-
Now that we have constructed Hilbert
spaces and ordinal
-
bases we can see some of their
distinguishing features.
-
One of them is norm conservation which is
-
called Parseval's theorem and is an
extension of Peter
-
[INAUDIBLE]
-
famous ordinality theorem to infinite
dimensions.
-
Then we will see the orthogonal projection
theorem.
-
This is a powerful method to take a vector
from
-
possibly an infinite dimensional space,
and project it onto a subspace.
-
Once we have this, we will consider
-
some examples of approximations and
othonormal basis.
-
Module 3.3 Hilbert Space and
approximation.
-
Your review of this sub module as
following:
-
first I'm going to talk about the norm
conservation, when
-
we have expansion into orthonormal basis,
and a
-
so-called equivalence formula, which is
known as Parseval's formula.
-
Then we talk about approximation by
projection, and we give an example.
-
Parseval's theorem has a very simple form.
-
It says that if you expand x in an
orthonormal basis, so we have a vector
-
in a Hilbert space, we expand it in an
orthonormal basis with this vector w.
-
Here, we are on the finite dimensional
case of dimension capital K.
-
Then, for orthonormal basis, the square
norm of x
-
is equal to the sum of squared
coefficients, okay?
-
This of course a generalization of
Pythagorean theorem, which says that if I
-
have the vector here, this is x, this
-
is, so length in an orthonormal basis or,
that would be alpha 0
-
in e 0, alpha with respect to
-
e1, then x square is equal to alpha
-
0 square plus alpha 1 square.
Okay?
-
Very old result, very useful result.
Let's actually verify it algebraically.
-
We start with the canonical basis, e0 and
e1, just as before.
-
We write x as a linear combination of e0
and e1, and we go to a new basis.
-
The new
-
basis is v0 and v1.
It' s an autonomal basis.
-
Okay.
-
You can verify that these guys are at the
right angle.
-
And this basis is given by v0 is cosine
-
theta sine theta v1 is minus sine theta
cosine theta.
-
You can see this here on the projections.
-
And so x in the new basis is equal to beta
0 v 0.
-
Vector 1,
-
V1.
-
What are the expressions of this, well we
know this because the
-
basis is orthonormal so the expansion
coefficients beta 0 is equal to
-
[UNKNOWN]
-
product between V0 and X beta 1 the inner
product between V1 and X.
-
Or in compact form.
-
We can just write these inner products as
row, column, scaler products.
-
OK, so the the zero is cosine theta.
Sin theta d one is
-
minus sin theta cos theta as we have seen.
And so we write this as R
-
times alpha where R is the rotation
matrix.
-
Okay?
-
A unitary matrix that corresponds to these
basis vectors.
-
Alright, and it's not hard to verify that
the rotation matrix times this transpose
-
is equal to identity, meaning it is equal
through transposition to its inverse.
-
So it's a unitary matrix, as we probably
well know by now.
-
Okay.
So let's look look at norm conservation.
-
So the square norm in the canonical basis
is just as we announced,
-
so x squared is equal to alpha 0 squared
plus alpha 1 squared.
-
The square norm in the rotated basis is
the same, but with respect to beta.
-
Okay.
-
So we are going to show that these two
things are the same.
-
Or, verify Parseval's formula.
-
Well, one way to write beta 0 squared plus
beta 1 squared
-
is to say it's a vector beta transpose
times beta, right, okay.
-
I'm just, so I'm making myself absolutely
clear It's
-
the scalar product of the vector beta,
okay, with itself.
-
Okay.
-
That's the thing here.
-
Now, of course, beta is equal to R times
alpha.
-
Beta transpose is R times alpha transpose.
-
You go to the extra step of reordering
here the products, so
-
the alpha transpose comes in front, R
transpose here times R times alpha.
-
This of course simplified to identity that
we know because R is a unitary matrix.
-
So this is equal to alpha transposed as
alpha.
-
And of course we verify what we set out to
do, okay.
-
Now we did this in two dimensions,
-
it's obvious that this will hold in n
dimensions for an arbitrary n.
-
It turns out it also tell it also holds
for infinite dimensional canonical basis.
-
Okay, so that's Parseval's formula, very
important formula in signal processing.
