But in truly high dimensional space it has a very strong utility.

Here's a 2 dimensional space.

It's truly a three dimensional space.

We take this curved 12 dimensional space, and transform it into a flat, four dimensional space.

Remember that now this state space is 3 dimensional, not 2 dimensional.

We have a 3 dimensional state space.

That's usually a problem, because if you have a high dimensional space, you just can't apply it.

Our state space is, I guess, 4 dimensional.

One idea is that our three dimensional universe may be embedded in a high dimensional space, just as you can imagine on these sheets of paper.

They don't really scale to high dimensional spaces.

Images like these are very high dimensional statistics.

In fact, they live in a 12 dimensional space.

So, a one dimensional space is just a line.

This is a vector field in 2 dimensional space.

That's been recognized because the number of particles you need to fill a high dimensional space tends to grow exponentially with the dimensionality of the space.

Kismet has an underlying, three dimensional emotional space, a vector space, of where it is emotionally.

Now, take into account the fact that we live in a three dimensional universe, OK, a three dimensional space.

So the only difference, class, is that our object now moves in a 2 dimensional space, where as in class, it moved in a 1 dimensional space.

So far we've looked at the state space of cities in Romania a 2 dimensional, physical space.

Suppose we're given the following data points in a 2 dimensional space.

Or in the high dimensional space, we might just write W times X plus W0, where W is a vector and X is a vector.

And now it starts moving in three dimensional space so up or down.

So it's a deep think about it in a big three dimensional space, very deep three dimensional space with herds of tuna, whales, all kinds of deep sea marine life

I create mathematical objects, symmetrical objects, using Galois' language, in very high dimensional spaces.

And that four dimensional space consists of X, Y, Z, and then the yaw angle.

A two dimensional space is defined by two perpendicular lines, which describe a flat plane

I can draw, really easily, a nice 1 dimensional space that follows these data points.

If I am able to project all the data points on this 1 dimensional space,

The trick, of course, is to find the nonlinear 1 dimensional space and describe it.

Realize that now the state space is 3 dimensional, just like in our localization example.

like a 4 dimensional or 5 dimensional figure.

Typically a dozen or so will suffice to make a face completely reconstructable, which means you've just mapped a 2500 dimensional feature space into a, perhaps, 12 dimensional feature space on which we can now learn much, much easier.

And the power of Galois' mathematical language is it also allows us to create symmetrical objects in the unseen world, beyond the two dimensional, three dimensional, all the way through to the four or five or infinite dimensional space.

So now I'm going to show you a number of projects that we've built, from one dimensional, two dimensional, three dimensional and even four dimensional systems.

We have progressed from a zero dimensional point, to a one dimensional line segment, to a two dimensional triangle.

Sorting multi dimensional array

Insert 2 dimensional vector

Toward the Dimensional Gate.

So you can basically program any three dimensional shape or one dimensional, two dimensional up into this chain completely passively.

And a three dimensional space adds a third perpendicular line, which gives us height and the world we're familiar with.

So it's essentially a way of measuring how much space something takes up on kind of a two dimensional surface.

Three dimensional shapes and the space they create in the middle creates a new shape, the answer to the sum.

A 2 dimensional gaussian is defined over that space, and it's possible to draw the contour lines of the gaussian.

So these minimum snap trajectories in this flat space are then transformed back into this complicated 12 dimensional space, which the robot must do for control and then execution.

However, I printed out a 2 dimensional table, not a 3 dimensional table.