K a uh had a of on and the thank you for it and that is saying to my presentation of max is very their is the regularization at a selection and uh basically it's about uh a of based estimations generally but uh we are going to i i explain our results in the context of of uh you a estimation and uh for this reason you a problem and how we can solve by uh a sparse representation and then yeah we address the regularization parameter uh estimation problem and we introduce a method to solve uh this problem and finally we compare the different method oh uh particularly a problem we have a set of sources and the a set of uh a small area of sensors and we are to uh estimated directions of the sources direction of arrival of the signals uh to the sensors only by having uh uh the uh at the uh area point so what our assumptions are that uh the signals are narrow and and uh they are and the the sources are very for so that we can write the uh there is a a a a as the in your model of the sense that that so if you assume a source from a certain direction fit in the space and we can write uh the received data uh as a multiplication of a is to as you knowing the car which is dependent on the configuration of this uh uh sensor are ready for a linear or and do with and that the at the a a big to are we introduce here so if we have any source uh uh the total received that will be uh uh by a super per uh was super imposing posing a different elements will be a a linear combination of uh uh the terms of which can be with in a matrix form oh of course uh are we have a always a noise there so are many ways to solve this problem but uh and many of them like subspace so as what we have able but a for uh the problem that the are not general for example in one assumption option and one that are at case it they fail so uh uh the most general a solution is the loss of a solution and that we are going to explain now uh for uh a base estimation we need to discrete time this space and the uh then uh we have some that the sources are uh i mean uh we with a very fine estimation are you will get a a a a a a a a estimation of so this is the quantization noise introduced by discrete i in the space uh but that we introduce many many in real sources for each grid point in the space uh most of them are sending zero except to the the the sources that are we are interested in so we are introducing a very lying uh a vector of sources and most of them or zero and then we can write a model that method by the previous uh a a a a uh as a a linear combination of the source and now a G is the vector a is that matrix use of all the steering vectors and a for the degree and it is very five trees so even if there is no noise this model cannot be solve this get it because a is not a veritable so we need a a a a constraint obviously as about uh uh the number of sources number of sources uh uh or uh a constrained to a a a to you know i mean the to signal is uh a sparse and the number of men and nonzero elements are know uh uh to express it in a general and situation when we have many a snapshots is the usual way is that uh we look at the a extended sources in the direction direction wise can we introduce yeah average average power from each direction of a space given "'em" by their average energy of the sources from that direction for the direction that there's no source yeah a it will be zero so is your norm of uh uh this come a vector which is long in itself uh gives the the number of source then the maximum likelihood is to can be written as this this is the usual max some like to this is uh related to the model and the constrained about the number of nonzero L so it's it's a hard to solve this problem is as hard as solving a use maximum like to in in nineteen your a square and uh what has been done is that you can write it in this uh equivalent form and then replace this zero norm which one or and uh this is a a an approximation but it works uh and the reason is that a which is explained by teacher and so uh the reason is that with so in its optimization it's more probable to heat for uh this is corners like here uh in the diamond shape to the damage that uh sort is introduced by the one or so if it's not this one or you can buy use of and that so here we have a lot line one skit hand uh the number of nonzero elements here well it will see that on the controls a number of the sparsity of the source so uh if you have a more uh uh i i highest that higher value of long and then we have a less source the active uh in in the model so uh the question a would be phones estimation point of view uh the question would be uh a how to choose the proper value of land a for example in this example of we have a three i sources and uh probably this point should be chosen chosen by a method but before going to the but a uh i need to and reviews review some a a a points about the last so so uh two a estimate the number of sources if you want the simulation will get such an eight spectrum of or this space and then we need a special thing to choose which ones i active and which one is the psyche but is not a problem because difference is very high and uh it the lasso so based estimation is a good estimate or or of the direction but uh it's biased for or sources and the so well if you want to good estimation of the sources is that the product of the i is the part of the main problem but if you are interested in that then you have to solve a and uh another estimation again by assuming and known direction as and it will be a usual uh max and uh usual the uh it is a square my and now we are uh to the problem of the regularization parameter selection uh first like just we use some concepts in a a estimation to especially for the maximum a posteriori estimator so i uh if you write the log out exactly scale bombs can maybe like and uh the combination of the prime your and the the model we have a so uh the question of how to choose the prior there are two ways to think about it a one way is that the the prior is the uh the point you're is given by the physics i i a it's given by the nature and uh it's a part of the model but the the way that uh the prime or is the two two apply some properties to the estimation so how we uh uh do the in the second way of thinking uh a to illustrate it uh a will show this uh graph which well uh one of the axes