0:00:13 | so that are known everyone |
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0:00:14 | my name is sophia K the any and and come from a P of L as non |

0:00:19 | did they i one present a this work that this so work with my adviser professor a press from site |

0:00:24 | only now a with approximation based and different |

0:00:27 | i |

0:00:29 | this is the overview of my though |

0:00:31 | in the beginning i will talk briefly about the motivation |

0:00:34 | and i will present the formulation for the problem |

0:00:37 | and the uh that we used to sell |

0:00:39 | and find and to one so you the results that we having |

0:00:42 | so i guess the first question that this is the answer is why do we bother to use mine |

0:00:47 | as we all know transformation by that's is important uh in many applications like classification on me |

0:00:54 | a i used in human C are in the position of the relating the transformed versions of a signal easy |

0:01:01 | however computers are are not do the same thing and even in simple case is like the one on |

0:01:06 | so |

0:01:07 | like a one so there where we have this more traditional for a model |

0:01:12 | uh the pixel values of the immense change a significantly and therefore for computers and not in a position of |

0:01:18 | directly relate the to signal |

0:01:21 | uh and it turns out that manifolds are a natural way to hold all at the transformed very zeros of |

0:01:26 | a signal |

0:01:27 | this is true because manifolds are to sense any low dimensional structures and they in higher dimensional space |

0:01:34 | and when we are having transformation |

0:01:36 | what where are actually having use a set of uh |

0:01:39 | small |

0:01:39 | a small number of parameters |

0:01:41 | a during the appearance of a high dimensional signal |

0:01:44 | then for there is a direct link between the transformation parameters and the money for team |

0:01:50 | and in things in this figure here we are seeing a a one D manifold but these embedded in |

0:01:55 | but a three D space |

0:01:57 | and since the dimensionality of the manifold here is lot |

0:02:00 | it could be used to model a transformation uh with one parameter like a limitation of these lemma |

0:02:07 | uh a on the other hand we all know that usually we want to have a linear models because they |

0:02:12 | are simple there is it to handle |

0:02:14 | and the other five computations and specially for distance |

0:02:17 | uh however uh the most to real world uh the signals don't have a a linear manifolds at in many |

0:02:25 | cases this money are us a know that we don't know the an only form |

0:02:29 | what we can do in this case um since we can just use one linear model |

0:02:34 | is to use many or models to model the money for |

0:02:37 | uh uh in this what we are going to use a set of a fine model which are just lean |

0:02:42 | are mark a models that are translated from your reading |

0:02:45 | and from the one no will call them fly |

0:02:48 | when you what we going you're using sets a representation for a manifold it's important to respect the manifold geometry |

0:02:54 | that means that you should place the flat since such a way that this big peak the data on which |

0:02:59 | a and they don't violate |

0:03:01 | a a so the situation user is that we are having a unknown manifold them or they mention E D |

0:03:06 | D that's a lower than |

0:03:08 | the high that |

0:03:10 | and then which is the dimension that of the space where the manifold leaves |

0:03:13 | uh we have a set of samples |

0:03:15 | coming from this man for |

0:03:17 | and in order to model the underlying geometry we also have the neighborhood graph |

0:03:22 | which C is uh |

0:03:23 | but the graph that we get one we connect the neighbouring sample |

0:03:27 | what we would like to at C be stop proxy make them with D dimension of flat |

0:03:33 | uh |

0:03:35 | in general it's flat is going to be representing a set a a a a part of the money in |

0:03:40 | in our case is where having some was it would be present a set of samples |

0:03:44 | we know that the lowest dimension mention that finds space that includes uh all the points in in a set |

0:03:49 | is the affine whole |

0:03:52 | therefore are what we would like to let T Vs long cover sets of samples with d-dimensional dimensional a |

0:03:57 | so if we were having a the sum was that that are sewn there |

0:04:01 | uh what would like to separate them into groups |

0:04:03 | so that each group has one dimensional mindful for a one dimensional fine home because the underlying might is this |

0:04:09 | one payments |

0:04:10 | um |

0:04:12 | however is not that easy to compute the affine holes |

0:04:16 | but |

0:04:16 | what we can compute these is it's the tangent space at its sample the man |

0:04:22 | and |

0:04:23 | a a we can i'm sure also that the that tangent in space at each of the samples |

0:04:27 | uh a in the case where we have a set with that D them so a fine for is going |

0:04:32 | to be identical to these stuff fine whole so in the previous example will have these constants for each to |

