Speech Transcript - MAJORIZATION-MINIMIZATION ALGORITHM FOR SMOOTH ITAKURA-SAITO NONNEGATIVE MATRIX FACTORIZATION

0:00:17	so um this is a talk about uh if the class i two um
0:00:22	a non-negative matrix factorization
0:00:24	so the outline is uh as follows that will uh uh briefly recall some uh
0:00:29	previous work about uh i S nmf and uh
0:00:33	describe the
0:00:34	latent a statistical model um
0:00:37	a two D is uh
0:00:38	specific uh and N F L
0:00:41	uh then that will uh present some uh a way of actually uh smoothing the activation the question
0:00:49	and
0:00:50	the major contribution
0:00:52	work
0:00:53	uh
0:00:53	going along with of course an algorithm
0:00:56	which is uh going to be a majorization minimization
0:00:59	algorithms
0:01:00	and the
0:01:01	before giving a few entries
0:01:04	so this is only to uh introduce my annotations of them
0:01:09	the data here the non-negative data here is going to be a view of dimensions F by and so
0:01:16	a stack frequency here and and is a number of frame that
0:01:20	and
0:01:21	and the dictionary matrix W the activation matrix H at the number of components K
0:01:27	and um so uh
0:01:30	and then F A
0:01:32	usually involves a
0:01:33	minimizing in
0:01:35	uh
0:01:36	a quaternion of uh this form and are non negativity constraint of the value
0:01:40	H
0:01:41	with specific uh
0:01:43	cost function here which in our case will be it a quite effective that mentions that that will
0:01:47	intra justly
0:01:49	so the applicative context here is a unsupervised the music you have representation
0:01:54	uh so we will deal with a um
0:01:56	audio spectrograms
0:01:58	okay
0:01:59	and the idea is to uh
0:02:01	learn some uh representative the spectral patterns of thoughts on the spectrogram
0:02:06	spectrogram
0:02:08	uh and of course uh so there's which you should questions about uh how to choose who V uh whether
0:02:16	you should choose the magnitude or
0:02:17	or spectrogram uh how to choose the measure of fit that using the
0:02:21	in the decomposition
0:02:22	and also and then estimate can did you was um
0:02:26	uh a wrong K approximation you approximate the spectrogram is the son of
0:02:30	uh a rank one spectral grounds and
0:02:33	question is uh if i want to retrieve a uh the comp the corresponding components in the time domain
0:02:38	uh how should i in this or rank one a spectrograms an
0:02:42	how should i did phase in
0:02:44	so well
0:02:46	um
0:02:47	a generative approach uh for in serving this the questions that
0:02:51	is uh
0:02:52	what every shot to a tech wise i two and then S um
0:02:55	uh
0:02:56	the uh with a latent model that is as follows so
0:03:00	uh let X
0:03:01	B you or uh complex value the uh stft "'em"
0:03:06	oh the signal you want to decompose so
0:03:08	X is different from V O X is a a complex a complex value of data
0:03:13	it's you assume that um
0:03:15	yeah data so a comp the a time-frequency coefficient
0:03:18	uh it's uh uh F and that
0:03:21	uh
0:03:22	he's a sum of uh components C K S and so okay the uh component index set at feast can
0:03:28	see and is the frame
0:03:30	uh complex value the such that this coefficient a
0:03:35	as uh
0:03:36	complex a circular a complex uh a gaussian distribution a
0:03:41	uh meaning and uh you have to run done the phase them
0:03:44	uh we such a structure on the variance and so basically a rank one uh a structure in the on
0:03:49	the variance
0:03:50	then you can it very easy to show uh uh i assuming that the components are independent
0:03:55	uh that the log-likelihood function and
0:03:57	easy quite two
0:03:59	uh uh it that quite set to they've jones between the power spectrogram of of data so that's
0:04:05	uh absolute value of X to the square
0:04:08	um
0:04:10	where the it why quite side to uh had a chance is uh defined a
0:04:13	by this equation
0:04:16	um uh so that's um
0:04:20	quite a nice uh thing to i mean to this model is quite a nice thing uh to high because
0:04:26	it's a
0:04:27	to truly we uh
0:04:28	it's a it's a proper uh generative model of the of the spectral grand
0:04:32	and in particular
