In this paper we investigate the convexity of the relevance-compression function for the Information Bottleneck and the Information Distortion problems. This curve is an analog of the rate-distortion curve, which is convex. In the problems we discuss in this paper, the distortion function is not a linear function of the quantizer, and the relevance-compression function is not necessarily convex (concave), but can change its convexity. We relate this phenomena with existence of first order phase transitions in the corresponding Lagrangian as a function of the annealing parameter.
翻译:在本文中,我们调查了信息瓶颈和信息扭曲问题的关联压缩功能的细度。 这个曲线是率扭曲曲线的类比, 也就是曲线。 在本文中我们讨论的问题中, 扭曲函数不是量化器的线性函数, 相关性压缩功能不一定是 convex( concave), 而是可以改变它的共性 。 我们把这一现象与相应的拉格朗吉亚 存在的第一阶转换相提并论, 与对应的拉格朗吉亚 存在的第一个阶级过渡相提并论, 成为了 Annealing 参数的函数 。