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The rate-distortion curve captures the fundamental tradeoff betweencompression length and resolution in lossy data compression. However, itconceals the underlying dynamics of optimal source encodings or test channels.We argue that these typically follow a piecewise smooth trajectory as thesource information is compressed. These smooth dynamics are interrupted atbifurcations, where solutions change qualitatively. Sub-optimal test channelsmay collide or exchange optimality there, for example. There is typically aplethora of sub-optimal solutions, which stems from restrictions of thereproduction alphabet.  We devise a family of algorithms that exploits the underlying dynamics totrack a given test channel along the rate-distortion curve. To that end, weexpress implicit derivatives at the roots of a non-linear operator by higherderivative tensors. Providing closed-form formulae for the derivative tensorsof Blahut's algorithm thus yields implicit derivatives of arbitrary order at agiven test channel, thereby approximating others in its vicinity. Finally, ourunderstanding of bifurcations guarantees the optimality of the root beingtraced, under mild assumptions, while allowing us to detect when ourassumptions fail.  Beyond the interest in rate distortion, this is an example of howunderstanding a problem's bifurcations can be translated to a numericalalgorithm.

Root Tracking for Rate-Distortion: Approximating a Solution Curve with Higher Implicit Multivariate Derivatives