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