In the beginning of Men in Black Detective Edwards (who would later be known as J) was pursuing a suspect who turned out to be an extraterrestrial.
What was Edwards chasing him down for?
We don't know, but there are certainly some clues we can get from context as well as the script and the novelisation
1) The "Perp" was hired by the Bugs to locate the Galaxy. His failure prompted them to send Kerb (the Bug) to finish the job.
"No! He's coming! He'll kill me! I failed and he'll kill me!"
2) He tried (successfully) to get hold of a weapon suitable for assassination.
KAY: A carbonizer is an assassin's weapon, Jeebs. Who was the target?
3) On arrival, the Bug heads straight for Rosenberg's shop. Clearly he has intel on his location.
4) The novel indicates that the Perp was fleeing a "bust".
Edwards put the pain into a holding area and started after the perp again. This guy sure was going to a lot of effort to avoid a bust for what would probably get bargained down to a misdemeanor. Didn't make any sense.
5) Rosenberg takes a short cab ride from his jewellery shop to his final destination, Leshkos Restaurant (on 2nd Avenue), itself within walking distance of the Guggenheim Museum (on 5th Avenue).
So, what can we ascertain? It's very easy to imagine that the 'Perp', panicking about having failed in his assignment, was hanging around suspiciously late at night outside Rosenberg's dwelling place or jewellery shop, having decided to take a more direct approach and kill him (or at least threaten him) his fancy new gun in order to reveal the location of the Galaxy before Edgar's arrival. He was approached by a policeman as a probable drug dealer or would-be burglar. He fled on foot and that's where our scene opens.
For an unknown offense, a Cephalapoid is on Earth running from the police, who quickly wince and pant. James Edwards, at the time a member of the NYPD, is the only one who is able to chase it.