they don't uh uh and i then you are there oh i right and see uh you know i be overwhelming majority work has has a a they'll with construct a matrix that are sub gaussian it's are made are gaussian or or a a binary and in some some fashion or or or or partial uh for matrix and and the like so that special export here is uh thank you so the but the the question away is what happens if that the the matrices that we using compressive sensing uh are not so cows and there some uh uh interesting a statistical and and uh a geometrical characteristics that but it will in march out of of a of of those uh prop so we will the first point we will motivate the we will uh some right of characteristics when these matrices are are not so gaussian uh we will talk about stable run process that are to one class of four or for all variables that are are not sub gaussian in the uh we'll will go for some uh rap equivalent characteristics of a piece of compressive measurements and show some computer simulation um so in compressive sensing the uh fundamental property the alright you the right uh uh shows how these dimensional reduction that is achieved when you look pressure upper side information in space right in there and several people in this uh so i should have talked about uh information per server uh_huh the different space and uh technical sub gaussian the uh measurements you got you have these uh i some a three you have the a some entry let you go from uh the L two space and when you of the projections you're are you have it's as a matter of preservation on L to specs as well um but i has to be uh gaussian or or something like that uh a chart tried of and the calling of is uh a generalized this idea to say okay well uh how modified look at this as a sum to still on the L two based but i could the regularization right in an L P and uh you could uh get that more sparse characteristics and in the in in this just so using this this came right yeah what we will do is we will uh generalise it to apply to a um compressive scenario where R is no longer so gaussian in in in fact be a very uh i'm whole save or are rich wrote a novel impulsive also characteristics it will will show that the use somewhat tree that is preserved here is uh and that a L to space it's and um and uh and L P space L L P uh metric going to uh and L L from metric and these are very well very distinctly related uh key will be less less in L two that's a correct so of and to uh can go from one to two um and and a stable distributions uh how have determines the uh uh the level of the degree of um that gaussian at if you will uh to meaning gaussian uh um and the the lower the out of the more impulsive characteristics you will have a and of course the that the ripple also give you at uh uh probability of of reconstruction and also a number of measurements you need i yes well it's of this a i i a more now this this is the geometric interpretation um um in a in compressive sensing typically you have a uh that the L two is uh if you have the L two distance in uh in the original space you are going to preserve these L two distance in the in the project space uh in in in a i in the image on a reduction actual literature uh there's a lot of interest in doing that this the mitchell i reduction in uh uh uh north that are not L two uh maybe the L one that that out maybe more uh a truck before the application and B for the image processing literature is well known that the L two um norms based consider are are very and you may wanna go to other other more so that was a motivation here you know can we get a other uh L two distance preservation in this in in this compress sense so in this case for instance of were talking of the L one norm here how i how maybe preserving in that the projections on the and this L P more and the associate with this the main char with action then you will have the uh a or the restricted isometry property on the reconstruction you would like to get the construction back so uh the that two to the result that we will look at two is basically this result here which is in since a uh there similar to the traditional additional right P when set the L two norm few we're gonna have the L L P a P more so what are the what are the things that this will will uh will get gain if you use this uh projection rather than that are not to a L C uh one is a you will preserve distances in in space is are not L L two uh but there's also the properties that are interest um uh we know when you pray when you process data with that uh a more that are lower than L two but the processing a lot is a lot more robust so that's form dress as one of the parks of what's given here by or real and bar that if you have a is in your in your observations that are not uh gaussian then you were uh a reconstruction gonna be much more robust than what you would get with can in projection uh you use L P therefore you you per are more the sparsity of the reconstruction and but um as i said that that the but distant reservation is is a different or space okay so in in a stable when we talk about a a stay uh a sub gaussian projections what is key is that if you print if you're generating a round on uh make trick rather than having a a a a uh the entries in B of a gaussian distribution they're gonna be an alpha stable distribution which uh have pales we're than than the gaussian so for instance here the uh as you change the parameter uh i'll cycle two with this this black of which is the gaussian if you have a a a a a i'll think with one point five is gonna be used blue curve and so on so as as you lower the L are you gonna have a have your heavy tails uh to generate that that the projection matrix um be a characteristic function of all stable the distributions have a this have the shape