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From the standpoint of loading down Reddit today, yes. But, if we want to talk computer science theory and what one compression algorithms one could build, that's not really true.
There are two classes of compression -- lossless compression and lossy compression.
Lossless compression retains an exact copy of the original data. Compress and then decompress and you get back the original.
So, okay. How can you compress data? I mean, if I have a byte of data, eight 1s or 0s, how can I use less than eight 1s or 0s to store those? For lossless compression, the answer is that you have to have some knowledge of what information it is that you're storing. If you know that information of a given length N with certain characteristics comes up more-frequently than others, then you can assign a shorter pattern M to represent that pattern and then use the old pattern of length N to represent something else less-common. Lossless compression is just the art of reordering representations of data to more-closely fit the frequency with which they arise: shorter for things that are relatively-more-common.
If you're wrong about that order, then lossless compression can make the representation of your data larger.
Now, technically the noise in there is actually probably very predictable, because it's likely based off a pseudorandom number generator (PRNG). That isn't really "random" -- it's just making numbers that look random from a single number that's hard to predict. If the PRNG isn't having more entropy injected into it over time, then all of the noise generated during the session comes down to that one small number. If you were clever and could figure out the seed -- a small number, often something like 64 bits, and often seeded off something like the Unix time at the time that the random numbers started being generated, which makes it even more predictable than that -- or at least the internal state of the PRNG, maybe 256 bits -- you could basically store the content of the whole video in just a few bytes. However, it's not always easy to determine that original state -- in the case of cryptographically secure PRNGs, it's specifically intended to be impractical.
However, we generally treat pseudorandom noise as if it were actually truly random, rather than just pseudorandom, which means that it's totally unpredictable, and if that is the case, then you cannot losslessly compress it and make it smaller, not over a sufficient quantity of noise, because you can know nothing about the frequency with which a given pattern arises.
So, depending upon the source of noise used, we might be able to do lossless compression of noise, if pseudorandom noise was used (probably) and if we can figure out what that number is that was used to generate that noise.
Okay, enough about lossless compression. Can we do lossy compression of noise?
And there the answer is...yeah, probably yes.
The way lossy compression works is that we have to know something about what information is actually "important" when we get around to actually using it. That lets us throw out some of the less-important information. What we get back, unlike with lossless compression, is not true to the original, but it's a lot closer than if we just threw out information without regard for what's important and what's not. Lossy compression is often used to compress audio and video.
For a lot of things, there's a lot of not-very-important information.
Let's say that we're looking at a video of noise. Your brain doesn't care about every exact pixel there. It's looking for shapes that remain across multiple frames, move together, so it can pick out objects and the like. Your brain just basically sees the noise as one big field of stuff of an approximate color changing at a given rate. None of the specifics of that noise matter. Basically, regardless of what seed was used to generate that noise, pretty much all noise with the given properties (black and white, 1 pixel size, changes every frame, N fps) looks pretty much identical. So a good form of lossy compression in a video codec would be to detect anything that looks like noise -- and then just replace it with generated noise using a fixed seed. All noise looks pretty much identical to a human. So you'd get pretty much identical-looking output. As it happens, existing video codecs don't have a noise detector, but they could.
So video of noise is lossily-compressible.
Now, I will grant that this is unrelated to putting load on Reddit, but, hey, might as well start filling the Fediverse with useful information now.
This is a good post. I don't really have much more to add than that but I want to boost you with the interaction. Very well written and informational, and as far as I can tell, accurate.