-
Okay, what it really means is that if you
have a vector X,
-
and you look at this vector in, you know,
this basis,
-
and you look at in some other basis, as we
just did, which is a rotation, because all
-
[UNKNOWN]
-
basis are essentially rotations of each
other and maybe
-
some symmetry, then the length of the
vector doesn't change.
-
Okay, so that's of course norm
conservational so means distance
-
conservation through these transforms or
-
through the expansion into orthonormal
basis.
-
Alright.
-
The next step we want to do is
approximation.
-
We had briefly mentioned this at the
beginning of this module.
-
So, we have a vector in R3 here.
-
And, the vector X should be represented in
a subspace spanned by e0 and e1.
-
Well, that's a subspace S of V.
Spanned by e0 e1,
-
now shown in red, and the approximation is
x hat.
-
It belongs to S, and it's the orthogonal
projection
-
x to the plane spanned by e0 and e1.
-
Okay?
-
And that orthonormal projection we'll
denote by x hat.
-
How can we do this?
It is very simple.
-
We take an orthonormal basis for the
subspace.
-
Okay, so remember we have the big space V.
-
WE have the subspace c.
-
So we take an orthonormal basis for the
subspace and
-
the orthogonal projection is simply going
to be given by.
-
X hat expanded in the ordinal basis that
spans the subspace, Okay?
-
So the set of vectors sk, is the
orthogonal basis for
-
s and here is the expansion form well in
this ordinal basis.
-
This orthogonal projection is the best
approximation.
-
Over S to the vector x.
And it's best in the l
-
2 sense, or, it will minimize the
quadratic norm of the error.
-
Okay.
-
So in a word, orthogonal projection has
minimum-norm error.
-
So.
-
Among all vectors y that belong to s, the
one that minimizes
-
the square of the difference here, so the
square norm of the
-
difference x minus y is this vector x hat
that we have
-
written out in terms of an ordinal basis
for the subspace s.
-
Very important property is that the error
x
-
minus x hat is orthogonal to the
approximation, okay?
-
So we'll see it in the next slide, but
this is an extremely important formula.
-
It's called the orthogonality principle.
It is used all over signal processing.
-
When we minimize quadratic error.
All right.
-
So let's see this very graphically; we
have
-
s a subspace, this guy is subspace, so V,
V is the ambient space
-
in this particle case R2 as is R1, it's a
one dimensional subspace.
-
Okay.
-
So we want to find the closest point to x.
So x is what we want to approximate.
-
We'd like to find the closest point
somewhere here in S.
-
Okay?
-
How do we do this?
It has to be closest in the 2
-
norm, so we create a circle around the tip
here of x, and we grow the circle.
-
Okay.
-
We are still not at s.
-
And at some point, we reach s.
-
This is cl-, either it is a closest one,
right?
-
Because the circles measure quadratic
distance.
-
And the first time we hit s, it is exactly
here.
-
Okay.
So we have x-hat in blue.
-
And we notice that x minus x-hat, the red
vector is orthogonal to the blue vector.
-
That's your
-
[INAUDIBLE]
-
principle we have seen on the previous
slide.
-
Let's look at a very concrete example.
It's polynomial approximation.
-
So we go back to our favorite interval, -1
to 1.
-
Okay, so we will get these guys, and for
this
-
interval, we define.
As subspace pn minus one to one
-
which are polynomials up to degree n minus
one on the interval minus one two,
-
okay?
So a basis for this is simply to take
-
as the successive monomials tk for k going
from zero to capital n minus one.
-
Okay, so a naive basis here.
-
Is really 1, t, t squared, t cubed,
etcetera.
-
Okay.
-
Now, these guys are not orthonormal to
each other.
-
Okay?
-
So this naive basis is not orthogonal.
-
Okay, which is self-evident, because, for
example, On the
-
interval -1 to 1.
Let's take this interval.
-
We have the first element.
-
That's this guy.
Okay.
-
The second guy, he's orthogonal, because
the first one was symmetric, the second
-
one is antisymmetric.
But the third guy is a quadratic function.
-
And it's not properly scaled.
-
I apologize.
But the quadratic function,
-
of course, is also symmetric.
So for example, the inner product between
-
1 and t square on the interval minus 1 to
1 is not equal to 0.
-
Okay?
