Tim Poston - Keynote: Data Comes in Shapes

I feel very honored to be invited here.

Thank you very much.

I like to, I think I've seen one

maybe two other people with gray hair here

[audience laughter]

The last talk I gave a few weeks ago

was to a meeting of ophthalmologists

and that was a bunch of

much older people, okay, and

a homage here to the Fifth Elephant

This is the novel, the cover of the novel

from which it was taken and, well actually

I'm using this as a connection for a

little bit of boasting because

Terry Pratchett wrote the book

I am a co-author of a co-author of
Terry Pratchett

and I actually signed a publisher contract

on my 70th birthday

a few weeks ago to publish a

science-fiction novel with Ian Stewart

and I mention that not just
as boasting but,

ok, this is a Data Geeks meeting rather

than the Graphics Geeks meeting, but if

anybody has graphics enthusiasm, there is

all kinds of stuff that would be fun

to build for the website we are putting

together for that novel. strange things

happening on that planet, so do make

contact if you're interested in drawing

strange and beautiful things because

I have some strange and beautiful things

to draw and some to interact with.

What I don't have is a budget

you have to just like it

ok, that is pure digression

I was originally a mathematician and

that was my PhD back before almost anybody

here was born, and I've kinda wandered

around the world and the sciences and

I've turned into some sort of an engineer

But what I'm going to talk about here is

the power of particular mathematical

point of view, which is that numbers are

not just numbers

They belong together in shapes, so

What are data?

Mostly, they're numbers

I know there are fields and things

We've been hearing about that, but

then you keep counting

Lots and lots of it is numbers

But are numbers only numbers?

Well, no they gather together in things

They come in patterns

and really big data is all about

the arrangements those things make

just knowing the numbers, you don't
know anything

You got to know how they fit together

Patterns are shapes

So, studying shapes, data shapes,
any kind of shapes

Space-time shapes. That's Geometry

But not the kind that I was doing when I
was 13 or 14 years old

Mind you, I had some taste for it and it
was quite fun

but it was all flat in the sand,
just like that

and here is Euclid

Stuff we would write in little triangles

and fun things

This I remember as a remarkable theorem,

but I have never ever, ever, ever
seen a use for

[audience laughs]

It's weird, it's very much something

about the plane, it's strange

and I have never encountered it or

referred to it in anything useful since I
left school

It's a bizarre theorem,
which is occasionally useful

Everything is so much in the plane

Data shapes don't live mostly in the plane

Geometry doesn't mean that you replace

now this, by the way, is highly superior
pointer technology

Much better than those twinkling little

red things that you lose track of
where it's pointing to

and 10% of your audience can't see red

Now, here is something serious

Children think in 3D

They think brilliantly in 3D

They naturally work in 3D

They are connecting how their vision

is working with their hands

they can reach out and grab your nose

If you watch a small child, it's doing
a lot of practice at building

a 3D model of the world, and then,

and these days that continues into
primary school

a hundred years ago, ugh..

but now primary school's good

but their secondary schools suck rocks

It's still, if you get any geometry

it's flat, flat stuff.

It can get more and more complicated

yeah.. but,

I just grabbed that off the web

as one particular complicated 2D diagram

but fix your mind in 2D, you
get to the point where you can't think

Are the x- and y- axes this way
or this way?

I found my UCLA students

if I switched drawing on the blackboard

from this way to that way

because something could be seen better
that way

they couldn't turn it in their heads.

Well data doesn't live in the plane.

It's not flat.

If we have three variables, we have
three dimensions.

That might be how far this way,

how far this way, and how far up.

And if you're doing graphics, it's three
very directly spatial dimensions.

But if it's, if you've just got numbers
about people

I look at everybody here, I know their

height, well I don't know their height but

you guys do because you're big data guys.

Know the height, know the weight, know
the age.

Three numbers, that's a three
dimensional set.

And the pattern that you make, that's

three dimensional geometry.

But of course, you typically have

a lot more. So you've got 'n'
dimensions

and 'n' can be quite big.

So you need to think about 'n' dimensions.

And there's two ways to do it.

One is to turn it all into algebra, which

is what people spend a lot of their time
doing.

And in this talk I'm only going to talk
linear algebra

which doesn't mean it's the only kind
there is,

but I've only got a few minutes.

Or you can practice thinking in 3D and

build up insights that help you very

seriously in 'n'-dimensional thinking.

I took up carving things when I was a grad

student because I realized that my mind

had been flattened by my high school

and my undergraduate. I needed to loosen
up

my mind and think in 3D, so I started

using my hands. You got, this is the

visual part of the brain, this is the

motor part of the brain, and the motor
cortex

just has to be 3D because you've got to

pick things up, you got to twist them,

connect it all up. So seriously, for 3D

thinking, take up sculpture.

So practice thinking in 3D, and the more

3D you can think, the readier you are to

think in other dimensions, general
dimensions.

But 2D? Nah, it's not enough.

So, question for you guys.

Most people here have done things - are

doing things with matrices sometimes,
right?

What does a matrix even mean?

Whats it represent?

Blah, you have told me the data structure.

It's, yeyah, it's an array.

This is the data structure.

It's an array this way, and this way.

But, at the level of algebra, and geometry.

It's something a bit more.

It's something that operates on vectors.

Transforms vectors, and in particular

there was a rule that they taught me

yea back at first or second year of graduate

of which way you multiply the
matrix and the vector.

And, I swear to you it took
me a year to remember

When I'm multiplying 2 matrices
do I go along this way

or along that way. Because it
was a damn silly rule.

