Hopefully this title with this approach.

So I will first start by explaining
why I call their grandfather.

Or some how some people some people you

can of we live upon is C.C.
many promise you see

your family if it can be returned
now we form the into a correspond.

To a lot this hour we
just saw some palm so.

So here is a list of increasingly
different cocoanuts Palm.

Is suffering in their system.

And then the more difficult one
is suffering lean their programs

suffering Cherno call as Palm Africa and

the coerced when saying and
this is the most genuine sample from.

Saying.

Is you see if you check for example you
few can soft outby you can use your LP

software it is awfully nice system and
if you have our form for

sampling from the concave you can use
that to solve all the palm before.

And as you can see the running time is.

Basically re still don't know the answer
for even for suffering then this is

this is the most important part
among all those palm We didn't know.

Before the first free prom the one time
is roughly peachy in free end tool.

But the last one day's a huge gap so

you if you want to complain
you can complain to sent us.

Yeah.

One way to figure out of our
one of our ultimate goal of of

the whole area is carry stuff
all of them in and squared town.

OK.

You see you go you.

Can practice.

So I have to see impressive

many spots a linear system in fact
we can solve in the near term.

For linear pull where I am.

Yeah we can also solve them
by using your promise or

you just need to send ten to twenty man.

Paul.

We don't know the answer but
one way to do is to to create in the sand.

Today we will focus on the last
one how can we sample fast as so

this talk will half a company practice for
fear you I would tell you we

still don't know the answer Pettis I would
tell you yeah basically we can solve in.

So less with the palm.

So we want to solve this this
less a we have a Congress set

the goal is to sample a pond according to
uniform distribution on the Congress set.

And so tasty from way to assess that one
way is to is in there although setting

given the parts you can't pass to read
it upon the sets are a lot yes or loath.

Explicitly for
example confirmed by a point.

I would discuss both setting.

He is the Chris for you guys.

Bowing so.

Can you find now a way to reduce
the palm of sampling from the general.

To sample a comma set.

They say fifty percent chance I would
tell you they laid the answer at the end.

Sue but why we care about sampling so
as I already give you a lot on this

of the law with the law on this of calm
West Palm in some sense sampling is.

One of the most difficult cost palm but
they are so

many other reason for studying this for
example you if you look.

At this rate of software outby is
by far and you do a poem F.R.U.

more evolve you are possibly
O'Connor set by some contests.

And currently depressed way to
concert those comments when saying

yes by sample a.

Few feeling about try to minimize it
general commerce Fund saying why it's

the best for Ethical this way in terms
of our CO turns out the fantasy way is

to compare the sense of quality
of certain comm I sat and

computing center of gravity
again you involve sampling so

you in some sense this is
at the heart of the for

at this hour from four
different call West Palm and

also four am Al is where you imparting if
you have a model you want to sample our

calling to more though for
example when you do base in stuff.

Also this sample then give
is a basic cost for many.

Are pushing on call assets for example
you can use sampling to compare wore them

center of prophecy matrix diameter
many different type of are pushing.

And also you can use that to us or

to my sitting com West banded
are different type of the problem.

Because many of you guys
this method met to sing.

The last recession you sent this gives
you a way to understand call my set

I will discuss that more later.

But it seems those is a leaper abscess so
I will give you a practical problem why we

care about sampling so
imagine where every year is F.

we call the knee and then wet every year
is the rate of a certain chemical we as

Union of body then those chemical we are
asking will satisfy some mass conservation

or you also satisfy some upper
bound on on those weight and

they will form a pile tome so
you if you and

we can funnel that part by
by looking on the D.N.A..

If you want to assume they have part of
the human body there are how can you do

by sampling X.

will answer this your question on part of.

Why you want to seem they are part
is because you want to fund shark.

For example this picture is a different
type of chemical we are a party but

this is just a picture.

Human reality it looks more like this.

Basically for
example you look at your body

all the chemical we're seeing
including the cats bet their.

Sampling from a million sized.

But previously the largest part of we can
sample use a thousand die Mansingh so

use them Sam We still don't know
how to sample from this part.

And this is one one one reason why we want
to sample from a super super large scale.

In some sense this
problem lies in the sense

that is there wiring spectrum you
can't do it really really freely and

you can also with the way Patrick I
totally don't know how to do the last one.

I will I will focus more on this I.

So the less you are used to progress for
this poem.

So.

So far there always cause that thing.

Can I get the first hour off them
first time out of our phone and

over many years in time is going to.

Expand.

Soon.

