Speech Transcript - MULTIDIMENSIONAL ICA AND ITS PERFORMANCE ANALYSIS APPLIED TO CMB OBSERVATIONS

0:00:22	but dimensional ica its performance than of this
0:00:25	applied to observation of the cost make microwave background radiation
0:00:29	this is joint work with the job was like all those all model yeah and and again ms
0:00:34	the motivation for this work is that's follows
0:00:37	that is not or piece of the cost like drive back on radiation also known as a C M B
0:00:42	consist of the small temperature fluctuations in the black body radiation
0:00:47	a left over from the big bang
0:00:49	the C E and is not P is depend on cosmological parameters
0:00:53	and that's mapping that correctly you'll ski key quantitative information about the formation of for universe and it's structure
0:01:00	line set to like of the european space agency which was launched in two thousand and nine
0:01:04	is taking images of of that whole sky it's simply frequency
0:01:08	we can proceed and frequency coverage and a a resolution and sensitivity
0:01:13	and this dog
0:01:14	we apply a second order multidimensional ica criteria and its performance and of the
0:01:19	to esther physical data
0:01:20	we extend form a results to deal with over to me data
0:01:24	and and i don't a mention of a component
0:01:26	finally we show a good match between the empirical and the predicted results
0:01:31	in refuse physical terms
0:01:33	what our paper is to take a set of images
0:01:35	such as this one
0:01:37	and
0:01:37	the the set of images into a there a to the C be the galactic emission and other component
0:01:47	we use a um uh uh the the that data model which use for the multidimensional ica yeah as follows
0:01:53	we use that component model in this model the observations are a sum of N C contribution which we do
0:01:59	not component
0:02:00	each component
0:02:01	X i with index six i
0:02:03	denotes component number i
0:02:06	at the sample index i work side goes from one to L
0:02:09	but i of each of these that component vectors to did not lead does and D where and D denotes
0:02:13	the number of detector
0:02:16	the components can be reconstructed from the observations by the oblique projection
0:02:20	the oblique projection is a matrix which projects on the subspace
0:02:23	in which a component because the is
0:02:25	orthogonality to all the other components
0:02:29	we use a latent model for the components
0:02:31	in this model each component can be regarded as a product of a or extended matrix metrics a high
0:02:37	and the short vector S i
0:02:39	each each as i
0:02:40	is denotes a piecewise stationary stochastic process
0:02:44	with the covariance
0:02:46	and i for each index I the covariance is
0:02:48	indexed by Q
0:02:50	for each side in main Q
0:02:52	where did you do you don't the partition of the induces one to L A to Q domains where Q
0:02:57	smaller than a lower or equal to a
0:03:01	for each the bank you
0:03:02	we can define a block diagonal matrix
0:03:05	this matrix is created
0:03:07	uh by taking a it's block of that gonna the conference batteries is of the process as i
0:03:14	we require that the set of these block to add a no matter says
0:03:18	can cannot be jointly block that can into smaller block
0:03:22	the total length of the vectors as i is the dimension of the signal space which we denote by and
0:03:29	uh by a um
0:03:30	concatenating side by side of that tall rectangle are or says a i to construct a matrix
0:03:35	which we did notice a a and required that these metrics be full column run
0:03:40	the dimension mention of the signal some of the signal this space
0:03:44	is larger than or equal to the number of components and smaller than or equal to the number of detectors
0:03:49	where a quality on both sides texas back to the simplest ica model
0:03:54	we perform component separation via maximum likelihood
0:03:58	maximum likelihood can be obtained from the approximate chunk block diagonalization
0:04:03	of the empirical localised covariance mattresses of the observations
0:04:07	by the inverse of the matrix a
0:04:10	the localized in jericho covariance recall conference mattresses of the observations