That came from the Algebra book.

But, trying to avoid spending
too much time on this.

You do know the rule most of you.
So, if you have this 3x3 matrix.

And, apply it to this vector.
[column of '1,0,0']

You get this. Which is this column.
[points to 'a,c,f' columns]

And, if you apply to this vector.
[column of '0,1,0']

You get this column.
[column of 'b,d,g']

And, if you apply to this vector.
[ column of '0,0,1']

You get the last column.

Now, '1,0,0' means lets suppose
this is the x direction.

X this way. It says anything
that is purely in the

X direction goes to 'a,c,f'.

Wherever that is.
Which is a 3 dimensional vector sum.

Anything that is in the Y direction.
Like, '0,1,0' goes to something else.

Specifically, it goes to 'b,d,g'.

And, anything that starts vertical
goes to 'c,e,h'.

So the matrix is actually a list of vectors.

It's saying, where does the first one go

where does the second one go,
and where does the third one go.

And, believe me if you're doing 3d
computer graphics;

understanding that point
will make it much easier

than anything I've ever seen
in an OpenGL manual

of what you ought to be do.

They don't explain matrices very well.

It's just a list of where
these 3 things go.

And, in the case of a rotation
that's particularly tidy.

Right angles, things at right angles
go to things at right angles.

And, so on. But, not every matrix
is doing something

as simple as a rotation. Unless you're as

simple-minded as an IQ theorist.

And, they rotate things they've
no justification doing.

If you remember that, you can always
clarify, see more definitely

what the algebra is doing, and
if you know what the algebra is doing.

You can make it drive better code.

What should the code be doing?

So, I'm just going to illustrate this point
of view.

With a very top down glimpse at
some of the things people do

when they got a lot of data. One of them

is Principal Component Analysis.

Now this very sketchy, very, very 2d.

I didn't have time to do wonderful 3d animations.
I'm sorry.

But, I got this variable, this variable.

Put them together I got datapoints.
Which are pairs of variables.

And, roughly speaking I can say.
Just looking at this.

That as this increases, that increases.

But, if I want to compress my data.
Than, I rotate my axis.

I put an axis along this way.
And, another axis along this way.

So, it's going to be a matrix that says
this guy, goes to there

and this guy, goes to there.

Finding that matrix; you had this chance
of algebra

not fitting into this time. But that's
the idea of Principal Component Analysis.

And, if you're applying it to a machine.
That does wobbles here, and here.

And, makes squeaks there, and
all sorts of things.

You've got a lot of numbers,
and you got to do the algebra

a bit more complicatedly than
that 2d picture represents.

But, that is really what's going on.

You're finding the way to move your axes.

And now, if you know where you are
on this axis.

You know most of what you want
to know about a datapoint.

Is it here or is here, one number.
How far along it is.

And, then you say well you can expect some
errors in that direction.

Where in the previous picture
you had to know

2 directions, 2 numbers.
So Principal Component Analysis

is beautiful technique for
information compression,

reducing the amount of arbitrariness;
handling all sorts of things

excellently; as long as things
are reasonably linear.

Which, is a very, very big if.

Small variations are more often linear.
That's the whole point of

the calculus, the calculus is a linear
approximation of small changes.

But, um, larger systems
with more variation.

Be wary, they're generally not linear.

There's an expression
"non-linear mathematics."

Which is a bit like
'non-elephant biology.'

You shouldn't be defining everything else
by what it isn't.

When what it isn't, is so special.

Mm, yea, you look to me like
an unusual elephant

with some missing teeth, and going
round on two back legs

for some reason.
[audience laughs]

That's not really a good starting
point for saying what you do.

So there's other mathematics
than linear

but linear is very powerful,
particularly when variation

is reasonably small.

And, that is the whole idea

of Principal Component Analysis.

Managing the machinery of it,
I know people who spent

their entire lives doing nothing but
crunching those matrices.

But, there's technique.
But, you need the idea

of how it's all working. Okay?

So, let's try another thing.

How many of you have done linear programming?

Okay... what is the geometry
in linear programming?

Actually the first course I ever
gave back when I was a graduate student.

I was told, "Teach these economists
from this book."

And one of the things they were supposed
to learn was linear programming.

It was all matrices, and you pivot this,
and you shquiggle that.

Is that the kind of linear programming
you had?

Right? What the matrices do.

Yeaa, but what's really going on
is what the geometry is doing.

So, first of all what does this mean?

Suppose this was just 'X, Y, and Zed'.

That if I put equal to 0; that's a plane.

I'm positive on one side, and
negative on the other.

I say I got to be positive.

Oh, how would I define a cube?

I'd say I'd got something
positive on this side of the cube,

something positive on
that side of the cube,

something positive as I go down
from the top,

something that is positive
as I go up from the bottom.

With 6 inequalities I've got a cube.

Same idea in N-dimensions.

But 3 is plenty for thinking
about this one.

2d figures don't give the magic at all

of what you need to do.

But, the 3d problem does.

You see here is where one
of those limiting planes is.

Here's where another one meets.

So they're doing all kinds
of things like this.

Inside this polyhedron you're
satisfying all those constraints.

Go outside, cross any one of
those planes; you're not.

And, the algebra problem is
that the planes meet

in a whole lot of places.

It looks like a sort of weird,
3-dimensional, hedgehoggy thing.

This plane, and this plane,
they don't meet on the surface;

but they meet somewhere here.

So you want to make sure you're
staying inside this region.

So about 1950 comes the
'Simplex Method'

That confused, because simplex is a word

that mathematicians use differently.