The white robe is a secret object that you can obtain by obtaining the Transcendence trophy, which is to acquire all Scarf upgrades hidden in the game. You can do this over multiple playthroughs.
Once you acquire the trophy, head to the start of the game via "Chapter Select". From the center of the area, look to the right and you will see several large cloths sticking out from the ground. Go over to them and simply walk into them. Wait until the cloths turn red and then walk out. Your cloak will now be white and the scarf you start off with will be longer. Unlike the red scarf, the white scarf will automatically regenerate scarf power once you land after jumping. Your scarf power also drains at a slower rate than normal. If desired, you may also walk back into the several large cloths to change back into your red cloak. Wait until they turn white instead. You may change between red and white as much as you wish at any point in the "Chapter Select" area after obtaining the trophy.