0:00:13so that are known everyone
0:00:14my name is sophia K the any and and come from a P of L as non
0:00:19did they i one present a this work that this so work with my adviser professor a press from site
0:00:24only now a with approximation based and different
0:00:29this is the overview of my though
0:00:31in the beginning i will talk briefly about the motivation
0:00:34and i will present the formulation for the problem
0:00:37and the uh that we used to sell
0:00:39and find and to one so you the results that we having
0:00:42so i guess the first question that this is the answer is why do we bother to use mine
0:00:47as we all know transformation by that's is important uh in many applications like classification on me
0:00:54a i used in human C are in the position of the relating the transformed versions of a signal easy
0:01:01however computers are are not do the same thing and even in simple case is like the one on
0:01:07like a one so there where we have this more traditional for a model
0:01:12uh the pixel values of the immense change a significantly and therefore for computers and not in a position of
0:01:18directly relate the to signal
0:01:21uh and it turns out that manifolds are a natural way to hold all at the transformed very zeros of
0:01:26a signal
0:01:27this is true because manifolds are to sense any low dimensional structures and they in higher dimensional space
0:01:34and when we are having transformation
0:01:36what where are actually having use a set of uh
0:01:39a small number of parameters
0:01:41a during the appearance of a high dimensional signal
0:01:44then for there is a direct link between the transformation parameters and the money for team
0:01:50and in things in this figure here we are seeing a a one D manifold but these embedded in
0:01:55but a three D space
0:01:57and since the dimensionality of the manifold here is lot
0:02:00it could be used to model a transformation uh with one parameter like a limitation of these lemma
0:02:07uh a on the other hand we all know that usually we want to have a linear models because they
0:02:12are simple there is it to handle
0:02:14and the other five computations and specially for distance
0:02:17uh however uh the most to real world uh the signals don't have a a linear manifolds at in many
0:02:25cases this money are us a know that we don't know the an only form
0:02:29what we can do in this case um since we can just use one linear model
0:02:34is to use many or models to model the money for
0:02:37uh uh in this what we are going to use a set of a fine model which are just lean
0:02:42are mark a models that are translated from your reading
0:02:45and from the one no will call them fly
0:02:48when you what we going you're using sets a representation for a manifold it's important to respect the manifold geometry
0:02:54that means that you should place the flat since such a way that this big peak the data on which
0:02:59a and they don't violate
0:03:01a a so the situation user is that we are having a unknown manifold them or they mention E D
0:03:06D that's a lower than
0:03:08the high that
0:03:10and then which is the dimension that of the space where the manifold leaves
0:03:13uh we have a set of samples
0:03:15coming from this man for
0:03:17and in order to model the underlying geometry we also have the neighborhood graph
0:03:22which C is uh
0:03:23but the graph that we get one we connect the neighbouring sample
0:03:27what we would like to at C be stop proxy make them with D dimension of flat
0:03:35in general it's flat is going to be representing a set a a a a part of the money in
0:03:40in our case is where having some was it would be present a set of samples
0:03:44we know that the lowest dimension mention that finds space that includes uh all the points in in a set
0:03:49is the affine whole
0:03:52therefore are what we would like to let T Vs long cover sets of samples with d-dimensional dimensional a
0:03:57so if we were having a the sum was that that are sewn there
0:04:01uh what would like to separate them into groups
0:04:03so that each group has one dimensional mindful for a one dimensional fine home because the underlying might is this
0:04:09one payments
0:04:12however is not that easy to compute the affine holes
0:04:16what we can compute these is it's the tangent space at its sample the man
0:04:23a a we can i'm sure also that the that tangent in space at each of the samples
0:04:27uh a in the case where we have a set with that D them so a fine for is going
0:04:32to be identical to these stuff fine whole so in the previous example will have these constants for each to
0:04:37the samples
0:04:37which is the same line as we had for the uh
0:04:41uh based on this observation what we can do is that uh we can chart on feature
0:04:47we some so having similar tangents and then group them together and represent them by the mean times
0:04:54following doing this formulation we can you to ban mess of the cost of uh a of grouping
0:04:59by a sound mean over all the samples in a set
0:05:03all of the squared distances between the times and but it somewhat and the mean tangent and over the group
0:05:09the mean that and
0:05:10as in general D dimensional or subspace
0:05:13and that's that's the they want the grassmann manifold which is the space of a dimensional are space so far
0:05:19a common to use there is the projection these stands which is computed as from here
0:05:24it based on the price for and does between the two sub-spaces
0:05:27and we test some the principle of or canonical correlations and uh and
0:05:31sept charge them from the dimensionality and that of this space
0:05:35then the mean time and can be from like those the car to mean which sees the times and that
0:05:40that minimize is the sum of squared distances over times as
0:05:44or using this cost
0:05:46we can for the from late of problem on shown here
0:05:49the input of the problem is a set of sample coming from the on fall and the neighborhood graph that
0:05:54is used to model the on it
0:05:56and then what we are trying to figure out is a partition of these samples C two groups
0:06:01so that we mean my as the sum of the costs for each school
0:06:05we do that uh anything or to verify that there we respect the underlying geometry
0:06:10we have also uh i'm not another constraint that says that
0:06:14at the sub graph corresponding to each group has to be egg
0:06:19uh yeah the is not we used to solve this problem is that be a bottom-up approach
0:06:24and these uh it can consists of two basic steps
0:06:28they what is the sample set the the same neighbourhood size K and the number of that cell
0:06:33uh with a uh we want to used to approximate the man
0:06:37in the first that what we do is the to compute the tangent space in the second we do the
0:06:42basic region merging procedure
