Really exciting stuff here! I’ve been working on an alternate formulation of circuit discovery in the now traditional fixed problems case and have been brainstorming unsupervised circuit discovery, in the same spiritual vein as this work, though much less developed. You’ve laid the groundwork for a really exciting research direction here!
I have a few questions on the components definition and optimization. What does it mean when you say you define C components ? Do randomly partition the parameter vector into C partitions and assign each partition as a , with zeros elsewhere? Do you divide each weight by C, setting (+ ?)?
Assuming something like that is going on, I definitely believe this has been tricky to optimize on larger, more complex networks! I wonder if more informed priors might help? As in, using other methods to suggest at least some proportion of candidate components? Have you considered or tried anything like that?
Love the way you laid things out here! Lots to discuss, but I’ll focus on one specific question. We’ve communicated privately so you know I’m very bullish on PD as a potential new paradigm. Don’t take the below as general skepticism!
I don’t understand this claim, except perhaps in a trivial sense which I’m assuming you don’t mean. My confusion stems from my intuition that we don’t have a good reason or evidence to assume that the model never needs the full magnitude of a particular parameter in two different mechanisms that do different things that are not related functionally, semantically, nor in any but an ignorable sense geometrically.
Is the statement above implicitly conditional on the assumptions of PD, one of which is that my intuition is false? If not, then in the worst case it seems to me the only other option is that the block quote is only trivially true in the sense that if there are many overlapping mechanisms where the full magnitude of a subset of the overlapping parameters is needed, then the PD degenerates to a single mechanism that is the original model. Would be interested to hear your thoughts!