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