The assignment flow recently introduced in the J. Math. Imaging and Vision 58/2 (2017), constitutes a high-dimensional dynamical system that evolves on an elementary statistical manifold and performs contextual labeling (classification) of data given in any metric space. Vertices of a given graph index the data points and define a system of neighborhoods. These neighborhoods together with nonnegative weight parameters define regularization of the evolution of label assignments to data points, through geometric averaging induced by the affine e-connection of information geometry. Regarding evolutionary game dynamics, the assignment flow may be characterized as a large system of replicator equations that are coupled by geometric averaging. This paper establishes conditions on the weight parameters that guarantee convergence of the continuous-time assignment flow to integral assignments (labelings), up to a negligible subset of situations that will not be encountered when working with real data in practice. Furthermore, we classify attractors of the flow and quantify corresponding basins of attraction. This provides convergence guarantees for the assignment flow which are extended to the discrete-time assignment flow that results from applying a Runge-Kutta-Munthe-Kaas scheme for numerical geometric integration of the assignment flow. Several counter-examples illustrate that violating the conditions may entail unfavorable behavior of the assignment flow regarding contextual data classification.
翻译:J. Math. Imaging 和 Vision 58/2 (2017)最近引入的派任流是一个高维动态系统,在基本统计多元上演进,在任何计量空间提供的数据上进行背景标签(分类)。一个特定图表索引的副点,数据点和界定邻里系统。这些邻里加上非负权重参数,通过信息地理测量的平离电子连接引出的几何平均数,定义了将标签派任发展到数据点的正规化。关于进化游戏动态,派任流可被描述为一个大型的复制方方方程式系统,以几何平均相配合。本文为权重参数设定了条件,保证连续时间派任流与整体任务(标签)的趋同(分类),直至在实际使用实际数据时不会遇到的可忽略的一组情况。此外,我们还对流动的吸引者进行分类,并量化相应的吸引力盆地。这为派任次流动提供了趋同的保证,这种调流将延伸到离异时间分配流,其结果通过应用Ring-Kutta-Munthe-Muntales- assalviol assal assilling assing exactalviolviolgilling exalgilling magilling exalgalgalgalgalgalgalgalgalgalgalgal exalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalgalginginginginginging,从而导致对数分算算算算算算算算算算算算算算算算算。