it seems quite cliche in this day and age that a self-dubbed 'composer' would "improvise" on the piano. Satisfied with his or her new composition, he or she then names it something alliterative, like "the mourning moon" or the "the moon over the lake." It is posted on the internet, and peolpe say, "ingenious," "very good," "introspective." One way to look at this is that the person who has little or no musical training is making an attempt, though in honesty the work has no place next to rachmaninoff, liszt or chopin. I admit that I am guilty of doing this, thinking i am a genius, though I think that chopin, mozart and beethoven and all the other greats probably thought the same way, that there pieces were stellar. This said, I think it takes constant practice with composition, arrangement and practice to become any good at improvisation, or composition for that matter. A bachelors in theory and composition is not going to gaurantee your compositions are good, but I think you have a lot more authority to make a piece that is listenworthy if you know about intervals, inversions, major and minor scales, sustains, add6s, add9s and being able to identify chords. IF you have heard of ryan stewart, he is perhaps a cliche composer, while some of his works seem to be improvisation. IF you have the time i recommend looking him up on youtube. I bought and listened to his album "equinaminity" it is very good. He has some music talent in my mind.
And this is for everyone now, post up your improvisations! Show your musical expression, not just though written music! Even if most will view them as being "unmusical' - who cares??!! Music history has always been full of music where people considered it as being unmusical!! Let your musical spirit shine.
I agree with Mr. Ondrich that it is best to post ones dabblings, to recieve feedback, but also to understand where you are in relation to other composers. I don't recommend constant performance, but I think that it is part of the process if you hope to improve as a composer/ improviser of piano music, whether or not one calls it classical or nonclassical