the Q for you a production and
that you
all you hear
in this talk um talk uh we go but a lower be just completion a john magic approach
and the corresponding performance going
on this work
is a joint work with a come and and a an echo beach
both school as those from your you C
in the introduction hard
oh would discuss
what is missing
for the run could be just compression
once we figure out what is missing
we would like to to you the gas
however the the natural approach
does not a work at all
to address this problem we propose a new more
but john much come home
and based on this john can norm
we are able to obtain some strong them is going T
what to get to me for go image of completion
ms start waves
compress a compressed sensing
comes at the is taken on its that the hand of sparse signals
a signal X
a set to be case sparse if they are owning kate on the those in X
and being we perform you i'm and
why are to five X
uh the reconstruction problem is should be come from the eggs
based on our my and the back to one
the ninety but coach
but compressed sensing
it is to do every the also search
oh exhaustive search basically we
try all possible locations
for long rose
but i or all
but as a we know that the complexity is huge
so it was you search is not to go
to reduce the complexity
we can use a a one minimization of greedy i
lower run image is can be seen
is also a tech that the and a sparse signals
now this
signal
is a matrix
and the sparsity
is in the space
now the X
yeah and by and matrix
that's consider use
think low value of decomposition
if you could do you tend to ace them
we each has close
yeah a
is that and image case
containing
singular value
we say X has to right are you and only of they are exactly are mining
singular values
not they don't
has the stigma sparsity it's in the egg space
the lower an image of completion
is great are we assume that we do not know the order entries
we only know some of them
yeah of media
is the index set
of all of the of the of the entries
it's a me is the part of the vision
we would like to you for the missing entries
based um
the of the the entries
and the low rank structure
so mathematically we would like to find a estimate if hack such that the right
is at the most are
and we have the data consistency
because of the similarity be um between
come processing and a low of images is seem many methods
can be used it for example a one we have a why mean magician we we replace the
it when norm where is the nuclear more
which is just a
the L one norm of the signal as
well so can try some ah a can use then some greedy rooms
for comes sensing to the mitch is seen
a a second will however
the is uh one thing missing
in this pure
that you
L would all search
income sensing with screen we can to all possible locations for non the hours
we can do exhaustive search even though the compressed these huge
but he me to of in problem
as we were sure what is shortly
we are looking at
a complete space
it but it makes sense to talk about exhaustive search you a in this space at all
that means even do we can is then to and Y mean an and a greedy i them from comes
thing to low up average accomplishing we can not extend tend the all search
and a a this search is
the main topic
of
this top
to define a of those search are we need some definitions
is is U you are the set
containing all the matrix a
containing
exactly are are in also normal colours
for any given
matrix you
in this but to you in a market guess bad and are T in no subspace
in this to okay are we have to just you and you plan from from this but you uh and
you and R
and they is spent to different
subspace a
no
given a image to X
suppose all the columns of X
like in the subspace spanned by you
the in this span you can be viewed as a colour space
of eggs
and the right of X
it's added
uh use exactly are
with this can to a we are able to if why it would all search
is a point in given just you in this but to you M R
we look at of all possible six
generated from you
and which choose the wine
that it is more uh than most
cost isn't
with all possible of the visions
then the uh object function if a if you it depend as
this a out of will be as one
of the different
the lower the is are comforting
it then you couldn't to minimize this object function
on on you
as if you they don't
was as if if the real in
you means a we know that
the could the be meetings
has rock are
and
it is consistent and with all part of the vision
in this talk we'll focus on how to fly it this
global minimizer used star
we were lost talk about
under which conditions
the global mean meant is unique
and a K O i would like to as a fundamental question why we have about
L was your search for she's completion
as a first space because you know
in sensing in it the all search
is used this
but here what mission of completion problem
and we can see this may this site you but in you and R is that
can the space
it is actually a smooth manifold
so we are doing of them addition
all a sum was mine for
the come by C time know
actually a to our team can use your ins
would your search in most cases i would also so she can be finished in just twenty of P D
O fifty iterations
it just uh
the was station
uh i want to playing the details but a P a we look at a modified a it will new
search
and ah
the right the like he's
the performance of the modified it would also so she and the house a kobe is the to the performance
is
and you can see
the modified a L the also it the actually um P mining as in the are true
so the key message to pay i is that
for me completion problem they was search
mel be very good
pixel
okay
however
it just mentioned some ones of every this that's but
it does not what
why that's and card
um
a some example
recall that the that from thing is so whether probably small
it the can be written as
a sum of money atomic functions and each at coming function "'cause" the two
one column
of the a low of the vision
alright right
in this example we only look at the one column
we suppose that are we know that
second and is that and entry the for centuries miss
do we assume that we know the rank it's one
the for the comes space
can be friend
to to by a contract
and we are interested in
the comes space
um permit
part of it to by T in this
by by four
note that we have a one one here we have to keep hey
as long as T is nonzero
then
the object function
it would be don't
we can choose a probably as well that E
but
if a you personally L
no matter what probably we choose
observe function it's a two
i
that means
the object function defined in the problem is more
a is not continuous added to see if they
as we all know in the optimization problem if the objective function is not continuous
then we instill strap
in most cases we can not get any of them is currently
to address this problem
we propose a geometric objective function to replace
the previous work is more
to define the
oh and so much more
that's use the reason is that okay
we look and one column
and we even have to be the key and the subspace
spent about all the vectors
