The very first time you encounter an edged weapon expert will be in Chapter 5. After your first sniper fight, you will fall down to an area. Some enemies will then come running through the doorway, and your team mates will tell you that one of the enemies is an edge weapon expert. He will be sprinting at you with his hand up in the air. Once he's close enough, take him out with whatever weapon you have, although I recommend a shotgun. If he gets to you, it's instant death. If you miss the opportunity, you can reload the checkpoint or attain it later in the game.