0:06:44and then the output are the groups that we have a a and cover and the flats that we used
0:06:48to read
0:06:51and the first that
0:06:52in order to a we
0:06:53firstly we use the parameter K to construct the K nearest neighbor graph
0:06:58then based on these graph and by doing is D D O need sample neighbourhood what compute that and
0:07:04uh i order to not account for possible noise or outliers in our data set we find there's small that
0:07:09times and by using gaussian kernels on the grassmann mine
0:07:14the second step is the basic procedure of the are really is that greedy optimisation
0:07:19you the basic idea is that we start with "'em" groups
0:07:21each representing a uh one sample at the beginning
0:07:25and that and then that detect a race and we met the neighbour in groups with a minimum cost
0:07:31was the from thing is uh in fact that the difference between the cost of the mets group
0:07:36my knows the costs of the groups before the marriage
0:07:40if we use that the form that it have on before for these colours
0:07:43we will see that it depends upon the mean tons and over the
0:07:48um and group
0:07:50and since these mean time this the outcome of an optimization procedure it wouldn't be got the feast in to
0:07:55compute it for every possible manner
0:07:58what we do instead is that we compute an upper bound for this quantity
0:08:02that does not depend and more on uh the mean down of over the max group but only on the
0:08:07mean tons and of the groups as the are so far
0:08:10uh using this upper bound
0:08:12we start with "'em" groups and then we do our iterations at each a race and we compute the co
0:08:18uh a course for possible mode things
0:08:20we decide which with thing has the minimum cost
0:08:23we perform a and then we find the flat uh
0:08:26to represent the you group as the mean can and of the most group
0:08:30then would say how many groups we having and if we have a the uh the desired level we stop
0:08:36in fact we don't count every single group but only the ones that are significant and uh a significant we
0:08:42define a groups
0:08:44that's how a more that two percent of the total samples
0:08:46that's something that we do it to
0:08:49to be sure that we don't pay too much as
0:08:52ten and a outlier
0:08:54uh at the end of a a a a a of the other read we out to put the the
0:08:58groups that we have found and rate a representative flat
0:09:01but there are a compute it as the sub-spaces corresponding to the D largest eigenvalues of bits groups that them
0:09:08a a in order to to uh figure out to have the put in order to see the performance of
0:09:13a are an we have compared it with three different others are schemes
0:09:17the first one is the high rack of B basic plastering on three
0:09:20which is a top down roads
0:09:22and the linearity measure that is used there is that the deviation between that you played in and the job
0:09:27see this
0:09:29the second one is the hierarchical agglomerative clustering
0:09:32at this is a bottom-up approach to right oh
0:09:35and the linear the method that is used as again the deviation between you played them into this these
0:09:40and and we have so uh compared with the N Q flat
0:09:44this is an one that comes from the family of subspace clustering method
0:09:48is that but it's and not directly applicable to money falls and that's because
0:09:53use use lead they don't have any constraint that would lead them to respect the manifold geometry
0:09:59that's an example is shown here where would have a P are um a for the K
0:10:04and as we see the and the flat that uh the i'm board in the fines for representing the groups
0:10:09the not for all that you much of the minor
0:10:12so for hours
0:10:13that is to have used to that this that the three this swiss roll and the S
0:10:18a training set that was consisting of two thousand points randomly sample
0:10:22and that that thing set from five times and random some
0:10:26for the testing what we are doing is that for each testing sample
0:10:30um we compute the K nearest neighbours
0:10:32and then by doing
0:10:34from the training set
0:10:35and then by doing majority voting or over the we find two weeks flat we should project
0:10:40then by protect yeah finding that distance between uh
0:10:44the sample the actual some when the projection we have a reconstruction error for these
0:10:49one and by summing over a or all of the samples of the testing things we compute the mean square
0:10:56so the results for this this roll or so here
0:10:59are are of this only with uh or
0:11:02and that's we can and on the horizontal axis we have the number of that
0:11:06and then the part of the goal of it's the mean the reconstruction they're
0:11:10as we can see in general the performance of our scheme is better in space in in the meeting a
0:11:14came she's from fifteen to thirty five
0:11:18um the picture is the same as for the S your data set
0:11:22where the difference is even more all views
0:11:24uh and again we see that the for all the number of lots our scheme performs better than the rest
0:11:31uh and you yeah also plotted the groups that we get for the S two for the case of base
0:11:37to have flat
0:11:38just to verify that each group uh uh is uh and connected region of uh one
0:11:45or what they have presented to you today use that greedy bottom up are going to approximate manifolds meant to
0:11:50be with low dimension how so that
0:11:53the linear you to measure that we use is the differences of dancing
0:11:56and we have seen that
0:11:58for our experiments itself performs a manifold approximation problem
0:12:03possible applications for this scheme could be data compression on multiview view classification in general
0:12:08cases is when we have to recognise a signal from transform
0:12:11for any transform versions of
0:12:14and you
0:12:20so question
0:12:28okay okay a question uh what is actually the regularity assumption on your uh any for your a processing of
0:12:40shown because i image
0:12:42you to learn a menu for a summer get one
0:12:49so we can i
0:12:50some some irregularity because we are having a smooth this thing on the tangent space computation but of course for
0:12:55what would have to be able to compute meaningful meaningful dungeons
0:12:58so that it's means that it has to be
0:13:01difference different sample or we have to smooth it at the beginning but for sure we are
0:13:05a a the case is that we can handle a cases of money false that have a uh where are
0:13:09where be states
0:13:11so we cannot have something like to really rely you like that we got or have to smooth it out
0:13:16in the beginning
0:13:17or to a of small thing
0:13:18during a
0:13:20if you there is just go for that
0:13:23for to approximate
0:13:26you mean like a my folks of real thing mouse
0:13:29but we are trying to do that now it's uh our future work
0:13:37and questions