参数对应于几何积分方案的每一步并由之产生。数值结果展示了该新方法在数据表示和分析方面的潜力，包括通过纠缠和张量化表示图上数据相关性。