In Rate Distortion (RD) problems one seeks reduced representations of a source that meet a target distortion constraint. Such optimal representations undergo topological transitions at some critical rate values, when their cardinality or dimensionality change. We study the convergence time of the Arimoto-Blahut alternating projection algorithms, used to solve such problems, near those critical points, both for the Rate Distortion and Information Bottleneck settings. We argue that they suffer from Critical Slowing Down -- a diverging number of iterations for convergence -- near the critical points. This phenomenon can have theoretical and practical implications for both Machine Learning and Data Compression problems.

中文翻译：

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速率失真问题中的临界减速近拓扑转变I

在速率失真（RD）问题中，人们寻求满足目标失真约束条件的源的简化表示形式。当这些最佳表示的基数或维数发生变化时，它们会以某些临界速率值进行拓扑转换。我们研究了Arimoto-Blahut交替投影算法的收敛时间，该算法用于解决此类问题，在速率失真和信息瓶颈设置的临界点附近。我们认为，他们在临界点附近遭受临界减速（收敛迭代的数量不同）的困扰。这种现象对于机器学习和数据压缩问题都可能具有理论和实践意义。