0:04:13	are that
0:04:14	normalized the sum of the outer product of the observations at each the
0:04:20	note that because that a joint block regularization
0:04:22	is possible only if you use
0:04:24	that
0:04:25	exact or parametric or the X
0:04:27	expectation
0:04:28	of the localized empirical covariance mattresses
0:04:32	therefore for the maximum likelihood procedure consists of a
0:04:35	first
0:04:35	joint a block diagonal i think of empirical localised covariance mattresses of the observations
0:04:40	from these we can
0:04:42	estimate the metrics say from this matrix
0:04:44	we can construct a the set of oblique projection is
0:04:48	on all the component
0:04:49	and by applying of this
0:04:51	uh estimate of the a of the oblique projections on the observations we can estimate
0:04:57	but components
0:04:59	we obtained component separation as desired
0:05:02	note that this month multidimensional ica procedure is a generalization of the well-known ica
0:05:10	in order to evaluate the performance of this a multidimensional ica criterion
0:05:14	we use the mean square error
0:05:16	the empirical mean square error is calculated for each component
0:05:20	in each detector
0:05:21	and in each domain
0:05:23	and it is
0:05:24	defined as that
0:05:26	a normalized uh
0:05:28	some of the
0:05:29	squared differences of the
0:05:31	empirical yeah the exact component
0:05:35	note that if the model then the mean-square the testing the estimated mean square error is larger than zero only
0:05:42	due to finite data that is due to the fact
0:05:45	the the sound but the number of samples in each domain is finite
0:05:49	if the model holds
0:05:51	then the S they the
0:05:53	for
0:05:53	that expectation of the empirical mean square error at each component to the detector and domain
0:05:59	is given by the following expression
0:06:02	this is an expression which is a function only of the exact or parametric covariance matrices of each of the
0:06:08	component
0:06:09	we shall not go into details about this
0:06:11	and complicated expression but we should validate
0:06:14	it
0:06:15	in numerical simulations later
0:06:19	we now define the ester physical data in terms of the multidimensional ica model in order to separate the cosmic
0:06:26	microwave background radiation from the other components
0:06:30	we use a statistical model for the stand for the C B temperature and is entropy
0:06:35	according to just referred to go theory
0:06:37	the
0:06:38	cosmic microwave background radiation
0:06:40	is modelled as a zero-mean is zero-mean gaussian stationary process on the sphere
0:06:45	with an angular power spectrum C L
0:06:47	this is a a typical you much of the cosmic microwave background radiation
0:06:52	and this is
0:06:53	an illustration of the angular power spectrum
0:06:55	where we should explain just terms in the next slide
0:06:59	in terms of a lot of the C B
0:07:01	is that one dimensional component we do not it and C N B Y
0:07:05	yeah
0:07:05	the covariance mattresses of the simply at each uh in excel or angular frequency
0:07:11	are given by the
0:07:12	product of the angular power spectrum that these index L
0:07:16	times the outer product of vectors acm we're vector A C B
0:07:20	reflects the C B emission log in all the energy detectors
0:07:25	the angular power spectrum is defined as follows
0:07:29	two dimensional function on the surface of the sphere which is indexed by the as well
0:07:34	and the polar angle
0:07:36	i
0:07:37	transformed using the spherical harmonic transform into
0:07:41	a set of coefficients indexed by two induces L where L is larger than or equal to zero and N
0:07:47	is between minus plus and
0:07:49	i of them are the coefficients of the spherical harmonic transform
0:07:52	therefore
0:07:53	the angular power spectrum of a random stationary process on the sphere
0:07:58	for each
0:07:59	angular frequency
0:08:00	is given by the
0:08:02	average of the expectation of the
0:08:06	um
0:08:08	of the uh
0:08:10	uh
0:08:11	sorry
0:08:13	of the square of the coefficients idea that
0:08:17	i told em mode
