so that are known everyone my name is sophia K the any and and come from a P of L as non did they i one present a this work that this so work with my adviser professor a press from site only now a with approximation based and different i this is the overview of my though in the beginning i will talk briefly about the motivation and i will present the formulation for the problem and the uh that we used to sell and find and to one so you the results that we having so i guess the first question that this is the answer is why do we bother to use mine as we all know transformation by that's is important uh in many applications like classification on me a i used in human C are in the position of the relating the transformed versions of a signal easy however computers are are not do the same thing and even in simple case is like the one on so like a one so there where we have this more traditional for a model uh the pixel values of the immense change a significantly and therefore for computers and not in a position of directly relate the to signal uh and it turns out that manifolds are a natural way to hold all at the transformed very zeros of a signal this is true because manifolds are to sense any low dimensional structures and they in higher dimensional space and when we are having transformation what where are actually having use a set of uh small a small number of parameters a during the appearance of a high dimensional signal then for there is a direct link between the transformation parameters and the money for team and in things in this figure here we are seeing a a one D manifold but these embedded in but a three D space and since the dimensionality of the manifold here is lot it could be used to model a transformation uh with one parameter like a limitation of these lemma uh a on the other hand we all know that usually we want to have a linear models because they are simple there is it to handle and the other five computations and specially for distance uh however uh the most to real world uh the signals don't have a a linear manifolds at in many cases this money are us a know that we don't know the an only form what we can do in this case um since we can just use one linear model is to use many or models to model the money for uh uh in this what we are going to use a set of a fine model which are just lean are mark a models that are translated from your reading and from the one no will call them fly when you what we going you're using sets a representation for a manifold it's important to respect the manifold geometry that means that you should place the flat since such a way that this big peak the data on which a and they don't violate a a so the situation user is that we are having a unknown manifold them or they mention E D D that's a lower than the high that and then which is the dimension that of the space where the manifold leaves uh we have a set of samples coming from this man for and in order to model the underlying geometry we also have the neighborhood graph which C is uh but the graph that we get one we connect the neighbouring sample what we would like to at C be stop proxy make them with D dimension of flat uh in general it's flat is going to be representing a set a a a a part of the money in in our case is where having some was it would be present a set of samples we know that the lowest dimension mention that finds space that includes uh all the points in in a set is the affine whole therefore are what we would like to let T Vs long cover sets of samples with d-dimensional dimensional a so if we were having a the sum was that that are sewn there uh what would like to separate them into groups so that each group has one dimensional mindful for a one dimensional fine home because the underlying might is this one payments um however is not that easy to compute the affine holes but what we can compute these is it's the tangent space at its sample the man and a a we can i'm sure also that the that tangent in space at each of the samples uh a in the case where we have a set with that D them so a fine for is going to be identical to these stuff fine whole so in the previous example will have these constants for each to the samples which is the same line as we had for the uh uh based on this observation what we can do is that uh we can chart on feature we some so having similar tangents and then group them together and represent them by the mean times following doing this formulation we can you to ban mess of the cost of uh a of grouping by a sound mean over all the samples in a set all of the squared distances between the times and but it somewhat and the mean tangent and over the group the mean that and as in general D dimensional or subspace and that's that's the they want the grassmann manifold which is the space of a dimensional are space so far a common to use there is the projection these stands which is computed as from here it based on the price for and does between the two sub-spaces and we test some the principle of or canonical correlations and uh and sept charge them from the dimensionality and that of this space then the mean time and can be from like those the car to mean which sees the times and that that minimize is the sum of squared distances over times as or using this cost we can for the from late of problem on shown here the input of the problem is a set of sample coming from the on fall and the neighborhood graph that is used to model the on it and then what we are trying to figure out is a partition of these samples C two groups so that we mean my as the sum of the costs for each school we do that uh anything or to verify that there we respect the underlying geometry we have also uh i'm not another constraint that says that at the sub graph corresponding to each group has to be egg uh yeah the is not we used to solve this problem is that be a bottom-up approach and these uh it can consists of two basic steps they what is the sample set the the same neighbourhood size K and the number of that cell uh with a uh we want to used to approximate the man in the first that what we do is the to compute the tangent space in the second we do the basic region merging procedure