no

to that i'm not going to talk about

that a problem and that inverse problem

uh i'm actually look at the much simpler problem

but i have a perfect set

i know exactly but am looking for

i put the device in the water

okay

and then i'm looking for nothing

basically i one to reconstruct

the what

okay

um

so if i you

if i do it before sold december paul inverse problem but they're we'll get its this

pretty shocking right

it's like finding can walter

um

in the

if you look

more careful that if picture of what you see

is that with a trace

from green to blue

we show a basically says that

this a in the time of flight

probably if we remove

this are you know these delay of time of flight we can get

to the correct picture do so

and what to get

is not quite

that effect

is a lot of uh fluctuations run sense

and that this file suggest that the positions

um

or not correct we have to

um estimate this position

so or two is to go row uh about it we can send was back to the manufacturer

ask them

to put them correctly

on the circle

and um you know it a distance

and that they would probably say well uh you know

uh on the piece of paper ever these simple but guess what these a physical devices

is the best it can

so we are

start but these do watch

okay

and then the goal is to find

the positions

all of these sense

um

uh so

how can we do that

um

if you are you know

homogeneous medium

do the simple religion should be didn't time of points and the pay was the senses

basically

um

through this a constant C zero

so if you know the time of flight

you know the there was sense

and what do so than we talk about their what distances

um because

there is uh

very nice there i mean out to a nine john you that says

if you put and

a a points

on the surface

and if you for this particular metrics

distance squared metrics

which is basically

the pairwise distances raised to the square

that's all only the rank of this matrix is going to be for

independent of N

it is very easy to actually prove this is always like to three long

but um so we're not result how can we use it to is another celebrate celebrated

algorithm called multi dimensional scaling

which basically through this uh

matrix L

if you put it on the left and right hand side of this and scored matrix

uh applied

no single value decomposition can find the exact positions

of the sense

so uh

when i say that you can find exact positions what i really mean

is that a can find them all to

rigid transformation

basically

if you only have a pairwise was distances

there is no difference between your topology

and the one that is reflected

okay

well the one that is translate there

or

the one that is rotated

so um

it seems that the problem is solved

because

um

i know the time of flight

i can't D do use the um

the pair was is

a can use M the S

i find a position

actually continue stock

well apparently um i should

and or a couple of challenges head of first

which prevents also

from using go

this met

okay

first thing all you know

the or in you know and

oh joint to do signal processing and in signal processing not think he's less

so well time of flight are actually not

that the first

and source of

certainty

that's not

actually

the big deal because um

well and the F is actually a robust against a a not

but the more interesting ones are come

the second

source of on L certain certainty actually structured missing in trees

and what happens here is that

when the transmitter here a signal

the ones

which are in its vicinity can not hear the signal

this is one of the limitations of this what

so

all of these red lines are going to be

you right

we don't have this information

if i put it in a mid trick for

well you know what have this use metrics

two hundred back to hundred so i can up put actually

uh you know numbers here

it to take all my

so instead i put

colours here

is top you know these numbers

so on the left hand side

basically what you know what we should

on the right hand thought however

the um

the central band and these two corners

are going to be race and we put zero because we don't know

the value

another source of uncertainty is actually what we call called right and the missing singing tree

what happens is that

if you plot the time of flight

with the respect to the sensor in texas

what you should get

um is this news mandy four

but

what you will get

uh

we'll have

you know a couple of sparks which do not make any sense

and these are basically down years

you have to discard them

so you put zero again here

no um

if i you list the game means that some almost

um

pair was these sense are going to be a randomly this on

well we seen that are going to be a riesz random of like the model let B C

okay

now um

so we all left be such a picture

you no longer have all the pair was this

and if that's not enough

uh well basically you know

well in a

okay

oh you have to wait

um

so in a matrix form

what it means

is that you know you will have a couple of dots

you know randomly

as yet or

you're

and if that's not enough

well you have these on known like that in the beginning of the talk i uh you know

oh i i mention

so well the reason here is that um but these are electronic get

right

and uh

oh when you fired the transmitter

it's not going to transmit the signal immediately each weights uh for a couple of seconds

mark second sir

and that but we don't know this V so we have to estimate these as well

okay

um

you know just to uh less first um

sort with these um

a missing entries forget about this that

um shift

and time of flight

um

um so we had

these amazing result that this stance squared matrix was ranked for we could use M T S

we can no longer use it because may of these that

there was these sense are missing

now

uh the question is there were not we can estimate these missing in tricks

well uh

this is actually a topic of matrix completion that had recently you know there are a lot of risk uh