0:08:18	well of the uh
0:08:20	uh
0:08:21	all the respective a modes
0:08:22	where the can mode
0:08:24	in this strike a money transform the indicate directionality and therefore for their are averaged out from these
0:08:29	i've i've out from this expression
0:08:33	in order to model the galactic emission
0:08:36	uh
0:08:37	we have to consider the following properties
0:08:39	the galactic emission
0:08:41	which
0:08:41	a typical image of it is that a given here in one of the C be frequency
0:08:45	is a superposition of several physical processes
0:08:49	you just passion correlation
0:08:51	this process is are cup out
0:08:53	therefore for in our model we can regard this galactic emission is one multidimensional component
0:09:00	we suggest to use the deterministic model for the galactic emission
0:09:04	since are in in our model price and requires
0:09:08	only that they
0:09:09	uh
0:09:10	strike a money a transform coefficients of the cosmic microwave background radiation and awfully galactic emission be uncorrelated for all
0:09:17	this is
0:09:18	then
0:09:18	note that if the galactic emission is deterministic and only the C is randomized
0:09:23	this indeed a hold
0:09:24	since there
0:09:25	C and B
0:09:27	is a random process with a zero zero-mean which is a from a physical re
0:09:33	therefore
0:09:33	we
0:09:34	two we choose to tape
0:09:36	that
0:09:37	it very cool
0:09:38	localised covariance france mattresses of the galactic emission instead of their parametric counterpart
0:09:46	we now use
0:09:47	a simplified yet close to realistic setup for our a very close study
0:09:52	first
0:09:53	we consider only two components the C B of course
0:09:56	and that that i think condition
0:09:58	this and
0:09:59	options are possible from desperate
0:10:01	from the ester physical perspective in the range of in this is a between two and nine hundred
0:10:07	since our model is free of noise
0:10:09	we all to noise our simulation
0:10:13	we now a partition the range of in this is between two and nine hundred in two consecutive non overlapping
0:10:19	is
0:10:19	each of length five L note
0:10:22	we take the number of detectors to be nine
0:10:24	which is the same as in the plant experiment
0:10:27	our simulations are based on the plants sky model
0:10:29	which is a soft or back and we can buy the black
0:10:32	uh component separation working group
0:10:35	i it creates a realistic images of this kind emissions missions at C
0:10:38	i the the C and the frequencies
0:10:41	further
0:10:43	the regions in the maps where the C B is to compare related are method out
0:10:47	therefore for the set of for data can it's of nine images is more or less like
0:10:52	this one
0:10:56	now
0:10:57	it turns out that number the multi dimensional i think yeah
0:11:01	but a model which we have just present that can to be applied directly to the data
0:11:06	this happens
0:11:07	uh due to the following point
0:11:09	are are as this requires that the the signal space they mentioned be what a number of detectors
0:11:15	since and they mention of the C B is one which is a
0:11:18	well known from mister physical
0:11:20	and know this
0:11:21	this means that we for the dimension of the galactic emission to be and
0:11:24	however
0:11:25	since the eigenvalues of the empirical localised go fast as of the galactic emission with a channel orders of magnitude
0:11:34	that the joint loved the notation of that
0:11:37	observations of the it trickle a guys covariance friends mattresses of the observation
0:11:41	with the with block men's is one and eight
0:11:44	is in ill conditioned
0:11:46	this means that in practice
0:11:48	the the dimension of the signal space is smaller than the number of detectors
0:11:51	and and are were determined problem
0:11:54	since we do not want to ignore detectors
0:11:56	then we have to answer the following questions
0:11:59	first
0:11:59	given the correct dimension of the galactic emission how can we applied the maximum likelihood
0:12:04	and the joint block localisation
0:12:06	this set and what is the correct them mention for the galactic emission
0:12:11	so
0:12:12	for
