Investigators believe the is tied to two similar cases in Los Angeles.
In each of the three, non-fatal incidents, a knife was plunged into the victims' backs as they slept, and notes were left behind. The Los Angeles Times called the notes "death warrants."
"We are working together to try to solve these cases," Santa Monica Police Sgt. Richard Lewis said of the Los Angeles Police Department.
In Santa Monica, a 10-inch knife was recovered from the scene of the stabbing in front of the Goodwill at the corner of Sixth Street and Santa Monica Boulevard.
The man remains in serious, but stable condition, and is expected to survive, Lewis said.
The two stabbings in Los Angeles occurred Thursday near Hollywood High School at the corner of La Brea and De Longpre and on July 4 at Third and Main streets. The assailant left notes at the scenes that were typed and signed.
Police want to question a "person of interest" who goes by the name of David Ben Keyes, said Officer Karen Rayner, a Los Angeles Police Department spokeswoman.
Keyes, who may be homeless, like his victims, recently came to Los Angeles from the Santa Barbara area, LAPD Cmdr. Andrew Smith, a department spokesman, said at a news conference Thursday.
The LAPD on Thursday released two photos of Keyes and described him as a "person of interest" in the case.
Smith urged homeless individuals to take steps to protect themselves.
"If possible, get yourself into a shelter. Get somewhere where you can be protected," he said. "We know that homeless people are vulnerable, they're some of the most vulnerable people in our society. But it's even worse when there's an individual out like this who's conducting these random acts of violence against homeless individuals."
— City News Service contributed to this report.
(Editor's Note: This story was updated at 6:50 p.m. to add that photos of the "person of interest" sought by police were released.)