率失真问题中拓扑转变附近的临界慢化
Critical Slowing Down Near Topological Transitions in Rate-Distortion Problems
AI总结：
该研究针对率失真问题中最优表示的拓扑转变临界点，分析Arimoto-Blahut交替投影算法的收敛特性，发现其存在临界慢化现象，为机器学习与数据压缩领域提供理论与实践参考。
摘要：
在率失真（RD）问题中，研究者寻求满足目标失真约束的信源简化表示。这类最优表示在某些临界码率值下会发生拓扑转变，其基数或维度会发生变化。我们研究了用于求解这类问题的Arimoto-Blahut交替投影算法在率失真和信息瓶颈两种场景下，在这些临界点附近的收敛时间。我们发现该算法在临界点附近会出现临界慢化现象——即收敛所需的迭代次数会发散。这一现象对机器学习和数据压缩问题均具有理论与实践意义。
英文摘要：
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.
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