such that
the second and as the entries assume as our part of the vision
and we choose
the first entry actually
because we do not know the centre
so we can treat it actually
and we look at the subspace spanned by
this time
for given
column space
represented by you
we look at the
minimum principal angle between the subspace E
and the column space
they'll
i
what about the detailed
definition about the principal and go but
basically a principal and go
just a the and does
between two planes
and we only and at of the minimum present fine go
because
this and but you've to the know if and only if
to subspace
in the
non triple
now we are able to define the john mentioned function point to call we define
a is john mentioned function as sense well
overall all comes in the thought of this are coming
function
and that it was this
the of those search
problem becomes
minimize this
john match
object function
i to you if G
you was to deal
this much
from we we are give us many in S properties
but
it is can you know
yeah we simply to all the come tools
of the four B small
and the john at mall
and you can say
the probe is norm
a discontinuous
and as a region
well the job much norm its continuous
everywhere
more importantly we have the following zero
the set and the left
is the
it things
all the colour space ace
that a all magically
consistent
with all possible of the regions
yeah if a you put it
the set on a right
contains all that comes with bases
that are probably is
cost this and
with without that
part of the vision here if F you put a little
i'll rooms is that because
the for is small is not continuous
this set is not close
but
the level set
it is a closure
but with the right side
what does this mean is a means out john might function
can be viewed as a
some new supporting of the forty some more
up to a scale
is on this fact we are able to obtain some strong performance score and he's
what two scenarios
what's general
are run to one matrices
with up to re that thing happen
second as in not real for them any matrices with up to rewrite
for this tools
so on rows we are able to prove that
if we use a re didn't is in the mess it
to optimize to minimize of jeff from if G
then with a probability one
we are able to each
a global
minima
a first point i would like to mention that um
because you know a the object function is not a convex
but to
lyman than is
we are able to
a what are we are we are able to prove that there is no and the local minimum
oh set of all
second
what we see out of "'em" this drawn a performance guarantees are use one because
different from a standard to re
we do not require in queens
condition
oh and we dollars
how with the probably D one and the body after a image size
it does not require the image size is
so if you know a large
just
very improve fully close through the
a key ideas behind a to
so in a a a a new star be a global minima the
P are are we may have a much a global minimizer
the in which just choose one of them actually
because if a you the L
every
a a function should be uh you post to the zero at will
in this to to the what line it's uh
so we is that for the i-th column
the right the line it is a set of a just column
the
global minimizer of must lie in the section
of this one
not for P given um one than they choosing
can space
you "'cause" to by you the or we compute the we didn't
respect to every
a coming function
we project and then team weekend um
to the back to used are a you the deal
we are
it but to prove that this protection
is always nonnegative
if you but to the deal if and only if the
at time from if be do all right
know that
the overall all gradient
it the summation of the gradient and for every a functions
if we put down to the negative of of all weekend
to the fact that your stop minutes you the you know
then we are able to show that
this projection is also active
it it you but to the all if and only if
if a G you put the all that means
you the arrow is alright
a a you met
so
we do not have any local minimum
oh set up for
the greedy in you put it on price
we have already reach a global in
in summary
in this talk
the main has to the main message is
that in no so that each me it's
actually can be very good
for completion problem
and we propose a a that you go object from chain
to or of what is the technical details of difficulty ah a with the natural
formulation
and based on that we are able to
proof strong of and got he's for two special case is
we do not to weak well in with condition
our our problems and that's it has with probably to one
and a body
but a tree which to size
a to to work
we would like to prove some of similar results for
the more general okay
thank you for you
at
questions
like
are
actually to me
questions
for for this question is what's to prove but model
mm
because it is proved to one
and the second question is one regarding the performance can
what happens for example if you take only
one samples to match
okay
so on
first sparse was uh about the probability models basically will assume that a we only assume that
um
biz a we to all the in on the can space
we only assume assume that
as the E initial state we run a peak
a space
uniformly
on this
a a compact set on a all possible call
uniformly distributed
as the initial state
after that we just use optimization estimate message
so method is
what about we only have one them one and two of the from the matrix
as i as a mission
and we do not talk about the unique an yes
of the source of which and so in that case
actually we have a you a need to mining comes space that the can out of the day
that's white i the a given didn't machine before but
that's why a you can uh you can you you you look at the estimation we out
yeah
you can see it with a
then close
the number of them hoes either the by some more
and so we are able to
for
a column space
that match all of the vision
yeah
oh okay
uh oh that's about the um uh us space you in R
yeah um you have markham's consists of the spy of four
you know so the much is well as the columns are
or not
right
yeah
so much as
so
is just lose its is robust many many so
yeah yeah i i and emission algorithm i'm info photo he because or i wanna have any minute
so actually a a is you we back a comes space and that that's can space it's at and and
uh in the grassmann manifold
and all lined as is down
not
um you in R i is actually down on the grassmann manifold
but i just a a to those details
i'm sorry but yeah
or do so
it's it's about the runs was amount of use that to the images with the projects i
uh right
so you can define five your a fines is on the circle
yeah yeah
i
i was just wondering what the circuit to serialise my son
really
them or yeah yeah yeah observing the entire matrix
yeah exactly
i
okay two
this triggers but are
what is an antibody that's
we use the gradient descent
method are
the manifold then we would do fine
that right from space
that part is not true
because uh in terms of image accomplishing this it's you know we
we need to do anything
we we need to do nothing
because we have the ball under
i