0:04:34	if uh you
0:04:36	quantity of interest uh is the um
0:04:39	and the the the the components of
0:04:41	you can uh construct these components
0:04:43	uh in a statistically run the the way for example to take the the and in C uh estimate of
0:04:50	coefficient uh uh component a high frequency F and and
0:04:54	it simply
0:04:55	uh of in a filter out the time-frequency mask
0:04:58	applied to at of summation
0:05:00	uh and uh
0:05:01	the time-frequency mask mask to defined by the contribution of
0:05:05	uh that component
0:05:07	uh in terms of ions divided by the variance of all the company
0:05:13	so that's uh that's it a quite so in M F the the the the the the basic that don't
0:05:19	so
0:05:19	of course uh audio
0:05:22	exhibit some some a time cost stance some
0:05:24	are we didn't see
0:05:25	and uh i taking this uh information to account uh tan
0:05:31	lee to uh more the estimation of H and thus so the value of a reduced to uh i don't
0:05:36	see beach ambiguities a
0:05:38	and they're still to charlie
0:05:40	more present component reconstruction
0:05:42	so in this work
0:05:44	we we want to uh
0:05:46	so if a P the uh and then F uh a problem not
0:05:49	where we uh
0:05:51	at the uh
0:05:53	a and i'll see a function a which measures on
0:05:56	this looks nests of the rows of H
0:05:59	another the question is a how should we uh
0:06:01	choose all uh bill this uh this most knots constraint then
0:06:06	and again a we can take a generative approach uh which is what we did in uh in in previous
0:06:11	work as well where don't
0:06:12	where we propose to uh
0:06:15	model the smoothness of the activation coefficients in terms of uh markov of chain so
0:06:20	either our are in house gonna all an on of change could the present
0:06:24	and non negativity so the id simply
0:06:27	to assume a prior a for H T and now so
0:06:30	um
0:06:31	the activation creation of
0:06:33	uh component K at frame hand
0:06:35	to be searched that
0:06:36	the mode would of
0:06:37	this uh a distribution is obtained
0:06:40	uh i
0:06:41	the coefficient
0:06:42	in the previous frame
0:06:43	okay
0:06:44	so you can basically uh a black here a ga now and again a distribution and that
0:06:49	so you obtain
0:06:50	this kind of a questions
0:06:52	and you get a um
0:06:54	a shape parameter and five here
0:06:56	which controls the peak S of them over the
0:06:59	around around uh the previous value you uh H K
0:07:01	and minus
0:07:04	so it you do uh map estimation of uh using this uh this prior for for H
0:07:09	you will get them
0:07:11	uh and optimization problem of just formal case so this is your fit data
0:07:15	and that this down R is simply the the minus log of uh are you all the point on that
0:07:20	have just find
0:07:22	so in the case of in can an arc of change
0:07:24	you will get a function of
0:07:27	of this uh of
0:07:29	and and five is uh like pitch for the shape parameter
0:07:33	that you use in the in the prior distribution
0:07:36	so you get something which is very close to two
0:07:39	the it tech Y to measure between uh H
0:07:44	and its shifted the action uh from uh one frame a yeah i them
0:07:49	plus
0:07:49	a lock function here
0:07:51	and this like function here is quite annoying
0:07:54	because it is going to mean use uh and ill posed um
0:07:58	minimization problem
0:08:01	in the the uh because of that down a
0:08:04	if you look at
0:08:05	at uh you objective function or
0:08:07	for a given W and page
0:08:09	and if you risk a
0:08:11	this um
0:08:13	this is a a couple of W and H M
0:08:16	okay by you deck gonna metric tell to
0:08:19	we should with diagonal down to they'll take yeah then a
0:08:22	you can choose a the scale here
0:08:24	search sufficiently small so that you decrease a
0:08:28	this objective function okay so this will push the solutions of
0:08:32	to all
0:08:33	a degenerate eight a degenerate the solution the
0:08:36	uh like this one
0:08:38	so a natural question is a can i just
0:08:41	uh
0:08:42	removal this down oh
0:08:44	and the answer is yes of course you can and it's even something