and then the output are the groups that we have a a and cover and the flats that we used to read and the first that in order to a we firstly we use the parameter K to construct the K nearest neighbor graph then based on these graph and by doing is D D O need sample neighbourhood what compute that and uh i order to not account for possible noise or outliers in our data set we find there's small that times and by using gaussian kernels on the grassmann mine the second step is the basic procedure of the are really is that greedy optimisation you the basic idea is that we start with "'em" groups each representing a uh one sample at the beginning and that and then that detect a race and we met the neighbour in groups with a minimum cost yeah was the from thing is uh in fact that the difference between the cost of the mets group my knows the costs of the groups before the marriage if we use that the form that it have on before for these colours we will see that it depends upon the mean tons and over the um and group and since these mean time this the outcome of an optimization procedure it wouldn't be got the feast in to compute it for every possible manner what we do instead is that we compute an upper bound for this quantity that does not depend and more on uh the mean down of over the max group but only on the mean tons and of the groups as the are so far uh using this upper bound we start with "'em" groups and then we do our iterations at each a race and we compute the co uh a course for possible mode things we decide which with thing has the minimum cost we perform a and then we find the flat uh to represent the you group as the mean can and of the most group then would say how many groups we having and if we have a the uh the desired level we stop in fact we don't count every single group but only the ones that are significant and uh a significant we define a groups that's how a more that two percent of the total samples that's something that we do it to to be sure that we don't pay too much as ten and a outlier uh at the end of a a a a a of the other read we out to put the the groups that we have found and rate a representative flat but there are a compute it as the sub-spaces corresponding to the D largest eigenvalues of bits groups that them a a in order to to uh figure out to have the put in order to see the performance of a are an we have compared it with three different others are schemes the first one is the high rack of B basic plastering on three which is a top down roads and the linearity measure that is used there is that the deviation between that you played in and the job see this the second one is the hierarchical agglomerative clustering at this is a bottom-up approach to right oh and the linear the method that is used as again the deviation between you played them into this these and and we have so uh compared with the N Q flat this is an one that comes from the family of subspace clustering method is that but it's and not directly applicable to money falls and that's because use use lead they don't have any constraint that would lead them to respect the manifold geometry that's an example is shown here where would have a P are um a for the K and as we see the and the flat that uh the i'm board in the fines for representing the groups the not for all that you much of the minor so for hours that is to have used to that this that the three this swiss roll and the S a training set that was consisting of two thousand points randomly sample and that that thing set from five times and random some for the testing what we are doing is that for each testing sample um we compute the K nearest neighbours and then by doing from the training set and then by doing majority voting or over the we find two weeks flat we should project then by protect yeah finding that distance between uh the sample the actual some when the projection we have a reconstruction error for these one and by summing over a or all of the samples of the testing things we compute the mean square error so the results for this this roll or so here are are of this only with uh or and that's we can and on the horizontal axis we have the number of that and then the part of the goal of it's the mean the reconstruction they're as we can see in general the performance of our scheme is better in space in in the meeting a came she's from fifteen to thirty five um the picture is the same as for the S your data set where the difference is even more all views uh and again we see that the for all the number of lots our scheme performs better than the rest uh and you yeah also plotted the groups that we get for the S two for the case of base to have flat just to verify that each group uh uh is uh and connected region of uh one or what they have presented to you today use that greedy bottom up are going to approximate manifolds meant to be with low dimension how so that the linear you to measure that we use is the differences of dancing and we have seen that for our experiments itself performs a manifold approximation problem possible applications for this scheme could be data compression on multiview view classification in general cases is when we have to recognise a signal from transform for any transform versions of and you i so question i okay okay a question uh what is actually the regularity assumption on your uh any for your a processing of shown because i image you to learn a menu for a summer get one so we can i some some irregularity because we are having a smooth this thing on the tangent space computation but of course for what would have to be able to compute meaningful meaningful dungeons so that it's means that it has to be difference different sample or we have to smooth it at the beginning but for sure we are a a the case is that we can handle a cases of money false that have a uh where are where be states how so we cannot have something like to really rely you like that we got or have to smooth it out in the beginning or to a of small thing during a if you there is just go for that for to approximate you mean like a my folks of real thing mouse but we are trying to do that now it's uh our future work i and questions not i