that's been lot of activities in the past to years

and the question is

um you know pretty five this state here

we have

a rank K matrix of dimension and by and

some of the injuries are random me

and then it turns out that on the the road conditions you can actually

find the missing it

um the one uh so

right now they are you know a lot of uh and is out

when we started this work the a couple of them

so um

we actually

use um

of the space um um

the develop point want to now already and his students at stand for

and the way that uh

um this algorithm works

is basically by projecting

the uh at the metrics on the space of rank

Q mattresses

and then uh doing some kind of great great in these

now um

do is a catch

and them in all these out algorithms

that i know

um you have to use you

that the in trees or

you raise randomly

okay

um so probably do true before i guess

um i

you know probably you know from of for this problem

but uh well to the best of my knowledge

this was the case

and now um so

but as i mentioned we have this structured missing trees

these are in trees that we know we will never get

a any

observations about

right so

um i to space is not going to work for

as a T

um

so we have to redevelop develop again these all the space to make sure that

a a when if we have a structure missing trees

this is going to work

so before like a you know all these error bounds for

uh for the classical a to space is no use for a

um um and um well the theory a a is actually quite simple and you can find it in you

know paper

um i'm not going to bore you with that you know details of the proof sets order sartre

but uh let me just mention

you know the model that use

so we no longer assume that the sensors are are actually say sensor circle

you seem that the R

uh a on these and we

we

uh be a

and the way that we are going to capture the structure missing trees are great are are

but a bit through the use um

uh a a and with that if

there's the transmitter here

all the sensor

um in flight three kill or not going to your anything

okay

so

if the sensors

uh or

you know distributed uniformly at random you sound was

and if you see assume

that um hmmm the time of lights are going to be a random with probability P

fix number

and for the structure

uh missing trees if we assume that the are fine is going to scale

like school root of log over and

then our our or and reads as follows

that

the

distance

between the

um

this squared matrix and in its estimate is going to be bound but boy these two true

um

uh what we should mention is that

um

we we did a assume anything about the noise

so the noise can be deterministic

random

um you know you name

so these bound whole

you need full generality

um

the all that thing that they should mention is that

this term goes to zero as an goes to infinity all everyone we control controlled easter

in many many cases

it goes to zero

but i'm pretty sure you can come up with example take doesn't

for instance

for go in noise

uh uh that are

now a prior we were not

you know interested in a find a distance score metrics what you wanted to do side you know finding to

positions

but as i said we can find the positions of to transformation

and

we have to make sure that you know what and uh we we we basically one it

uh

uh

one to define the distance between the estimate

and the right one

in a way that doesn't depend on the rigid transformation it should be in

it turns out the right way to do it is basically

these four

um which is in barrie

on the different formation and

it's going to be zero these

diff the ins distance

even all if E X you equals X i

now

um if we apply

and the S

after

oh the space

uh we can actually bounded if then

basically the same rate that B D before

is going to be it's same expression

okay

no uh for the uh

so we had another other source

of uncertainty which was these to like these constant that to have to measure

here we assume that is

going to be

uh for every want for every transmitter is going to be say

okay

now

a there is going to be a need to about it and that's fine

these um this T zero

um

but the for the sake of them are not be true the details of these out with M

uh what they should mention is that is probably is nonconvex

so um i it's very difficult to find

uh you know to you actually out prove the convergence

uh we have if you're if the with and it converges numerically or we don't have any pro

uh and the idea is again to use this property of the these sense square metrics metric is rank for

wanna make sure that you know what we're fine is actually going to be as close as possible to the

right form

oh okay

so

um

unofficially we had access to real data what

um

oh i cannot report these these you know

these data as here so what we did is just some simulations that maybe the characteristic of uh the real

data

and then uh

um

a diffuse basically you know well what you still before it uh

you know the the a or is going to be in the twenties and D meter is the number of

them or two hundred and then

uh

the deviation is going to be half from that are is does that was the D is going to be

D the metrics

to real matrix if this is going to what

to to be able to have

and the

if the you

um you know you actually do well our them

uh a fee that there are going to be a lot of deviations

uh

um

from the from the circle so

if you got yeah that this is the prince function that it have

that all of these sense of are going to be on the thing bill but if we want that are

with them see that they are not going to be

a you know they're got not be to be place exactly the same

so

this the last

um

if you the the picture that we started that of

if we actually

remove the uh

do you lace

you know these constant you'll lay set we have to find out

we'll get if speech or are you it before

if we complete the distance a a three

with a a a space

you'll get is picture

it's not very different from the previous picture

a but if you find the positions and then you know a sold the inverse problem

you get back into one

the

it's really important to calibrate the system is really important to find

position

and the

even if all you know beforehand the the um

the range was from

uh one thousand uh

four hundred to one thousand six hundred before the close

from

you know these value this one

then you don't see any deviation a

so it eight towards um

you

yeah thank you very much and i'll be have to answer questions if true or german

thank you for this okay and

we have time for one question

please

i

um we just some on the might like best aging

and and was like given a set up and you make get that yeah so you don and i one

writing differentiation between you time at like measurement

and and just an eight station at the sense that the "'cause" what we found like if a code is

getting good trying to fight management

it "'cause" i things like multiple and more fundamental fashion

um so i actually the you're not have a seen the time of flights measurements so these there is this

yeah

these guys they have these estimators for the time of flight

right

and we seen that what were they you know what where we got from the is actually correct

as a how likely as i think you know

uh uh what is the and that kind of flight measurements

i i get to get good ones because you got a dispersive medium

um no i actually don't know the yours

K can i think you it seems out

okay so let's move to the the second work

no the to not be in me