Child to explain to you.

I will try to explain to you one
of the simplest Our from Paul walk

this child to explain what is the
difficulty of and I think those feeling.

In some sense we already know at
least in this all the course setting

in some sense we already know what type
of our form superior Automower on but

the main difficulty is to analyzing
the mixing come of such a process so

I will explain the simplest process for
sample thing is called Bach.

By the way or how or when is the N.

Three forty four forty.

OK could.

Somehow this talk is to our lawn so I will
do as when them sampled at the meter to K.

it half.

But anyway so yeah so
I was in Spain then the Bach hour for

me is kind of the most simple thing
you can humanity you sat there X.

and you want to go to
the next power walk and

you do you join up or and
then random sample points inside.

If the party is inside the car as that you
can check that by the membership local

then you go to that point otherwise
you sample again so this is.

This walk is well to fight for
and the set up for

example you can use staff or sample
laying on the interior of the set but

the poem years you if you for
your body have some part of that here

then you can you imagine when you do
sample then you mix in this region and

you left attached a gift this this
pattern that is where we are now.

And in some sense the one thing to
come out of this the parents are.

Meant to car chick a constant.

So the chick a constant is defiant this
way see you have for a comment set K.

and use just over on the SAT ass and

you compute the ratio of each
in the area of the pantry

over there warn them of the smaller
sign of the sats for example is here.

And you search over all the set and
take the minimum and

there is to check a constant for
example for this for

the more than we know
the minimum set is is just this.

He says you try to search over
the sestet the pantry areas as small

as possible compared to do all them for
example fall for

this this is because the area here
is more so the ratio is more.

And the.

Crossings.

Are firm like this so the mixing
time of the Paul awkward depends on

to Parmenter one year step
size you are taking for

the ball walk the second
part MATY is a constant.

For example it is set to check a car
that is where is more because you have

a bottom the here then
the missing time will be large.

Also if you take small steps I said in
the that the mixing time will be larger.

But then you may ask why don't we just
take a step size to be infinity yet

there were makes you mix in the one step
because you basically sample according to

do you found this interesting or

of the are cars that are only counting the
step of successful step so you if you set

up the doubt as large as possible then you
are simply tired out with the answer but

it is very difficult to implement this
step because this is the origin Paul.

So you can channel why you want to do
is you want to say less step sizes that

SAS probability of test
stably as constant.

One thing to know it is.

A constant Yes laugh I
am where when you if you

if you scale your set
the constant will change.

Also it can be operated more for
example imagine at two.

You can finger and he is one unit and
the wall then he is.

So took off line for an hour that you
can stand with you on the for Al.

To where we want to pound chicken cost of
a little more lies the body a little bit.

And one stand along with licensing is you.

You do a Lindy a chance on missing.

The.

CAR I said his identity.

One important consequence of this we are
more licensing is are we call those sets

is isotopic is means zero and we have
our identity co-heir and Smith chicks.

Turn So for any isotopic color set you can
take a step size waffly one of a square

and and so using this so if you picked
out is wonderful and apply the P.F.A.

slammer then you will get the mixing
Thomas and square over five square.

And you may ask How can you make
the chance on missing turns a.

Long way to do this this is
I can buy sample thing but

there's some way to avoid this chicken
parm So basically this is how we do so

in order to understand Paul Oct eighth
only wrong content he laughed is too too

chicken constant of isotopic
Congress set so days a famous by K.

our ass concert ticket consing
is at least some constant.

So if you are part this it would
translate to the ball case and square.

So they is tight.

Thank you.

Yeah.

And.

For hypercube the reason is because if
you're staff there is wonderful what and

then one of the speaking on any
direction you're only moved

roughly by one though for an and

because this is a random walk to move
from one punch you to and square step.

And so you if you have Care us.

Down we have to fight mixing tampon
of the bark and in fact people.

Suspect this is the best you can ever
hope for for the sample and calm for

the or CO setting.

Speaking is because.

In some sense you imagine to sample
a car set at least use it able to learn

the tricks of the car sat because
if you if the conversation

is where we lay all you some hit and over
to die messing then you can never find

the parts inside if you don't know the
Matrix and so they see in some sense and

square bits of information and
it's all we call cog if we want it.

And so is the best estimate for this so
currently to Bess's and to elect it if C.

or point two five.

We know a lot for this country or

we do is really try to push for
a stronger contour to sue.

Of funding the chickie constantly
pander law to get constant.

This is free to and
to elective zero point two five So for

a song you can't turn so the.