0:08:48	rather a reasonable to do
0:08:49	because
0:08:50	if you uh a re are of the expression of your or this a measure
0:08:55	actually you can see that this
0:08:57	uh a one of a i'll far uh
0:08:59	times like um
0:09:01	down or
0:09:02	actually uh uh when and five becomes a sufficiently super or you a greater than than one
0:09:07	it basic is going to can
0:09:10	so basically
0:09:11	uh is quite reasonable to replace
0:09:14	yup open that you function by simply
0:09:16	the uh uh a a it once i to update are in between that
0:09:20	H H and its shifted uh the uh one frame of being
0:09:25	okay that that gives you a natural sure of us skating valiant us nets the michelle
0:09:30	and uh there's no need to control a the norm of the value you uh in this uh in this
0:09:34	of to this and program
0:09:35	so it's it's rubber convenient
0:09:38	okay so
0:09:39	mm in let's talk about the i agree with that is now well um
0:09:43	i would skip the most of the details and you can uh brief of to do it to the paper
0:09:47	for more information on
0:09:50	so one approach them
0:09:52	um
0:09:53	to solve the all uh generalized the
0:09:55	and then F problem of
0:09:56	is to be the em algorithm none
0:09:58	uh well you could use them
0:10:00	this latent components that i introduced introduced number that a
0:10:03	uh as complete data
0:10:05	okay so this is we did the similar thing it um
0:10:08	in previous uh in this work for a another of our uh another and that G
0:10:12	function
0:10:13	uh the problem is that this i great i'm is quite slow because the augmented data the missing data is
0:10:18	very large uh
0:10:20	as compared to the uh
0:10:22	available data
0:10:23	so it is to a a very uh slowly converging agreed on the
0:10:27	and we here propose them
0:10:28	and new approach uh based on uh majorization minimization
0:10:32	uh which does not uh required to to men the data uh meaning to uh to use the the latent
0:10:38	component C
0:10:39	in the n-gram
0:10:42	uh hmmm so it works uh um as um as described so this is
0:10:47	our objective function okay and so we we produce an iterative algorithm one
0:10:52	which updates dates that will you given age so that's
0:10:55	uh
0:10:56	stormed down and on the and then S we note to do that
0:11:00	and then that we are going to update the columns of H M
0:11:04	sequentially
0:11:05	even a the current update uh uh the but you and uh given the and values of the neighbours
0:11:12	uh of H and so frame and minus one one and and minus uh and and and plus one
0:11:17	okay
0:11:18	so this problem here uh
0:11:22	boils down to this uh
0:11:24	so problem
0:11:26	okay
0:11:27	uh where you uh you want to uh minimize this the function which depend on the on the vector H
0:11:33	and for this uh uh we will use a um that and and a
0:11:37	uh i agree about no
0:11:39	which uh is just on the out the optimization a procedure one
0:11:44	uh so it's an to achieve a posted you
0:11:47	where uh given a
0:11:49	uh i rent a data
0:11:51	H T and there
0:11:52	okay
0:11:53	so in blue we have the the function that we want to to minimize that
0:11:57	so locally in the in the cure and update that
0:12:00	we simply construct a
0:12:01	a server a gate the all sherry function which is easier to minimize and uh the the the original cost
0:12:07	function
0:12:09	okay
0:12:09	and then we need nice this function instead of uh the blue one
0:12:13	and then we get a new date and then we to rate and it leads to a descent i them
0:12:19	which will
0:12:20	uh converse to the to the mean and
0:12:22	so the question is a of course how to be a little uh such an ox you are a function
0:12:28	um and i'm not going to give the details a a here but basically the principal as are
0:12:33	um
0:12:35	so in your function you have a uh a the fit to data and the and it down for the
0:12:40	fit uh to data you can actually uh match arise it done
0:12:44	you can match rise is comics part using a uh a jensen inequality
0:12:48	uh you can approximate much or right this can make part by a a first order taylor approximation and and
0:12:53	as a matter of fact
