The man followed a 24-year-old woman on Easter Road and made inappropriate remarks.
He chased the woman and subjected her to an indecent assault, but fled down Bothwell Street after the victim began screaming. Police say around 15 minutes later another 24-year-old woman was attacked in the same manner, as she walked from Easter Road and turned on to Albert Street.
The man grabbed the woman from behind, and struggled with the victim, before punching her in the head, leaving her with a cut above her left eye. He then made off towards Easter Road.
Both assaults took place in the early hours of Monday morning. The suspect in the incidents is described as white with fair/blonde hair, between 18 and 25-years-old with a slim build and blue eyes. He was wearing blue jeans and a long-sleeved top.
A spokesman for Lothian and Borders Police said: "Given the proximity of both assaults and the consistent description of the suspect, we believe these incidents were carried out by the same man."