Officers were called to Leith Walk at about 06:00 following reports about a man with a neck injury.
Detectives said it was believed he sustained the injury sometime between 05:30 and 06:00 in nearby Smith's Place.
He was taken to the Edinburgh Royal Infirmary where his condition was described as serious but stable.
Det Sgt Neil Spowart said: "We're working to ascertain the full circumstances surrounding this man's injury.
"We believe that he was in a business premises on Leith Walk, near the junction with Balfour Street, at around 5am and that the injury was sustained sometime between 5.30am and 6am.
"Anyone who remembers seeing a man in the Smith's Place area around this time, or has any information which may be able to help us establish the cause of his injury, is urged to get in touch."

The man was arrested on suspicion of preparing for terrorist acts and terrorist training as he stepped off the plane from Istanbul on Thursday evening.
He remains in custody at a south London police station.
Officers are searching a property in east London, Scotland Yard said.

Sales volumes increased by 1.7% in November from the month before, the Office for National Statistics said.
Compared with the same month last year, sales were up by 5%.
However, Keith Richardson from Lloyds Bank Commercial Banking said it was "too early" to say whether Black Friday was really a success for retailers.
"Retailers were better prepared this year and adapted the US-style Black Friday to better reflect British culture. Some chose not to take part while many others spread the discounts over several days to better protect margins and ease the burden on their websites and IT platforms."
The ONS said the amount spent by shoppers was up 1.4% in November compared with the month before, and was also up 1.4% from the same point a year earlier.
The value of online sales increased by 4.9% in November from October, and were 12.7% higher compared with last year.
"Retailers may see some pay-back after the Black Friday promotions led shoppers to pull-forward spending that would otherwise have taken place in December, but the underlying sales trend looks set to remain strong as we head into 2016," said Chris Williamson, chief economist at Markit.
He added that spending was not just being driven by discounts. "Households are benefitting from improved job security, low inflation and falling energy prices, the latter helping free-up more income to boost retail sales."
Police in Edinburgh have appealed for information after a man was found seriously injured in Leith.
[NEXT_CONCEPT]
A 30-year-old man arriving in the UK on a flight from Turkey has been arrested at Heathrow Airport by counter-terror police.
[NEXT_CONCEPT]
UK retail sales rose by more than expected in November, as shops offered promotions at the end of the month in the run up to Black Friday.