0:12:13	given the right that they mentioned for the galactic emission we want to turn the over determined problem into that
0:12:18	to one
0:12:19	we do this
0:12:20	uh V a man's production using a
0:12:22	principal component analysis
0:12:25	first dimension mention of the signal space is the dimension of the galactic emission last one
0:12:30	we now take to the first and S singular vectors of the empirical conference the of all the observations
0:12:36	in two
0:12:37	uh the full rate then a matrix U S
0:12:40	and the transpose of this metrics
0:12:42	project
0:12:43	the N D dimensional observations
0:12:45	onto to a reduced
0:12:46	and S dimensional vector space
0:12:50	we don't take in period a local go fast mattresses as
0:12:53	of the observations in there were just the vector space
0:12:57	and used them as the input to the joint block to globalisation
0:13:00	which now works
0:13:01	and and the output is an invertible matrix which we note by a
0:13:05	not a a and this one has indeed rank and is
0:13:09	we are used
0:13:09	the inverse projection to expand this matrix back to the and D dimensional observation space
0:13:15	and again we have obtained obtained an estimate of the mixing a sort of
0:13:20	and mixing matrix eight
0:13:22	metrics a a then from this matrix again we can
0:13:25	an estimate the oblique projection my says
0:13:27	and apply and all to on the observations
0:13:30	and obtain estimates of the components
0:13:33	a a to obtained
0:13:34	component separation as desired
0:13:37	we now discuss
0:13:39	the problem of
0:13:40	selecting the order or the they dimension of the galactic emission
0:13:44	in general
0:13:45	the considerations are that it's this they it should be the smallest one for which the model hold
0:13:50	with a certain tolerated error
0:13:53	if we choose a dimension which is too small this means that we give a wrong model for the data
0:13:58	and therefore are
0:14:00	you model our separation will not work
0:14:02	if we choose a no it in which is too large
0:14:05	then again to go back to the bad condition problem
0:14:08	and also we shall have a redundant parameters
0:14:13	so in order to select that the base and for the galactic emission
0:14:16	we use
0:14:16	no of the
0:14:18	experiments
0:14:19	and this figure
0:14:20	some this experiment
0:14:23	for each candidate they mention of the galactic emission
0:14:26	we can relate to the empirical
0:14:28	many square error twice
0:14:30	in this example we
0:14:32	uh is for the reconstruction of the cosmic microwave background background uh
0:14:36	component
0:14:38	once we got calculate the empirical mean square error
0:14:41	well the finite data or that is the normal way
0:14:44	and second we calculated it without out to find a day errors
0:14:47	this is obtained by performing the joint block globalisation
0:14:51	on the exact or
0:14:53	that that the uh the exact parameters which are the expectation of that
0:14:57	empirical localised conference and
0:15:00	but off the observations
0:15:02	this is a a this evening page
0:15:04	i think but in condition
0:15:07	in each case
0:15:08	the empirical mean square error is it is
0:15:11	averaged over forty but want to colour trials
0:15:14	so the results
0:15:15	for to um
0:15:17	first
0:15:18	the results are summarised in the following figure
0:15:20	first
0:15:21	uh a for that it can lead a mention of four
0:15:24	we see that the the graph the blue line and the red circles design
0:15:29	are separate
0:15:32	a this means that indeed
0:15:34	the the reason um
0:15:36	for this they for this and that for this dimension of the galactic emission
0:15:40	in the the a data error or is that is the dominant factor in the mean square or
0:15:45	which is the uh you is that the
0:15:48	seems that the model is okay
0:15:50	we compared it to the to be compared this to a they mention of three
0:15:54	in this case we see that the both results
0:15:56	that it's a blue line and the line was uh
0:16:00	it's a it's a a green line and a good and a with the red is
0:16:04	black circles
0:16:05	we that is lies all over overlap
0:16:08	so first we see that