0:12:54	you don't need to much or nice to measure i sorry
0:12:57	the been that sit down because uh uh you are you get um a tractable update without to
0:13:02	necessary doing this
0:13:04	and in the end of you get a very simple a date to a question okay so that's
0:13:09	really really simple to implement a
0:13:12	uh where the contribution of the the prior priors on the pin that it down on in the red and
0:13:16	so that's if you set long to
0:13:18	the to to zero you simply get the storm down the
0:13:22	it tech i set to and an F uh of day
0:13:24	okay
0:13:27	okay so um
0:13:29	now we can have a we can look at a few um
0:13:31	of to result
0:13:33	um so i basically applied this uh uh uh a penalized the
0:13:38	so this move like the quite set to an an F i grieve them to some uh
0:13:42	all the uh uh jazz to um
0:13:45	music the music signal so the
0:13:48	the the power spectrum i'm here it sounds like this
0:13:52	where X
0:14:03	it was
0:14:04	a
0:14:06	i
0:14:07	a
0:14:10	a
0:14:11	a
0:14:13	a
0:14:16	a
0:14:17	a
0:14:19	a
0:14:20	so and so on
0:14:22	and that um
0:14:24	so first term
0:14:26	let's compare
0:14:27	the
0:14:28	uh a convergence in "'em" of uh object objective uh a function value one
0:14:34	of uh this and then i agree about a
0:14:37	uh of us use the em algorithm that we could have a um
0:14:41	done so using that you could do using uh this uh
0:14:44	this component as a a late and by about "'em"
0:14:47	and you can you can see that the the improvement uh a of the and then allegory of i'm is
0:14:51	quite a significant a
0:14:53	so this is a a log scale a and this is a desired the iteration
0:14:56	and it trends a pretty fast on it's close to uh a C P U to real-time time
0:15:01	on the store now the still not compute
0:15:04	and the uh so this is the effect of uh uh the regularization for a a values values are
0:15:11	of of the the pin G uh weight them so the the parameter or um that
0:15:15	uh
0:15:18	and uh
0:15:21	so this is the baseline and and pin lies the and then F ten than on the was one um
0:15:26	they quantize ten and uh one and read them
0:15:30	and fortunately i don't have a magic uh
0:15:33	but a two uh what too much telly uh uh that i mean the
0:15:37	the right uh the right uh that you uh you have to be a
0:15:40	you have to
0:15:41	it has to be user defined a
0:15:43	a to could D in a in a us on the editing eating uh
0:15:46	uh sitting at this on is you know we'd have to uh tune this parameter according to do the design
0:15:51	does
0:15:54	a case i don't know uh do i have uh
0:15:57	quite something in it so that's
0:15:59	okay so i uh to to finish i um
0:16:02	i wanted to uh uh show you
0:16:05	um the structure of the time-frequency mask
0:16:08	that are around by the decomposition K because i think it's a
0:16:12	it's quite interesting to
0:16:14	to see here these uh the structure a and to see that actually
0:16:18	with a limited number of components
0:16:19	so take once ten
0:16:21	for that uh two minute uh
0:16:24	a a piece of um
0:16:25	of of music
0:16:27	you can learn some interesting things
0:16:28	okay
0:16:29	so the time-frequency mask remember all the other wiener filter or else that you know games that you apply yeah
0:16:34	uh uh to the observation to reconstruct a uh each of the the component
0:16:40	so this is the first uh
0:16:42	the first uh
0:16:43	time-frequency frequency a mass school
0:16:45	the values the zero is a
0:16:48	um
0:16:49	white and uh one is a back
0:16:52	and the uh you get different uh a structure so here you you get the rather wide the
0:16:58	uh
0:16:59	wideband the E major
0:17:01	and you get a so uh
0:17:04	more pitch structure so for example we can use sent to one of this
0:17:08	uh structure
0:17:10	to one of these components typically
0:17:12	uh
0:17:13	this is going to uh capture it's of uh notes
0:17:16	it sounds like this
0:17:32	i
0:17:34	and we now know that uh
0:17:35	this is not actually the
0:17:37	the time no
0:17:39	uh the component
0:17:40	it simply the mask
0:17:42	okay that you applied to the observation so
0:17:44	it means that even if you have some uh uh
