i i think you uh many was out slow be presenting uh paper and title the the group two two of for or as i don't because this problem uh by brian kind of a my adviser uh fess with peter image okay so um a well known problem that's been around a while is uh your a title for "'cause" this problem where we uh C to identify and orthogonal cycle transformation between two data matrices and uh this context we seek to minimize the frobenius norm squared of the difference of a in this case the rows of B with rotated rows of a and and this is useful uh uh it to serve as as a as a geometric uh transformation or a geometric rotation to to correct for uh rotational distortion as uh distortions between i mitch each a and B um uh going going to going to a second order version uh there was there's the two sided or because he's problem or you have to uh symmetric similarity matrices C and C are little are can to give is reference and in this in this problem we have rather than um uh de de de rotation during on one side we have it of on both sides and in the form are transpose C R and and again we're trying to match up um this this this to side rotation with with with some uh some reference uh similarity so uh the says a bunch of applications uh one being a a covariance or or correlation a matrix matching so we eight a data matrix to we've induced uh a double rotation on the covariance matrix um a more interesting problem would be a related to a graph matching problem so in a ice metric graph matching we're given uh two adjacency matrices again C and C R and uh this can be weighted um and and and and the at a is to find a a permutation matrix which best uh which we we which serves to to reorder the the nodes of one graph and to the other man in in the other graph uh again matching in in in the for been used um score different so this is a hard problem big given this combinatorial nature uh there are one one one popular method is when me i'm as method um and involves relaxing this problem to an orthogonal uh constraint and then and then finding the closest permutation matrix as the hungarian algorithm okay so if we go back to the of that don't is problem the set this has a known solution and um given uh the the the right circumstances it's a unique and this given by D uh single value decomposition of A transpose be where a singular values um have to be transform just a all ones oh have for the two sided however it's a it's it's a bit more complicated because uh we have a close form solution but it's really family of solutions so if the eigen decomposition of C is is V lamb V transpose and C are B R and R the are transpose then uh a solution to this problem is given by V V D V are transpose where D is a a is a diagonal matrix where each i an entry is plus one or minus one so um right away we see that we can have know to the N possible solutions at least uh so so so i was a bit of a history um in in in this talk we're going to consider scenario area where a for example a graph verdi sees don't just have similarity measures between nodes but um but also there some side information and and so this paper considers case where the a the seize our are parsley later in to go uh so so for example if uh let's say were modeling a face so so you know certain certain notes belong to the i certain nodes belong to the nose set um or we can think of a a spatial configuration so and uh in uh fmri analysis we might have certain regions of interest so we have kind activity is between of voxels but we also know uh what are our why those walks two okay so suppose are are are data dataset is parsley did in capital "'em" groups but is for each point i there's there's some group that i belongs to and that's uh will note that G of i and uh uh uh the goal is to again estimate an R uh a in in in in the two sided of course this problem but here are is gonna be block diagonal and and and those block diagonals correspond to the nature of of the uh problem so uh because problem the uh groups to sided or that process problem and uh of course when when when M equals one we go back to the original a standard to prior so get given T block tan'll constraint um it essentially introduces some some linear constraints that is the elements of this large matrix R a a such that uh the uh it in X P Q zero if um if uh notes P and Q of the and uh some some benefits of this approach are you know the the these constraints regular as the problem so uh we might be reducing the effects of fitting and also were were we're using prior knowledge that uh we know about a about this graph um also we have uh a few were parameters to estimate so it's uh it's a were dealing with a i a a a a a a lower dimensional problem okay so so for "'em" greater than one we so that it it it does have a a close form family solutions but for "'em" greater than one where we're gonna have to look at an iterative uh technique so that let see i i and C I R denote be with an region or within groups similarity structures uh and see i J where i is not equal to J we'll what we'll do not the cross somewhere structures so again given that this is uh going to be an iterative solution uh we'd like to start from a a a a a good starting point and so a logical starting point would be to just consider the you within group similar so we can start it each block element of our rotation matrix to be uh V M M D M V M M R transpose where uh the V and the Rs correspond to the eigen decompositions of oh of uh of uh a C M M and and and M R so and and so from the starting point our goal is to improve on this initial estimate uh by looking at across region similarities so it's broken out into two step process uh the first is trying to assign appropriate plus one's uh plus one or minus one entries for the uh D matrices and it turns out that um and in in in a in expanding the problem this is um reduces to a quadratic binary optimization problem um and so we can use uh uh we we we can get an approximate solution with the max cut uh but but but by by using the algorithms to solve max cut uh the second step after we've got an R are D matrices wouldn't be to uh fine the or that call estimates uh with a with a greedy uh strategy and the way we do that is by selecting um elements that that are within groups and applying uh appropriate givens rotations to decrease the objective and so this is of a form of a block or set i to just a a a a a bit on a the second step um we can we can view the matrix C as a yeah as a noisy rotated version of of of a C R and and because the noise the the the eigen vectors uh of with a group similarity our berger so um the the uh factorisation we had the for uh of our initial estimate of are M equals the M M and D M V M R transpose uh we introduce another matrix you to get the M M U i'm the M V M M R transpose uh and and we view the U M and the D M as a a corrective measure to decrease the object so the D Ms as that's so so the use in the D uh are uh a there to incorporate the cross region similarities to correct the uh perturbed eigenvector so if we let uh you be D block diagonal uh version of of you want through you M then uh what we do is is is in our really approach is is you generate these givens rotations so we we pick um uh elements uh P T in Q T that belong the same group and we successively apply givens rotations to um at at at iteration C to decrease the object um um and at a a P change you T R are found uh using uh and and and and approximate greedy and and the angle is the sample the line search so uh look at some results uh so this was tested on on two data sets the first is set was the N nist handwritten digits um and second was the L B face day so for "'em" nist we have i ten groups for ten digits and for face database we have fifteen groups for uh a fifteen uh people so each group was uh transformed by by a random orthogonal matrix it's of M and then and and corrupted with noise so and you see before you the the the D block diagonal uh elements of this matrix are the with group similarity structures and the and the off diagonal elements are do you cross group and so you'll see uh you know the covariance of uh digit three and seven here point by the okay case at test accuracy we looked at um the our air estimate to noted uh by what what we're each we're reach block of of that rotation matrix is the dated uh and had uh we looked at the the inner product of um a have with a and uh and uh we took the average so so so this as the figure between minus one and one um and we see that uh that at iteration zero uh this is the result if we just take in to account the within group similarities and then as as a T increases the number of iterations increase uh out knowledge of across region similarities uh help us to teen uh a better match and uh we and this is true for both you know of the M nist and the old um going to the more adjusting problem of graph matching so uh each group in this case was was transformed by by some random permutation matrix and then perturbed with noise and uh the rather than solve for the uh am after after solving for each i M at at every time step you then use the hungarian algorithm to get the uh corresponding um permutation matrix and we can and and and we plot the accuracy uh with the correct number of uh permutation matches and uh for the M this there's there's a uh uh a a a a steeper improvement between only looking at within region two looking at across region a in conclusion oh O we pose the group uh version of the you side for this problem orthogonal cycle is problem uh we write an algorithm that's come to fit efficient given D givens rotations um and uh a simple to implement and there's results show that uh effectively you utilizing both within group but across group structure allows as to the estimation of the um and known orthogonal transformation thanks i sure yeah i i to los the obvious question is what can you say that convergence of the skein i would say that um that there are probably local minima given given given the constraints and so the key starting from that that that good initial and and the initial point so um more than that uh i can say no no