The rate-distortion curve captures the fundamental tradeoff between compression length and resolution in lossy data compression. However, it conceals the underlying dynamics of optimal source encodings or test channels. We argue that these typically follow a piecewise smooth trajectory as the source information is compressed. These smooth dynamics are interrupted at bifurcations, where solutions change qualitatively. Sub-optimal test channels may collide or exchange optimality there, for example. There is typically a plethora of sub-optimal solutions, which stems from restrictions of the reproduction alphabet. We devise a family of algorithms that exploits the underlying dynamics to track a given test channel along the rate-distortion curve. To that end, we express implicit derivatives at the roots of a non-linear operator by higher derivative tensors. Providing closed-form formulae for the derivative tensors of Blahut's algorithm thus yields implicit derivatives of arbitrary order at a given test channel, thereby approximating others in its vicinity. Finally, our understanding of bifurcations guarantees the optimality of the root being traced, under mild assumptions, while allowing us to detect when our assumptions fail. Beyond the interest in rate distortion, this is an example of how understanding a problem's bifurcations can be translated to a numerical algorithm.
翻译:率扭曲曲线在丢失数据压缩中捕捉压缩长度和分辨率之间的基本权衡。 但是, 它隐藏了最佳源编码或测试频道的基本动态。 我们争辩说, 源信息压缩后, 这些光滑的动态通常会沿着一条小线顺畅的轨迹。 这些光滑的动态会中断于两侧, 在那里, 解决方案会发生质的变化 。 亚优化的测试渠道可能会在其中相撞或交换最佳性。 通常存在大量亚最佳的解决方案, 源于复制字母的限制 。 我们设计了一套算法, 利用基础动态来跟踪在率扭曲曲线中跟踪某个测试频道。 为此, 我们用更高衍生器变速器在非线性运营商的根部显示隐含衍生物。 为Blahut 算法的衍生物变压器提供封闭式公式, 从而在给定的测试频道产生任意秩序的隐含衍生物, 从而让其他对象相近。 最后, 我们对两面变法的理解保证了根部的优化性, 以温和的假设为基础, 并让我们在更深处检测出一个误判问题。