-
Please check this more explicitly if
-
you like, but it's fairly geometrically
evident.
-
Okay, now we're are going to try to
approximate
-
something that does not live on the
polynomial space.
-
So that would be for
-
example trigonometric functions.
-
So take x, what we are going to
approximate as
-
sin t over minus the interval minus one to
one.
-
And we would like to approximate it on p
3.
-
So up to polynomials of a third degree
-
Okay, so the way to do it is, we build an
orthonormal basis from the naive basis.
-
We project x over the orthonormal basis.
-
We compute the approximation error.
Okay?
-
So same thing as usual.
So S.
-
Here is this P3 space.
-
Our sin is somewhere out there, and we
want to compute this.
-
To do this, we first construct an ordinal
basis for the sub spaces.
-
Okay.
-
We can compare this to the
-
naive approximation, which would be Taylor
approximation.
-
It's well known, very useful, but it's not
optimal, as we will see in just a minute.
-
Okay, so from the naive basis, remember,
we have the naive basis, it's 1,
-
t, T square, t cube etc.
-
We can compute so that's a viral basis we
compute an orthogonal basis.
-
There is a procedure to do this which is
called the Gram-Schmidt algorithm, okay.
-
You explain in the appendix of this
lecture, we
-
are not going to do it in the main
lecture.
-
And it's a recursive procedure where you
take the first one You
-
normalize it.
That's fine.
-
You take the second one, and you make sure
it's ordinal to the first one.
-
You normalize it, and you keep going.
-
And the result of this is that you get
ordinal null
-
vectors, u 0, which is just a scaled
version of 1.
-
The second one, these two ordinals, you
don't
-
have to change much except for the
scaling.
-
The third one is a transformation of
t-square.
-
T-square,
-
I mean, like this, more explicit,
t-square.
-
Into, you know, a second order point on u,
which
-
is constructed such that u2 is orthogonal
to u0 and u1.
-
And you can keep going like this.
It's a standard construction.
-
It's called Legendre polynomials.
-
And just from the name, you know this is
-
a 19th century construction, so it's
extremely well known.
-
And the appendix gives the details.
Okay, so now we have an
-
ordinal basis for the subspace.
-
And, let's just watch these Legendre's
polynomials, they're sort of cute,
-
so the first one remember, it's, 1 over
square root of 2.
-
So here we go.
That's 0.7 something.
-
It's the black line.
-
The second one is proportional to t, but
it has been scaled.
-
The third
-
one, which is quadratic, has been moved
about.
-
So now it is actually orthogonal to
-
It's automatically, the yellow color is
automatically
-
orthogonal to the red curve because one
-
is symmetrics the other one is
anti-symmetric.
-
But the shift that was added if we go back
here, let me just show you if I
-
shift here, this shift makes sure that the
inner product between the yellow curve and
-
the black curve is zero.
Okay?
-
And it's a fourth, is a third degree
polynomial, is the green guy.
-
It is antisymmetric, so it will be
automatically orthogonal to the black one.
-
The yellow one, but it has to be adjusted
so that is orthogonal to the red one.
-
And we can keep going.
-
Okay, so that's a four story guy.
Four order polynomial, and so on.
-
So Legendre polynomials go off to
infinity, but
-
we'll just look at a few of them.
-
Okay?
-
Here's a phase one.
-
And it's a very cute picture.
-
And this set of polynomials on this
interval
-
-1 to 1, okay, they're important on this
interval.
-
It's defined in such, it's constructed
actually
-
in such a way that the inner product
-
of two of these functions is equal to
delta i minus j.
-
So it's equal to zero when i is different
from.
-
j, and it's equal to one when, i is equal
to j, okay?
-
It's not self-evident when you look at
the, at the functions
-
except for the symmetries that I pointed
out, a minute ago.
-
Alright.
-
So now we can compute
-
our expansion coefficients.
Remember, we want to write the formula
-
where x hat is going to be some sum of
alpha-k.
-
u k we call these guys here, and k goes
from 0 to capital K minus 1.
-
So that's the orthogonal projection onto
the subspace spanned by the u k.
-
Alright, so we have to take the integrals
between -1,
-
1.
-
Of the function, these polynomials, we
have just defined
-
[UNKNOWN]
-
polynomials and the function we want to
approximate sine t.
