Both are great speakers. Here is how I would break them down, given my experiences of course.
Paradigm Studio 100 V4
- These speakers give the music a more up close and personal presentation. The treble is more forward and detailed than the LSi-15's. They also sport a more open sounding midrange, and can throw out a wider soundstage. I found the Studio 100 V4's to be quite a bit more dynamic than the Polk LSi-15's as well.
With the Paradigms, the music can feel fast and exciting. The compromise to it all is that the Studio 100 V4's can get a bit forward/bright in the top end, and if you aren't careful with matching them to the right amp, the midrange can sound a bit hollow as well.
If you go the Paradigm route, I'd highly suggest throwing some good tube pieces into the system. Good tubes + Paradigm is always a solid match.
Polk Audio LSi-15
The Polks are more laid back than the Paradigms, and will sport a warmer, thicker/meater sound - particularly throughout the midrange. Their imaging, though not as "wide" as the Studio 100's, is more pinpoint and centered between the speakers. Both speakers have decent bass extension and suffer a bit from a "bump" down low.
With the Polk's, you get a more romantic sound. The caveat here is that the 8 inch side firing woofers can sound a bit slow/bloated if you do not pair the speakers with a high current amplifier. If you plan on running the Lsi-15's, I would advise you to go with a good solid state amp that has a real good power supply and control over the bottom end (think: Marsh, Odyssey Audio, Bel Canto, etc..)
Well, there ya have it! Both speakers are good. Both speakers are fairly adept at playing all sorts of music, yet both are polar opposites in terms of how they deliver the musical goods. Good luck!