Lead to answer for the Palm.

Yet.

When we send care is important
because it is the first

to many contrary to your mathematics for
example one famous.

Is to slicing country to
base many other countries.

Care as equal friends to care as for

example one really Kuiken gesture I
really like is this imagine you have for

commerce set you randomness that
direction from the sphere you project

the you computer matching the on that
are saying you care one time messing.

And you want to ask why is that this
verse in the like where is the answer.

They slam with this you were saying and

the context is how close
is that to long with this.

And turns out that question
yes you could first to chaos.

Yeah soon after talking about some theory
I will go do some more practical for

now assuming someone know it
the proof the care ask contest so

we get the ball walk number for
you to sing is and square.

And because each step off
Paulk involve And and Tom so

the total one time of the Bach will
be NK and A is still too much for

four of of four for example sample from
your body of a million sized Partow so.

You can you do better than ten so usually
where and when you do if those problem one

thing you you child to may try
to avoid ALL course setting

because all we can set in is come kind of
give you a lower bound for the Palm so.

That the sample and from the point tell.

Now is that for all co we're
explicitly good friend M H A N The red

hippy we want to sample from
points inside a point so.

Yeah I forget to yeah so.

Basically the day some progress
on this Palm first by can.

Yeah.

They get into a sing M.

and and

each iteration you involve suffering
some involve matrix multiplication.

Yeah later on someone
improved a little bit.

Currently the best is AM and

two zero point six six if this is as
impressive at all because the total

running time is still really bad
you can still have the hope of.

Doing that for me and I'm answering but
but the funny thing is and

and I would just give you a picture or use
those out from the way you can sing Porky

is is each time you do is check
NY on the top you go away in the.

Small chick lie on your part
Is that what we do use to be

we sample from a random
curve on the part of.

Our those.

Look horrible but turn so
in practice is pretty good so.

Really type palm We tried to
sample from upon the most NG for

this you can think this roughly speaking
is a Bonomo thing because basically

just summing up a bunch of random
wearable beachy and acted on them one.

Of our days a typo here less than one.

Yeah so so you if you look at.

All of the existence of form.

Those free is a popular sampling
our form so walk here and

one then call and they hear one they
are led to take supercoach I think Tom.

Or cogitate Tom.

Our from in the last
life this woman turned so

you can proof for this part of
the running time yesterday near.

The experiment and the funny thing
is is lot just lean there for

those part of which I fall for

if every child falls standard
library on Alki this P.K.

we tend to one hundred times lower than
four for something alkie So you say.

Our form.

We hope their one time will
be closer to software LP.

And more interesting the most interesting
epic aspect of those hour for

me is those owl form only
to take many steps and

it's established just evolving
suffering some leanness system so

you if you have a supercomputer
you can paralyze the awning Tom So

you says in our sample in from
a billion size part of the is possible.

The problem is past if
the matrix A S past.

So yeah still so they turned
the minister left so they choice.

Him walk so and they went to scare us.

Hammock.

Chaos.

Yeah.

It's so because I talk about care so
they smile consequence.

Turns out care I see a sequence first
a man a man a Palm many of those formulas

really can tell you intuitive at at
the first moment you see the statement for

example to stay minister is you
look at I so top we remember

isotopic Lacan case this abusing me
in the dish abusing is means zero.

With call where is identity.

Because off this you should expand
the expectation of the alarm square is an.

Then you want to simmer the expectation
of Llambias or saw waffly school and.

Letter you want to ask
how about the variance.

Of the content he is asking
we expect this is all one.

This to me saying If you look at the topic

disabusing most of the war led me
is lying on the relief in shadow.

And the reset relay
this statement is more.

So Bass out often force offering
L.P.'s rely on this statement and

the commerce that is.

And.

If you maybe I just join
any car set in the is is

containing a you lives I guess I fit the
same M.P.'s same if you and thus the units

are by square annual counting their
whole lives are of course the magic for

symmetric commerce set and
then need a cause to their kids.

And and this statement is I'm
saying kind of looks like this but

this is the published text Damon is seeing
and the COM S.S.E.'s lot just look at

the top and the car I say is not just
look at you live by a square and

fatter is look like we're fatter
one pass one therefore what an.

So and the square and
year is roughly speaking how do I am

what I am come from also maybe it's also
how the panel for several seven come from.

First to you if you have some idea of
how to use testament to software out

there maybe you will get something better.

Yes or
Anyway days many are the contests about

equal friends conjecture about care as.