0:16:09	the mean square error
0:16:10	for dimensional three is significantly larger of that the error for the mint for four which is it's not a
0:16:16	very favourable
0:16:17	and second this implies
0:16:19	that that for the dimension of three the model error is done in
0:16:24	we have also run this experiment with a the think they mentioned candidate of five which is not to the
0:16:28	big
0:16:29	and the results were very very similar to the results in the of four
0:16:34	we have also a similar trend
0:16:36	with there
0:16:37	uh
0:16:38	running the send experiment on uh the mean square error in either uh
0:16:42	detectors
0:16:43	as well as a with the reconstruction of the galactic emission component
0:16:47	and the results uh we're always so a similar
0:16:50	so we conclude
0:16:51	that the best uh they mention to describe the galactic emission
0:16:55	is a for
0:16:58	now out using a a a can i of four
0:17:01	we would like to compare our
0:17:03	um
0:17:04	that's your right decoder prediction of the mean square error in which was the complicated expression which we have shown
0:17:10	several slides before
0:17:12	we want to compare to the empirical results
0:17:14	so in this figure
0:17:17	the blue colours
0:17:18	do not that the empirical mean square error again calculated for the C B component in one of the detector
0:17:23	channel
0:17:25	there are that blue line denotes the average of the empirical mean square error or over the of trials
0:17:32	and the for called the lines and do not to how the standard deviation of of is fourteen want to
0:17:37	color trials
0:17:38	the red line the big
0:17:40	the predicted as you're code mean square
0:17:43	so
0:17:44	and
0:17:44	we normalize normalized
0:17:46	this don't mean square errors as the precoder then that your a by one they are normalized
0:17:50	for each index L for each and a lower frequency yeah
0:17:53	by the end what our power spectrum
0:17:55	all that C M B component
0:17:58	so first we see that the normalized mean square error
0:18:01	is more or less
0:18:02	and at
0:18:03	at to the value of ten to the minus four
0:18:07	this means that the error is relatively small and it indicates one separation
0:18:12	second we see that the predicted value
0:18:15	is with that standard deviation margin
0:18:19	we we again
0:18:20	we obtain seem not trends at uh and the scent experience with different a
0:18:25	detector frequencies and also for the reconstruction of the galactic emission component
0:18:30	so these results
0:18:31	validate date
0:18:32	our theoretical prediction of the mean square error
0:18:36	for this one and for this data
0:18:40	so to summarise
0:18:41	we applied to multidimensional ica
0:18:44	uh method and its performance and was this
0:18:46	in terms of components
0:18:48	first or physical data
0:18:49	we extend and form results
0:18:51	to do with over the it's data and and and and a mention of a component
0:18:55	and we have shown good match between the pure ego and the predicted result
0:19:00	finally we we acknowledge to use of the plans a model developed by the components suppression working group of the
0:19:05	plan collaboration
0:19:07	and Q
0:19:15	oh
0:19:16	um
0:19:17	my first question
0:19:19	what do you
0:19:21	or
0:19:23	so with that might you have any performed
0:19:31	the horse yes
0:19:32	there is a and we have shown this in paper which we showed in and
0:19:36	a conference
0:19:38	as as be conference
0:19:39	yeah you're right there is a
0:19:41	then
0:19:42	i mean
0:19:42	a what in this in this data that which does not where the model does not hold exactly and we
0:19:47	have actually ignored the fact
0:19:49	we assume that the the
0:19:51	power spectral of are piecewise stationary but in the in in practice they or not
0:19:56	so there
0:19:57	the model does not hold exactly so in practice we compared our results to also to classical ica
0:20:03	with a rank of the data
0:20:04	and in fact the results are very similar
0:20:08	but
0:20:09	for a tech
0:20:12	and a
0:20:14	what
0:20:15	actually like
0:20:17	i
0:20:18	how the pen
0:20:20	that
0:20:21	for
