I wanted to leave some initial thoughts on the preamble section. I think it's really interesting, but that you could make it much sharper.
Toy Problem Setup
I wonder if you could make your toy problem a bit simpler? While it's fairly straightforward, there is a bit of effort in setting up the idea of these images where half are random noise and half constant and how to manipulate them... (For example, one way you could simplify it would be to just do it all with two-dimensional vectors -- so could I and J would be two intervals on the x-axis of a plane, e_theta just (cos(T), sin(T)), and so on. That's just one possibility to throw something out there!)
Of course, the slight complexity of the present example is fine if it is serving a good cause. But I'd encourage you to ask: is it paying rent? That is, is there an upside that makes the increased complexity worth it? For example, you may want to use your present version because you want something that can be seen as both high-dimensional and low-dimensional. Or because you want to be closer to your later examples. Or something else entirely.
Section Title
"Preamble" as a title doesn't communicate much of what the section is about. I'd consider something a bit more descriptive like "A Toy Example." This helps set the readers expectations and helps a reader quickly skimming over the article understand the narrative arc.
Acknowledge Example Concerns
Once concern I had while reading through this section is that the example seems unrealistic. Why would the angle be so unnatural? You address this later, but a sentence acknowledging these concerns in this section might help. "The decision boundary being tilted like this may seem unrealistic, but we'll see in the next section that it's actually a very common phenomenon."
I wanted to leave some initial thoughts on the preamble section. I think it's really interesting, but that you could make it much sharper.
Toy Problem Setup
I wonder if you could make your toy problem a bit simpler? While it's fairly straightforward, there is a bit of effort in setting up the idea of these images where half are random noise and half constant and how to manipulate them... (For example, one way you could simplify it would be to just do it all with two-dimensional vectors -- so could I and J would be two intervals on the x-axis of a plane, e_theta just (cos(T), sin(T)), and so on. That's just one possibility to throw something out there!)
Of course, the slight complexity of the present example is fine if it is serving a good cause. But I'd encourage you to ask: is it paying rent? That is, is there an upside that makes the increased complexity worth it? For example, you may want to use your present version because you want something that can be seen as both high-dimensional and low-dimensional. Or because you want to be closer to your later examples. Or something else entirely.
Section Title
"Preamble" as a title doesn't communicate much of what the section is about. I'd consider something a bit more descriptive like "A Toy Example." This helps set the readers expectations and helps a reader quickly skimming over the article understand the narrative arc.
Acknowledge Example Concerns
Once concern I had while reading through this section is that the example seems unrealistic. Why would the angle be so unnatural? You address this later, but a sentence acknowledging these concerns in this section might help. "The decision boundary being tilted like this may seem unrealistic, but we'll see in the next section that it's actually a very common phenomenon."