They are for example this one year is
a generalization of their few content

chasing defend I should
topic distributing and

then live just fine saying we want
to ask where did they live just for

seven years can send the mean
on the lock distributing.

Concept.

Sphere or something like this.

So you can figure is
a generalization of Chan Afghan or

cows in concentrating on those kind
of stuff and this one I imagine

if marching off can I say Look like I
was in you for a slice of Commerce as

if efficacy has the last lies
if we were Minkowski is tight if

the anchovy is of a lot concave disabusing
is a possum it by causing this abusing.

The independence of to look.

As if it is look like cows and
I guess banner it is a whole.

Equal friends contest for us.

This is the history.

First result they get
upon one over school and

by using Stand by using
the term mistake localizing But

this this contest is difficult enough
to people as we member chick accosting

is asking about you search all for
different type of set so

a simple question is how about if
you restrict your set is ellipse.

So you get any big turns out that Kondraty
is Excel it is the way in south lawn of X.

and people so
how to improve the square as.

People show how to do their patter but
the real point for

you is to surround by out that.

He basically what he is yeah
a surprise to only search for

the sets for a surprise to only
focus on the cut of the sphere.

He proofing fire you see
close to that county so

using the prefix by you get one for.

Basically there were all we do is just
slightly string from there on we get one

fourth this is the string for enough their
band we care and the constant is one for.

Ya currently resteal don't know why you.

Answer this but personally I believe
the carrots concert is true.

Despite the last we saw it.

The.

Main thing I will focus is just
explaining to our local guy sitting.

Down also you so how to use the F.

for our suffering different type of
question for example the last talk.

Before doing the stuff little I say sing I
live to explain the localization first so

this is a super cool them or.

If you don't remember anything from this
talk this is the only use to remember.

So so the statement is this
statement is really weird

this to me is this week different
to Qantas when saying F.

and G.

on our N..

We assumed.

They can be positive and

negative reaction there you take off
after you take off she is positive.

He shows is then you can fun though we

you can think this is a tiny needle or
you can fling is that.

What he says you can find.

Says that there you take off after in
this chunk a the Congress positive and

you take off G.

in this truncate the economy is positive.

Yeah just to explain the more so he it L.

to Amyas one is just a mere
of this this reaching.

And are you still enough and
saying so is that call.

And you are going to creating
a long two point eight M.P.

So it will be a chunky take home.

Take waiting there for a lawn this now
I with the waiting is out to the M.R.S.

one so you can free is just
a test among the small comb and

you are integrating the fencing on.

Why this is important you
imagine you want to prove.

If integrate off after his
positive day into a career of G.

cannot be positive then walk and
you do you just lead to prove this for

Effi leader.

So you say it with use proofing a year.

From end by measuring to one dimension so

I will give one example here so
probably still is.

Let's say we have a comma set where free
partitioning asked one as two as free as

is the meter.

Distance of as one as
two years is some P..

And also less a matter of the whole
conversation is kept at thirty and

the statement yes this.

The war Lemma of the media offering.

Is at least.

The distance of of the two
set over the time and

it comes to a small side of the wall them.

So you can think this this one
years suavely look like look.

Inequality but
the only difference is the chicken or

the chicken in the definition of chicken
constantly talk about the pantry area but

yourself talking we allow the punji
have some with the refuse T.

S.
one as two so

that this statement is more
general then the constant.

So you have this statement is
true then why you can so years.

It's so
warm basic facts about common assets is.

Different isotopic our set
you can chunky the set out

to outsized say to school and basically RW

only chunky explanatory small stuff and
you will up a fat chick a constant.

Most of what we want to do for
for one purpose.

Less A We already.

So the less is the common sense now have
the time it is less aid to school and.

Then applying the slammer you will get to
get counseling is lower bound by one of.

So so it turns out this this film can
be proved by directly applying the.

Localized.

Suit to a padded localise using them
at their first when you look to two

years convert this into a question about.

Doing integrating.

Namely your little concept the fencing so

the first thing you can
say is this let's say see.

We want to proof the deal for a capital T.

then you can set the fencing Afy is yes
you can shift the fencing affianced C.

on one side like to run Yes and.

Then you can check this same as the in
take off after large and C.E.O..

And similar destiny Carty is same
as in take off the largest you.

We want to proof by contradiction so we
assume or Lamar asked for years last and

while I'm off as one
as to buy some factor.

And then we convert this
two to two to spoofing.

Of.

Disproving this.

Then what Lex then now we can use local.

