Bring 4 cups of water to a boil in a saucepan. Add the ube cubes and boil gently for about 20 minutes, or until the ube is very soft.
Add the ingredients for the crust to a food processor. Pulse until the ingredients come together in a cohesive mass.
Line the bottom of a 9-inch springform pan with parchment paper. Press the crust evenly and firmly into the bottom of the pan. Place in the freezer.
Reserve ¼ cup of ube and ½ cup of the cooking liquid. Add the rest of the ube, cashews, and agave to a high-speed blender.
Add the coconut milk to a small saucepan. Whisk in the agar agar powder. Bring just to a boil, then simmer for 5 minutes, whisking frequently.
Add the coconut milk to the blender and start blending the ingredients on the lowest speed. Gradually increase the speed and continue blending until the mixture is very smooth.
Remove the crust from the freezer. Pour the filling over the top and smooth as necessary.
Add the reserved ube and water to the blender and blend until smooth. Drizzle the mixture in parallel lines over the filling. Drag a toothpick across the top of the cheesecake in the perpendicular direction to create feathering.
Place the cheesecake in the fridge for about an hour to set.