0:04:37 | the samples |

0:04:37 | which is the same line as we had for the uh |

0:04:41 | uh based on this observation what we can do is that uh we can chart on feature |

0:04:47 | we some so having similar tangents and then group them together and represent them by the mean times |

0:04:54 | following doing this formulation we can you to ban mess of the cost of uh a of grouping |

0:04:59 | by a sound mean over all the samples in a set |

0:05:03 | all of the squared distances between the times and but it somewhat and the mean tangent and over the group |

0:05:09 | the mean that and |

0:05:10 | as in general D dimensional or subspace |

0:05:13 | and that's that's the they want the grassmann manifold which is the space of a dimensional are space so far |

0:05:19 | a common to use there is the projection these stands which is computed as from here |

0:05:24 | it based on the price for and does between the two sub-spaces |

0:05:27 | and we test some the principle of or canonical correlations and uh and |

0:05:31 | sept charge them from the dimensionality and that of this space |

0:05:35 | then the mean time and can be from like those the car to mean which sees the times and that |

0:05:40 | that minimize is the sum of squared distances over times as |

0:05:44 | or using this cost |

0:05:46 | we can for the from late of problem on shown here |

0:05:49 | the input of the problem is a set of sample coming from the on fall and the neighborhood graph that |

0:05:54 | is used to model the on it |

0:05:56 | and then what we are trying to figure out is a partition of these samples C two groups |

0:06:01 | so that we mean my as the sum of the costs for each school |

0:06:05 | we do that uh anything or to verify that there we respect the underlying geometry |

0:06:10 | we have also uh i'm not another constraint that says that |

0:06:14 | at the sub graph corresponding to each group has to be egg |

0:06:19 | uh yeah the is not we used to solve this problem is that be a bottom-up approach |

0:06:24 | and these uh it can consists of two basic steps |

0:06:28 | they what is the sample set the the same neighbourhood size K and the number of that cell |

0:06:33 | uh with a uh we want to used to approximate the man |

0:06:37 | in the first that what we do is the to compute the tangent space in the second we do the |

0:06:42 | basic region merging procedure |

0:06:44 | and then the output are the groups that we have a a and cover and the flats that we used |

0:06:48 | to read |

0:06:51 | and the first that |

0:06:52 | in order to a we |

0:06:53 | firstly we use the parameter K to construct the K nearest neighbor graph |

0:06:58 | then based on these graph and by doing is D D O need sample neighbourhood what compute that and |

0:07:04 | uh i order to not account for possible noise or outliers in our data set we find there's small that |

0:07:09 | times and by using gaussian kernels on the grassmann mine |

0:07:14 | the second step is the basic procedure of the are really is that greedy optimisation |

0:07:19 | you the basic idea is that we start with "'em" groups |

0:07:21 | each representing a uh one sample at the beginning |

0:07:25 | and that and then that detect a race and we met the neighbour in groups with a minimum cost |

0:07:31 | yeah |

0:07:31 | was the from thing is uh in fact that the difference between the cost of the mets group |

0:07:36 | my knows the costs of the groups before the marriage |

0:07:40 | if we use that the form that it have on before for these colours |

0:07:43 | we will see that it depends upon the mean tons and over the |

0:07:48 | um and group |

0:07:50 | and since these mean time this the outcome of an optimization procedure it wouldn't be got the feast in to |

0:07:55 | compute it for every possible manner |

0:07:58 | what we do instead is that we compute an upper bound for this quantity |

0:08:02 | that does not depend and more on uh the mean down of over the max group but only on the |

0:08:07 | mean tons and of the groups as the are so far |

0:08:10 | uh using this upper bound |

0:08:12 | we start with "'em" groups and then we do our iterations at each a race and we compute the co |

0:08:18 | uh a course for possible mode things |

0:08:20 | we decide which with thing has the minimum cost |

0:08:23 | we perform a and then we find the flat uh |

0:08:26 | to represent the you group as the mean can and of the most group |

0:08:30 | then would say how many groups we having and if we have a the uh the desired level we stop |

0:08:36 | in fact we don't count every single group but only the ones that are significant and uh a significant we |

0:08:42 | define a groups |

0:08:44 | that's how a more that two percent of the total samples |

0:08:46 | that's something that we do it to |

0:08:49 | to be sure that we don't pay too much as |

0:08:52 | ten and a outlier |

0:08:54 | uh at the end of a a a a a of the other read we out to put the the |

0:08:58 | groups that we have found and rate a representative flat |

0:09:01 | but there are a compute it as the sub-spaces corresponding to the D largest eigenvalues of bits groups that them |