0:20:23	yeah but this method does not separate the pair that components it is only in tandem
0:20:27	to a to money
0:20:29	separate
0:20:31	but
0:20:32	independent components
0:20:34	and
0:20:34	it does not separate to and compose in
0:20:37	them so that is can not
0:20:41	what
0:20:43	oh okay you're basic to to the band
0:20:47	got like to call home
0:20:49	yeah
0:20:51	uh_huh so
0:20:52	i cannot see that
0:20:55	first uh that one advantage is that
0:20:58	a theoretical performance and all of this
0:21:00	which is derived from that multi dimensional model
0:21:03	i do not now a performance was which is drive
0:21:07	from the uh
0:21:08	classical ica
0:21:10	so
0:21:11	this is a set of rules
0:21:13	there are some
0:21:16	working on the
0:21:19	think all source
0:21:21	and it is also a
0:21:23	try
0:21:24	so
0:21:25	and
0:21:27	i
0:21:29	this your L
0:21:31	i
0:21:33	so one i mean
0:21:35	right now nice
0:21:36	okay
0:21:36	right
0:21:38	all
0:21:39	what should be
0:21:41	what you actually
0:21:43	we first you're a okay first this is a in this article working the very preliminary work using this um
0:21:49	algorithm
0:21:50	it is not so a it does not intend or uh
0:21:55	say that is going to
0:21:56	to be better than at there
0:21:58	in a method especially does not in it may be used
0:22:01	it does not intend to separate the
0:22:03	uh uh components but it may be used as a a perhaps a better to like to estimate
0:22:09	i separate the C M D from the galactic emission
0:22:13	using a blind some some which has an advantage at least in some parts of the data it is not
0:22:17	intended to be a better solution for
0:22:20	uh
0:22:21	instead of all the other uh
0:22:23	uh methods which are applied to ica
0:22:25	i don't mess may be better mean
0:22:27	uh separating the
0:22:29	for is that may be better separate the components and and the dependent components but they may be more sensitive
0:22:36	to some
0:22:37	and their properties of the data which this but that is not since it is long
0:22:42	one but we know that yes
0:22:44	only well
0:22:45	you know it's a maximum for forty
0:22:49	um
0:22:50	for ecological
0:22:52	but
0:22:53	what we actually or oh encroach on their mission
0:22:57	collect all conditions
0:22:59	here
0:23:01	what can be
0:23:02	for all
0:23:04	it is not right for you get it is not for one probably if we talk about
0:23:08	which was asking more of the galactic emission
0:23:12	then
0:23:12	yeah
0:23:13	we would have time to a more i is um
0:23:17	then the um
0:23:19	the coupling between is uh
0:23:21	this is really
0:23:23	galactic emission a sources which you have mentioned
0:23:25	maybe a
0:23:26	more negligible and
0:23:28	very likely if we had um
0:23:30	why mess
0:23:32	and we would of work on the of the um
0:23:34	higher are
0:23:35	and all of the higher
0:23:37	alright of of this not and we would have obtained to collecting the middle from this dimension is only intended
0:23:43	to run the separation it is not
0:23:45	it does not have a store physical meaning by itself
0:23:48	it is say a parameter
0:23:50	for this operation
0:23:52	i
0:23:53	but really to to what you
0:23:55	okay
0:23:57	but can think that the part in the middle
0:24:00	you're basically here so
0:24:03	are writers or you
0:24:06	okay
0:24:07	do like to compose or or the colour
0:24:09	way
0:24:10	joe are or like to say
0:24:13	you all there i guess is this is a very simple example we can also choose
0:24:17	different patches of the sky perhaps
0:24:20	only the galactic uh
0:24:21	play or
0:24:22	after their as which contain only part of the a a use of the sky which i have
0:24:27	i and use again this algorithm one these patches and you probably have
0:24:31	different results
0:24:33	this again this is just a a simplified example
0:24:35	okay thank you
0:24:36	thank you

MULTIDIMENSIONAL ICA AND ITS PERFORMANCE ANALYSIS APPLIED TO CMB OBSERVATIONS

Source Separation and Applications

Presented by: Dana Lahat, Author(s): Dana Lahat, Tel Aviv University, Israel; Jean-François Cardoso, Télécom ParisTech, France; Maude Le Jeune, Université Denis Diderot-Paris VII, France; Hagit Messer, Tel Aviv University, Israel