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