Invariance

 Invariance, as opposed to inflammable, means, does not vary.

The phrase “invariant in time” simply means “not changing over time”.

Einsteins theory of relativity is all based on trying to create a consistent view of the world where certain things shouldn’t change depending on who is measuring them and how they are being measured.

Invariance is important in artificial intelligence too. It’s incredibly important because if you can assume something won’t change, you don’t have to perform a calculation working out how it has changed.

Take the simple example of newtons laws of motion, summarised in the formula:

F=ma

The power of this formula lies it its invariance. It doesn’t vary with the day of the week or time of day. It doesn’t vary with shape, with the type of object, with who is in government.

It doesn’t vary with who is looking at it or where they are.

Physics is assumed to be invariant with things such as the political persuasion of the observer but why can we make that assumption?

One of the leading problems in robotics is the ability of a robot to predict, looking at an object, how the object is going to react to the robot’s actions.

Take mass, for example, one might assume that this is rotationally invariant (The mass of an object doesn’t change with orientation).

What about the centre of mass though, for a rigid body, like a fork, the centre of mass doesn’t change as you rotate the fork (Assuming a rotating reference frame).

Can you assume this though? What about a bottle of water?

Even given a rotating reference frame, the centre of mass of the bottle of water will change as it rotates.

One of the things that is important to consider when picking things up is the friction.

One might assume that friction is positionally invariant. It doesn’t matter if you put an ice cube nearer the north wall of the room or the south wall of the room, it’ll be slippy.

But what about holding something above a hot pan of water? Now friction is to some extent dependent on position, not just that but it is dependent on it’s historical position.

One invariant that might seem more sensible to question, that is included in F=ma is that the acceleration of an object is independent of the historical path of that object.

To know where an object will be in the next time step, you need only know the net force on it, it’s current position and velocity, and it’s mass.

It probably doesn’t strike you as odd that you don’t need to know where the object has been before you recorded these three things, but why?

Phycology is certainly dependent on history; to know what someone is likely to do, you need to know what experiences they have had in the past.

One of the many causes of the 2007/08 market crash was the assumption that there was independence between mortgage default rates in different parts of the US. This makes the models far simpler as what would otherwise be an incredibly complex model of how the default rates vary with respect to each other, you can treat them independently, looking at only one at a time and assuming invariance to out of local area crashes.

This independence assumption was one of the failings that lead to such an “unforeseen” crash.