So, a one dimensional space is just a line.

It doesn't work in a universe with four dimensions of space, nor five, nor six.

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

So we use a little trick.

like a piece of paper.

Why is that?

Let me explain it to you more intuitively.

Same situation as before. We'd like to go from here to here.

So the best way of viewing it is by analogy

This is a vector field.

Fly your ship in a 4d environment

In this case X is one dimensional which is N 1.

The second heuristic we're using looks at the 3D problem but without obstacles.

They are usually easy to implement.

Where this one is the orthogonal vector shown over here.

And the Phoenix Islands is very rich in those seamounts.

The other one is from the previous brick that was placed.

They have some deficiencies.

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.

The arms down, the arms up, walking, throwing a ball, and so on.

And so on. To render means to create an image.

The orthogonal dimension in this direction carries alomst information, so it suffices, in most cases, to project the data onto this 1 dimensional space.

Even though moving one over here and one over there would give a better solution.

And that's where I work.

Here's a 2 dimensional space.

It sees her, pays attention to her.

Further, if you re represent each data vector using those eigenvectors, you'll find a 3 dimensional space where original data falls into a varity of different places.

They are usually easy to implement compared to any other filter.

The answer is here.

They don't really scale to high dimensional spaces.

The dimensionality reduction looks a little bit silly when you go from 2 dimensions to 1 dimension.

Under extreme conditions, general relativity and quantum theory allow time to behave like another dimension of space.

You see the loop on the right side.

Let me explain k means by an example.

In that sense, it's actually a three dimensional state, so 3 is the best answer.

The Kalman filter is the only filter that does scale exponentially, so it's very nice.

When you study the mathematics of string theory, you find that it doesn't work in a universe that just has three dimensions of space.

I want you to plug in the f matrix.

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.

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

I capture the essence of the data.

looks like in three dimensions.

So, I hid in a time dimension.

This little black dot there is a 3D accelerometer.

So now we've got the stuff color sorted, angle maps onto color.