Local I say sing them a basically asked
us to disprove this statement for

every for every though but
why is Testament for

a leader who is saying is saying
is asking us to put this lemma but

on the Lido on the one by missing
comma set and we are no and

that mess in crisis is the interval
the same in skillet two for you before.

The hour so this finished the proof you
are in the question about this proof.

The Sudar we will go through the proof or
local licensing.

Speaking the proof is this.

You carry forth those you heard
that by your lips are meant for

you care for is doing those in lives.

So the hour is caught in
some sad concept to proof

you try to maintain a reaching
sestet a unicorn is positive for

power for F.N.C. you start with
the reaching is where we were a lot.

You can start with a super large for.

And while you do is roughly
speaking like this.

And then you look at the center of gravity
of of the reaching you are maintaining.

And now with the fire half Space X.

Faith A is one side of this half space.

And now we defy affray to you is.

Do you take off after on on the half
space intercept with the car set.

So one thing to notice is effete or

purse after like it if it is
the whole into go because one is

take on this side one use and as I
still so some is just a whole into go.

Because we know this is.

Because we know that this is positive.

One thing you can do is you try to walk
after half space until you hit a point

Boethius.

The reason why you want
to fund their half space

prophecy code is because
now Kofi is positive.

And they're lecturing you know if this is
on one of the side takeoff she must be

positive because some way off on both side
is positive so now you can recurse on

one of the half space and eventually
the statue becomes smaller and smaller.

Yeah I didn't tell you too much
detail where you can imagine

there was then becomes
more than a small one for

you really to proof he is
the object converts to a leader.

You can prove by this thing.

In this post a sea of man the freedom
of choice we are there only thing we

pick is center of poverty we pick a two
dimensional space to concept those have

space so you some sense you can pick two
dimensions of space istead the commerce

at level we do is to a point because
you can avoid a certain time answering.

But anyway so
to end the way I like to forget about the.

How of yes I'm too slow.

Yeah WAY too for

you know about the local licensing
them I used to say I really like.

Imagining this is how awful all.

Satisfy some you Carty.

Then turn so
the local guy sitting there my proof or

the extreme point of this car set
is I doubt that fencing or X.

will lasso fencing on
on seventy into four.

And why this is useful for
example if we want to.

If we want to minimize any concave
fencing on the commerce hat

then we know that Susan must rely on the
extreme point for example you if you want

to optimize a linear friends in our part
of we know the R.C. is on the word test so

you say it shows if you want to
minimize a quantity like those

there you'd only to check for
fun Tsingtao to that doesn't fencing or X.

will answer fencing.

So those.

Here are.

OK as I only have one minute so

I will only explain to you why
we expand we can do better to

experience this so in some sense
you can frame what we're doing for

local licensing them is this We tried to
put this inequality and we don't know

how to proof and the way we do it this
is by keep cutting your car stat.

And the problem year is in the worst
case you can't get the car set and

suddenly you've become the leader and
foreign leader

we know the chicken casting is one of
the T.V. and this is why the one of the T.

come up.

But but
one thing to notice is in the local that.

You have a lot of freedom how you choose
to dimension picking paying to do the cut

so one thing to do is you randomly pick
two dimensions of space and then try

to cut the body and then in that case used
to express the process look more like this

is of ecommerce to a leader and
this is basically what we do.

But the palm of two doing this process
this process is where we discrete

difficult to analyze so
out on to is basically

the inchoate there is a more smooth
worsening of cutting a calmness that So

the way this see here this is still.

Going on one time matching and
we are in the standard

localization while we do use
recap in the meter but instead.

Two years you multiply your
lock concave disabusing by X.

Ballina fencing at that into your.

So you can simply Finian
this this is the always.

The next step involve multiplying P.

with one pass some tiny linear fencing.

So for example for this one then
the next step will look like this.

And if you take the steps to see zero
then you will get the star classic.

You questioning.

This is basically how you avoid.

How you try to analyze.

When them local licensing.

Our form is roughly look at
this this is a car set and

if you keep when I'm cutting
distribution we are all.

At the end Co saying closer
to a lot more disabusing.

Turn So one nice thing is all we member or

witching localizing ever test to
Aledo before the stock localizing.

Worse to a lot more dispiriting and

they're goofing us we know how to
put care as far along with this.

Disabusing basically that this is how
the fear done Yeah scared to approve.

The I knew I would just stop.

Stayman I'm not going to come to the pool.

So.

They are recently really improve
their slowly for Carty Thank you.

Thank.

You.

For that I can then say
Harvey infanticide.