In the PCA algorithm, we may have an example like this, so

maybe that's my example x1, and maybe that's my example x2.

And what we do is we take these examples,

and we project them onto this one dimensional surface.

And then now we need to use a real number, say z1, to specify the location of

these points after they've been projected onto this one dimensional surface.

So, given the point z1, how can we go back to this original two dimensional space?

In particular, given the point z, which is R,

can we map this back to some approximate representation x and

R2 of whatever the original value of the data was?

So whereas z equals U reduce transpose x,

if you want to go in the opposite direction, the equation for

that is, we're going to write x approx equals U reduce, times z.

And again, just to check the dimensions, here U reduce is going to be an n

by k dimensional vector, z is going to be k by one dimensional vector.

So you multiply these out that's going to be n by one, so

x approx is going to be an n dimensional vector.

And so the intent of PCA, that is if the square projection error is not too big,

is that this x approx will be close to whatever

was the original value of x that you have used to derive z in the first place.

To show a picture of what this looks like, this is what it looks like.

What you get back of this procedure are points that lie on the projection of that,

onto the green line.

So to take our early example, if we started off with this value of x1, and

we got this value of z1, if you plug z1 through this formula to get x1 approx,

then this point here, that would be x1 approx,

which is going to be in R2.

And similarly, if you do the same procedure, this would be x2 approx.

And that's a pretty decent approximation to the original data.

So that's how you go back from your low dimensional representation z,

back to an uncompressed representation of the data.

We get back an approximation to your original data x.

And we also call this process reconstruction of the original data

where we think of trying to reconstruct the original value of x

from the compressed representation.