0:09:08 | a a in order to to uh figure out to have the put in order to see the performance of |

0:09:13 | a are an we have compared it with three different others are schemes |

0:09:17 | the first one is the high rack of B basic plastering on three |

0:09:20 | which is a top down roads |

0:09:22 | and the linearity measure that is used there is that the deviation between that you played in and the job |

0:09:27 | see this |

0:09:29 | the second one is the hierarchical agglomerative clustering |

0:09:32 | at this is a bottom-up approach to right oh |

0:09:35 | and the linear the method that is used as again the deviation between you played them into this these |

0:09:40 | and and we have so uh compared with the N Q flat |

0:09:44 | this is an one that comes from the family of subspace clustering method |

0:09:48 | is that but it's and not directly applicable to money falls and that's because |

0:09:53 | use use lead they don't have any constraint that would lead them to respect the manifold geometry |

0:09:59 | that's an example is shown here where would have a P are um a for the K |

0:10:04 | and as we see the and the flat that uh the i'm board in the fines for representing the groups |

0:10:09 | the not for all that you much of the minor |

0:10:12 | so for hours |

0:10:13 | that is to have used to that this that the three this swiss roll and the S |

0:10:18 | a training set that was consisting of two thousand points randomly sample |

0:10:22 | and that that thing set from five times and random some |

0:10:26 | for the testing what we are doing is that for each testing sample |

0:10:30 | um we compute the K nearest neighbours |

0:10:32 | and then by doing |

0:10:34 | from the training set |

0:10:35 | and then by doing majority voting or over the we find two weeks flat we should project |

0:10:40 | then by protect yeah finding that distance between uh |

0:10:44 | the sample the actual some when the projection we have a reconstruction error for these |

0:10:49 | one and by summing over a or all of the samples of the testing things we compute the mean square |

0:10:55 | error |

0:10:56 | so the results for this this roll or so here |

0:10:59 | are are of this only with uh or |

0:11:02 | and that's we can and on the horizontal axis we have the number of that |

0:11:06 | and then the part of the goal of it's the mean the reconstruction they're |

0:11:10 | as we can see in general the performance of our scheme is better in space in in the meeting a |

0:11:14 | came she's from fifteen to thirty five |

0:11:18 | um the picture is the same as for the S your data set |

0:11:22 | where the difference is even more all views |

0:11:24 | uh and again we see that the for all the number of lots our scheme performs better than the rest |

0:11:31 | uh and you yeah also plotted the groups that we get for the S two for the case of base |

0:11:37 | to have flat |

0:11:38 | just to verify that each group uh uh is uh and connected region of uh one |

0:11:45 | or what they have presented to you today use that greedy bottom up are going to approximate manifolds meant to |

0:11:50 | be with low dimension how so that |

0:11:53 | the linear you to measure that we use is the differences of dancing |

0:11:56 | and we have seen that |

0:11:58 | for our experiments itself performs a manifold approximation problem |

0:12:03 | possible applications for this scheme could be data compression on multiview view classification in general |

0:12:08 | cases is when we have to recognise a signal from transform |

0:12:11 | for any transform versions of |

0:12:14 | and you |

0:12:16 | i |

0:12:20 | so question |

0:12:28 | i |

0:12:28 | okay okay a question uh what is actually the regularity assumption on your uh any for your a processing of |

0:12:40 | shown because i image |

0:12:42 | you to learn a menu for a summer get one |

0:12:49 | so we can i |

0:12:50 | some some irregularity because we are having a smooth this thing on the tangent space computation but of course for |

0:12:55 | what would have to be able to compute meaningful meaningful dungeons |

0:12:58 | so that it's means that it has to be |

0:13:01 | difference different sample or we have to smooth it at the beginning but for sure we are |

0:13:05 | a a the case is that we can handle a cases of money false that have a uh where are |

0:13:09 | where be states |

0:13:10 | how |

0:13:11 | so we cannot have something like to really rely you like that we got or have to smooth it out |

0:13:16 | in the beginning |

0:13:17 | or to a of small thing |

0:13:18 | during a |

0:13:20 | if you there is just go for that |

0:13:23 | for to approximate |

0:13:26 | you mean like a my folks of real thing mouse |

0:13:29 | but we are trying to do that now it's uh our future work |

0:13:36 | i |

0:13:37 | and questions |

0:13:40 | not |

0:13:41 | i |