0:17:48	uh values uh a yeah to one at some place if there is nothing in at this
0:17:52	a time-frequency point in the data
0:17:55	uh
0:17:56	you you get a uh
0:17:57	uh S T a estimated the um
0:18:00	has spectrum which is a which is a which is zero okay
0:18:03	uh and for example we can know this is another type of uh a time-frequency structure which is a a
0:18:08	white band are
0:18:09	and uh this captures a uh
0:18:12	the at tax of the of the buttons
0:18:14	i
0:18:15	a
0:18:17	a
0:18:20	a
0:18:21	a
0:18:21	a
0:18:23	i
0:18:25	i
0:18:26	okay and so on a
0:18:28	so have ten components like to like this um
0:18:33	uh this one uh G to a clearly uh shows the the bass okay so it's just a low bass
0:18:39	uh for uh component and this one a shows that this noise which is present on the on the recording
0:18:44	so it's so
0:18:46	it's uh
0:18:47	hi pass we can see we can listen to it
0:18:53	a basic is just noise
0:18:56	so
0:18:57	you do don't some things them uh even and uh we've a limited number of of components are
0:19:02	and you can do uh
0:19:05	nice uh sound the um
0:19:07	well this this type of decomposition can have some a nice uh
0:19:12	uh oh joy reading a tradition for example am
0:19:15	you can uh uh use a so basically you have decompose your original "'em" the recording yeah
0:19:21	into a number of uh of components so this involves some manual grouping to actually a reconstructed this uh uh
0:19:27	the sources from from from the component
0:19:30	and speaker lean you can so remove the noise and do a denoising and there so
0:19:35	uh
0:19:36	remastered these different components are a a a on two channels to produce a as to re recording from them
0:19:41	on the recording
0:19:42	very similar to
0:19:43	uh the show and tell in more of what you know me just to the animal if you if you
0:19:47	which use so the the the demos it's is the same uh
0:19:50	same kind of it is
0:19:51	so typically am
0:19:53	uh
0:19:53	from this original no
0:19:55	you can create uh
0:19:57	and that mixed and you noise that
0:20:00	rations so will play that for you so for as the original mono
0:20:06	a
0:20:08	a
0:20:09	a
0:20:10	a
0:20:11	and and uh
0:20:12	because
0:20:13	notion
0:20:13	a
0:20:14	a
0:20:15	a
0:20:16	and
0:20:16	a
0:20:17	a
0:20:19	a
0:20:20	a
0:20:21	a
0:20:22	a
0:20:24	i
0:20:25	and if you want you can can and for example for the
0:20:28	for the brass uh components so
0:20:30	the trumpets carry net
0:20:40	so the interesting thing is that even if you have some artifacts on some of the estimated sources a uh
0:20:45	because you uh replay play the sources to give or
0:20:48	uh you actually uh uh don't to uh
0:20:51	this sent to the artifacts and you
0:20:52	it's you can run there are some the special uh a special pressure
0:20:57	and uh that concludes and my uh my talk
0:21:31	yes
0:21:33	yes
0:21:34	i i don't know
0:21:35	i mean um
0:21:36	you can you can be a L and and i agree an for the estimation of the and H
0:21:42	okay
0:21:43	uh
0:21:44	using a
0:21:45	this latent and components
0:21:46	as the complete data
0:21:49	okay
0:21:49	but is not shown here
0:21:51	okay but you can do it quite easily
0:21:54	offline line sure
0:22:09	uh_huh
0:22:18	what what a take K uh let less some more components
0:22:23	uh that's a good the question
0:22:26	um hmmm
0:22:30	ten components seem to be the the the proper or a number of components to use
0:22:35	because uh
0:22:37	adding more components
0:22:39	uh only uh
0:22:41	uh
0:22:42	tended it to uh reach find a decomposition of the noise
0:22:46	okay
0:22:47	so it it seemed like uh
0:22:50	uh after ten components um
0:22:52	you didn't a obtain a more interesting uh almost right it's all season
0:22:58	now to be honest i don't remember a a what does a uh when you take less than uh
0:23:03	then ten compare
0:23:06	i i don't remember
0:23:12	click you're

MAJORIZATION-MINIMIZATION ALGORITHM FOR SMOOTH ITAKURA-SAITO NONNEGATIVE MATRIX FACTORIZATION

Non-negative Tensor Factorization and Blind Separation

Presented by: Cédric Févotte, Author(s): Cédric Févotte, CNRS LTCI / Télécom ParisTech, France