uh is a class of uh uh density functions of distributions uh a characteristic function is very well to if is very compact very nice but that dish regions are not that's very interesting class um here the out that if you put of equal to two that's a characteristic function of a gaussian so in general this is the easiest way to cracked tries that the of or stable distribution and i think that's very interesting that with the stable distributions it it has a a a a stability property much like the gaussian has a a is gaussian central in your uh stable distribution have a a generalized central limit your or or stable wall you will and it's that that it it's interesting because if you have a um uh let this are be a stable distributions you stable distribution with some that this this this this five i'll of uh in this fight this partial and so you don't have the the if you have person uh if you have a a a bunch of them uh the output what also be stable of the same parameter alpha and but of this portion will be and and so like an L P uh a metric of of the dispersion right so uh for instance if the it if all of these have a a a a uh this person one and you are and you out all of this the N the the output the why which is the sum of all of these will be um a of the E the L A of a more of four of of of the data so that was partial power captures the that the of the other or um it is an example what you have a a a cat can data if you do the cal you random the projections in the output will be that cal she with that of the L one this this is a special case of i is stable distribution okay um so when you tack of a stable distribution since are heavy tails uh you can use second or or or second order moments are second or statistic because they're very heavy tails and the means and variances are that where will find so what you have to use is you have to have what you have to use you have to uh use fractional uh mode so uh if if uh X is a uh stable distribution then the expected value of X to the P so that X of the P has uh you compute the the the moments then you will this will be related to the dispersion the P and uh therefore if you have the L P more oh of these this is a of the sum of a a lot of these terms then the expected value of that the they'll P then it'll be another constant that term a variable P and alpha in L so the dispersion of the of the very so this can be used in order to do the analysis that we need for the for the uh are i P that would will go to so this is a a that they are we were gonna we gonna look for is uh we're gonna have a run the matrix that again but like you generate uh compressive random the to see that are gaussian that we gonna be generating i got matrices was is that are of the stable i at uh with out between one and two and you will uh a show that if you have a K A the dimension of is matrix B well sir it than the mentioned uh which is as uh a and of over S a very similar to that for a traditional are P where is that the sparsity then this this uh um distance card to station will be preserved and with a given problem a came to do that to prove that we will use this are similar steps as as as we do in the in the uh traditional are P construction except that we will be using these other uh different than for and the uh uh fractional moment and that the procedure is of course to look at the probability probabilistic approach that for a fixed X in uh that is sparse uh that the rip hold then we will look at the uh that the rip uh uh i is achieved for any X in a in our and the S and then for any submatrix of B R are so it's very similar to be other or just on the traditional on the traditional um alright are P the relations so we will just get how we do this um so the first lemma tell so that if you have the index of uh uh S being the sparsity and uh oh we're we're looking at this uh be a submatrix of K by ace uh of these uh projection matrix then uh this will hold this this will whole whole these are are some constants that we will derive in in a wheel with this will hold with a given probability and uh the problem that is is uh are related to the to them up the fractional moments we went uh we just discussed uh a minute ago we you have a i is equal to the projection major each of the interest of these why will be the linear combination of the X with the stable components therefore for this uh a random variable Y to inter a projection will be alpha stable with a zero mean but the dispersion of uh yeah uh the the L L right the uh the L P L of the of Y then is um we know that by this somewhere where you have each of these is the i is uh is given there so one um this is just a that the sum of this elements of of the why way of the of the people are and that you can then take the expected value of of the projection which um is just uh again it's a this are uh the P uh uh the the the out of an a normal of the of of of the back oh so we have a then is is we have the expected value of and then we have the variance in in the variance you can similarly divide there so you have a you have the mean and you have a variance and you trying to get this bound in order to detect the probabilities that that that the distance as will be preserved in that this um um then if you approximate of the distribution of of of these uh P more by an inverse gaussian then you will have a a of by bound that you can then relate how likely use how like there these distances to be in that in that in a given "'em" ball if you will in that will be a function well this parameter a to we'd be parameter at the the distribution of these me are and signal that we just review in the previous slide and the but the turn of band provides as that probability um which uh will be a function of the K right that will tell you for a