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Random forest is a machine learning algorithm created and trademarked by Leo Breiman and Adele Cutler. This seems to be a proper noun, however, when I search https://ludwig.guru/ or use Google's https://books.google.com/ngrams/ for common usage most articles favor lowercase. As in, "random forest machine learning," and not "Random Forest machine learning."
When I use "random forest" in a research article, I don't know if I should I capitalize it.
I doubt that anyone can give you a specific answer for your situation, because it's unlikely that anyone else has heard of "random forest" before. The answer, therefore, will really depend on whether YOU think that it's a proper or common noun. You seem to have done some research (noting that the term is trademarked but that most appearances are lowercase), so you'll have to make a decision about which is more likely. If you really can't decide, then I recommend following the common practice in your field.
Note that many terms that start as trademarks (e.g., "Aspirin") become "genericized" and used as commoun nouns (e.g., "take five aspirins"). Therefore, it is sometimes the case that either version is acceptable, especially when the term is in the intermediate stage of transitioning from a proper to a common noun.