the probably there are for an estimator uh when the the but the values of true parameter values or to come one and S one and the the other their as would be a problem to of error white this estimator i when the true by do the parameters of to to to an S two and that there is a tradeoff between a this to you cannot be good as a whole something and by choosing the prior are actually moving the score uh and the probably you need to a you want to be here if you don't have any preference uh you want to be good for all of them and that's why we choose the prior usually a a a from prior what works well but that's always and uh a here you can see that if if a a are good and one at them to the case where than for example the noise was a zero or the number of just go to infinity then uh it's possible to get a better estimation the mute a new this man i mean a at the same time you will get a good estimation for all the uh but so uh one uh a hard uh example a the selection with which we are interested because we want to know how to choose the right land uh which is equivalent to choosing the right model order uh uh and the uh and you can see uh a choosing a uniform will be and over feet there has been a and many uh discussions is about how to choose a prior a good prior and that there are there have been some would prior or as a based an asymptotic case uh for for example for no a but uh number of snapshots which given by a i I C or and the L and here we introduce the and the uh and which uh chooses the best model uh as the model that describes the date i mean less number of bits and this can be expressed is one uh a term is related to the number of bits of the parameters and this one is the number of they'd have it's with respect to the parameters we chose and that you can see there is a relation between a a maximum likelihood and uh the and here for our problem in be at given by this for me a no and you at them all as you can see for mean is not as work will work very about what uh for a low number of snapshots doesn't work so uh as we are are for a in the case of one a snapshot it's tends to well were fit the model again and yeah we need to one to choose and don't the prior the good way to choose or a prior is to look at the last it's so because lots of works so probably uh it will it is it gives us a good prior as well so if you we compare the maximum likelihood we a lot so uh formulation you will see that if we choose uh a laplacian prior then we can compare i some likelihood we are K and then you can use the maximum likelihood or uh choosing a that it give this estimator or it is is actually uh discuss in a in many papers that this doesn't work it actually works but the the problem is that only works for a high snrs where you are very close to the case where actually every prior may more uh uh and the as you can see now it follows the right answer the minimum is related to the true number of sources but it as the tracker so so what the problem the problem is that we are doing in a giving uh a high probability to not desired values uh uh actually a we we are not interested in be high dimensional uh speech but the laplacian prior gives us a uh the the main problem the probability is given to very high dimension space so remove that or and we constraint the a probability density function the for the prior to the don't a low dimensional space and is given my this is not there's one this an up there shown where the uh but only over though that a low dimensional space that and then uh solving for uh maximum likelihood would get such a a a a a a as which or but at something and and six more uh a deterministic but unknown parameters and then you can use the rubber the estimation of sick man and uh as explained before to get an image and as a result you can compare it to the and the uh so we have a uh the maps estimate i mismatch with at that is made by playing with the optimized actually a a you have we can have many queen wasn't forms of a optimisation and all of them can can interpret it in different friend uh by using in four so uh and uh i there is a long discussion the paper or i'm not going to be the that's really but then that you can see by plane with optimized the will get you different priors but all of them or working very well uh but as you can see uh there is like slightly different related to and the prior your that you play and you change the point in your career but and and he does not work and it probably works for i mission oh are there is a uh uh at the conclusion a uh we have the less so can and uh it's words and you can choose a uh two i uh use the you a model order selection but you also can choose the asymptotic here uh you can choose uh uh the law that so as a uh prior to the the model order and in this case what of what's much more it so thank you and yeah so yeah maximum this is me less but i guess you i think what right that look at all probably to should be max maximum of problem to you and arabic bic which are usually are so you're using the mike the negative like so you're writing then mean since your discrete C don't uh why is it that you to detect search clear peaks should be able to to sure should shouldn't yeah by cool fishing P R to kick or in one of much previous slide yeah uh actually a that there there are some proof that well i zero no and one or might but and some you but looks order to show that there is a low correlations between the columns of your a a or in this case your discrete choice in space so to see mark are conditionally all right that's actually i i i well as you proof that this case right image matrix oh from that man is K do you we have a sparse still okay and you have the question i do have a question oh it seems that you have some kind of mixture between uh that's a deterministic approach and a bayesian approach so what why don't you go to yeah a long to full by way assuming that um guys i by we some should yeah or way that i i mean i i the first approach that and right prior you know is a way to look at but the it doesn't work and in the literature are some paper that used for some or prior and and it's by jeanne a but uh i i want to say that uh there is no way to think you can go back what the back get and you always choose a of the prior you want is a higher elevation uh oh at some extent should i thought okay thank you