given probably that i i'm within the ball and need so many projections the on the uh set the but on the compressive sensing measure so a uh we then general that not just for a given X but for any arbitrary X that could but that preserves that uh it'll P norm in that does just changes this probability you just have to be a little more i don't the on the uh on the distances in then for any submatrix i you just change also the but the constants that an N to prove the that the or and then you just have to uh many P like the constants to put in to this form which is what we you what will do but in in and at the same time that in says the minimum number of measurements to it's to attain the rip which is again uh a function of these S slot and so this is an example what happens if you can if you if you um if you do these projections so you have the the shall reduction you you have this matrices that satisfy this is the this reason uh and then you can do the reconstruction that only just mitchell to but you wanna to the reconstruction but when you do the be a a reconstruction you're project thing with this uh not some gaussian sub gaussian entries in therefore for you can not use traditional compressive sensing algorithms uh like the L two a one or or uh methods that are rely on and to uh distance distances um point of the greedy out within to is you really with ends would fail uh because of the impulsive nature of that the measure so in this uh in this uh example uh oh we're gonna we getting um compressive measurements Y where are are are the alpha stable uh projections without equal to one point two uh well the noise is one what one point two and um and then you have this is a density function a number of measurements is four hundred K is the number measurements S is the sparsity and this is a uh uh um reconstruction algorithms that are not develop in this paper but in different paper that uses uh uh not L two data fitting uh uh terms uh but uses a a um uh i L L lorentzian regular say a lorentzian based metrics to do the the the data fit then you can see that the the data that is the original or the circles blue circles bin them cover data is is is well preserved if you the L one a is you will week we curve or we cover the sparsity terms you will have a lot of a ears uh it's star and then you can do that and you can test the um the number of uh measurements that the you require and again for these method we have a the medical uh K that you need to to fit the in this construction you you were able to uh do the inversion uh in a within the bounds of the tip that they L C so in summary O what uh what if you use uh if you don't use of gaussian a measurement uh we we explore that you can get a a a uh this is trees trees or these distances in a in a not an end to but another another other uh norms or and uh uh at the same time you will you will find that the they're more last you have a lot more robust against noise that very close it and uh uh also we use you yeah the sparsity inducing a a a more that but you want for question i well you're what you're but your uh you're using your use in the journal bounds on the L on that uh uh fractional uh tire so you're doing the close to the P so or you can use a approximations matt not couch in that right him where you can put inverse gaussian or other channel out some of that is true a well the a sure because you do your use using those on the on the L P type uh very he is uh yes yes yeah yes yeah what happens is a um you generate you generate your your matrices using a stable distribution right so you have some that is not gaussian you generate the projections for you gonna get a vector of measurements but these measurements are are are linear combination alpha stable distributions so they're not uh they're not so gaussian than i gauss right so the inverse a algorithms can use traditional uh L two uh data fitting terms with the regular right so you have to use norms that are robust you have to use norms perhaps like the ones they what they were mentioned the later or earlier but are are one or or a more robust and you're of so in this case uh this illustrates that if you if you norms of data fitting that are are there are more robust like the L L the or in that was uh yes yes uh yes that that will be that will double will you will get a good result if you do yes within that the in the algorithm the that we use as a lorentzian type but we could that derive an algorithm that uses and an L a people less than i i yeah well the L one is the approximation for the for the right um i we have really run run that experiment because are to turn right there there's the data fitting "'em" which is the that one that you were used put a zero point six in and there's the sparsity term which could be the L zero even the L one right so if if you if you if you can but without of them that was the and the data fitting normal zero point six and use the L zero we a very good or or L so point six of than then one maybe fine but he these odd with them the you we try was one that was somewhat to double that that we could go down to that as as was presented with a more in more uh in this here shows you that if you the the L two they are fading you can of the port you gonna get a lot of spurs result because of that because of the structure of the projection which is not some gauss yes that's a good question um uh well we explore where the fun that the goal um characteristics of of such mate uh a to generate them and how do you uh a user in practise that's that's something to be explored uh a sort you you could try uh for a gaussian mixtures are easy a very good approximation of of not sub gaussian uh majors but in general to be in practice you have to to come up that's an open open question okay thank