-
Okay.
-
So sin is of course an odd function as we
know
-
as so alpha zero is going to be
automatically equal to zero.
-
Alpha one is going to be different from
zero,
-
because both u one and sine are odd
functions.
-
So that's what you get if you
-
do the integral, 0.73 something.
-
And the third coefficient, alpha two, is
also
-
equal to zero, because this character here
is.
-
Symmetric, and this one is antisymmetric,
so it's automatically equal to 0.
-
Okay, so what do we get?
So, when we do the orthogonal projection
-
on these three basis vectors,
-
so, u0, u1, u2, we get an approximation,
which is
-
simply alpha 1, u1 And it's given by this
formula.
-
If we do Taylor series, then the first
order approximation of
-
Taylor series simply to take sin t equal
to t, okay.
-
And so we're going to compare these two
approximates, they
-
look very similar but one has been scaled
a little
-
bit, okay.
Alright.
-
So now we see that the approximation of
sine, which is the blue curve, the smooth
-
blue curve here, t is the red curve, and
green is simply a scaled version.
-
Doesn't look like a big deal.
-
It's, you know, 10% smaller, but you can
immediately see that it's
-
actually hugging the blue curve more
closely over the interval -1 to 1,
-
right?
-
So we are approximating this over this
interval.
-
If we change the interval So, our
approximation would look different.
-
Okay?
-
But over this interval, as we can see
-
here, we plugged the absolute value of the
difference.
-
The red one is sin t minus t.
-
Okay, it goes off quite a bit at the end
intervals here.
-
It's very nice in the region around zero
region, okay.
-
And we see that the
-
green approximation which is sin t minus
our projection onto
-
the subspace of the legorn-, the legendre
polynomials of orders zero
-
one and two That error is overall, it is
smaller.
-
Never goes goes off to these values, okay.
-
Sometimes it's bigger, but overall, it
actually turns out to be smaller, okay?
-
So, to compare this numerically, we can
compute the norm of
-
sine t minus alpha 1 t, and it's 0.0337 In
Taylor
-
serie, it's almost three times bigger.
It's 0.08.
-
Necessarily we have to be as good or
-
better than Taylor series because it's a
theorem.
-
It's the orthogonal
-
projection theorem we find the minimum
norm approximation.
-
Okay?
-
And with this, we have Showing on a very
practical
-
example how to do orthogonal approximation
using an orthogonal basis.
-
Now, this was all a lot of work.
-
We defined Hilbert spaces, we had a lot of
definitions, and so on.
-
So why do we do all this?
It's a great question.
-
So, first is at both finite-length and
periodic signals live in C N.
-
So we can use all of linear algebra and
-
all the geometry of linear algebra to do
this.
-
And
-
Infinite-length signals, that we like for
general signal processing live in
-
a more general Hilbert space, which is
small l2 of Z.
-
Okay so we have a common geometric frame
work for
-
both finite lengths, periodic signals and
infinite length sequences, okay.
-
So we have one way to think about the
whole bunch of different problems, okay.
-
Then we'll see that the expansion into
orthogonal
-
bases is very central to signal
processing.
-
So em, we can use different bases.
-
As different observation tools for
signals.
-
We're going to see something called the
Short
-
Time Fourier Transform, to be defined in
Module 4.
-
And the Short Time Fourier Transform as
the second half says, is
-
something like a Fourier transform, but
it's a very particular way to look
-
at signals.
It's very popular.
-
For doing speech analysis and the like.
Okay?
-
And when we do subspace projections, we
will
-
see that we can do filtering, which will
-
be explained, of course, in detail, in
this
-
class, and we can do, for example, image
compression.
-
Okay.
-
So the notions we have seen, one was to
build bases, that's important.
-
These are like our
-
tools to look at signals.
-
Okay, and another one was subspace
projection which was
-
something that will come a lot when we do
approximation.
-
And in particular when we do compression.
-
Okay, let me just finish this properly
here.
-
We have the origin, we have a sub space s,
we have
-
x and we have the orthogonal projection
and we have the orthogonality principle.
Claude Almansi
Revision 1 = provided subtitles
Revision 2 = same, but with title added
Claude Almansi
Revision 1 = provided subtitles
Revision 2 = same, but with title added