I appreciate the thought-provoking examples :-)
> Humans find and reject "low rank" correlations all the time ... steak and cookies ...
Sure, but the appropriateness of a low rank approximation depends on what you want to predict. Eg: for predicting basketball success, {height, wingspan} might be a very useful "low rank" description, while for predicting obesity the relevant low rank description might be the body mass index (BMI) or something like it. Wingspan = chest width + 2 * arm length and BMI = weight/height^2 could be composite features "discovered" by such a model+inference toolkit.
> Conversely, some doctors have an encyclopedic knowledge of various ailments, and either memorize or infer from experience a particular ailment from a huge collection of possible ailments. If the number of possible ailments is large, their "understanding" is not low rank, it's the opposite.
That's exactly why I would not call that understanding, especially if they have to memorize a book full of ailments and symptoms. "Understanding" would entail a (causal) model of underlying physiological problems and the symptoms they give rise to, with a recipe for inferring in reverse.
> Low rank does not necessarily mean "understanding", it's just a local minimum in some function of a random variable.
I don't understand that statement at all.
PS: I don't have any particular affinity to neural networks over other ML models, and don't mean to escalate the hype :-)