the wiretap channel was introduced by wyner  , in which the eavesdropper is assumed to have access to a degraded version of the intended receiver s signal .this pioneering work was later generalized to cover the non - degraded scenario  and the gaussian channel  .however , these results show that the secrecy capacity saturates in the high signal - to - noise ratio ( snr ) regime implying a * vanishing * value for the secure degrees of freedom ( dof ) .recently , there has been a growing interest in the analysis and design of secure wireless communication networks based on information theoretic principles .for example , the secrecy capacity of relay networks was studied in  , while the fundamental limits of the wiretap channel with feedback were analyzed by  . on the other hand, the multiple access and broadcast channels with secrecy constraints were investigated in  .finally , the role of multiple antennas in enhancing the secrecy capacity was established in  and the positive impact of fading on secrecy capacity was revealed in .  here , the frequency / time selective -user gaussian interference channel with secrecy constraints is considered . without the secrecy constraints, it has been recently shown that a degrees of freedom ( dof ) per orthogonal dimension is achievable for each source - destination pair in this network  .the achievability of this result was based on the _ interference alignment _ technique ( see also  ) , by which the interfering signals are aligned to occupy a subspace orthogonal to the one spanned by the intended signal at each receiver .however , the impact of secrecy constraints on the degrees of freedom in this model has not been fully characterized .in fact , to the best of our knowledge , the only relevant prior works are the study of the two - user discrete memoryless interference channels with confidential messages  and the one with an external eavesdropper  .the frequency selective interference channel adopted in the present paper is , however , fundamentally different from these _ memoryless _ models .we consider two distinct network models , namely 1 ) the interference channel with confidential messages and 2 ) the one with an external eavesdropper . in the first scenario, one needs to ensure the _ confidentiality _ of each message from all non - intended receivers in the network .since all users are assumed to belong to the same network , one can assume the availability of channel state information ( csi ) while designing the secrecy coding scheme .towards this goal , we employ an interference alignment scheme along with secrecy pre - coding at each transmitter .intuitively , the interference alignment scheme has two effects on each receiver : 1 ) it aligns the signals from transmitters to a small dimensionality subspace , and 2 ) it assigns the signal from transmitter to the orthogonal subspace . hence , while the signal from its own transmitter is _ received cleanly _ , the signals from other transmitters _ are mixed together_. our secrecy pre - coding takes advantage of the phenomenon to ensure that the resulting multiple access channel ( from the interfering users ) does not reveal any useful information about each non - intended message .this way , we show that * secure * dof per orthogonal dimension is achievable for each user in the three - user gaussian interference channel with confidential messages .we then generalize our results to the -user gaussian interference channel showing that each user can achieve secure dof . in the second scenario ,we study the external eavesdropper model where the fundamental challenge is the lack of channel state information ( csi ) about the links connected to it . despite this fact, it is shown that dof per user is achievable in the ergodic setting .this result provides further evidence on the diminishing gain resulting from knowing the instantaneous csi of the eavesdropper _ a - priori_. interestingly , by comparing our results with those obtained for the point - to - point case  , one can see the * positive impact * of interference on the secrecy capacity region of wireless network .the underlying idea is that the coordination between several source - destination pairs allows for _ hiding _ the secret messages in the background interference .the remainder of the paper is organized as follows . in sectionii , the system model and notation are introduced .section iii is devoted to the interference channel with confidential messages .the analysis for the external eavesdropper scenario is detailed in section iv. finally , we offer some concluding remarks in section v. the technical results needed to develop our proofs are collected in the appendices to enhance the flow of the paper .we consider a frequency selective wireless network comprised of transmitter - receiver pairs where the receiver output at time and frequency slot is given by ) and vectors are denoted as bold capital letters with bars or tildes ( for example , and ) .we define and denote for . a zero - mean circularly symmetric complex gaussian random variable with variance denoted by . ]  here , is the transmitted symbol of user at frequency slot during time , and is the additive white gaussian noise at receiver .we assume that the channel coefficients are randomly generated according to a continuous distribution and are fixed during the communication period .we also assume that the channel coefficients are known at every node in the network .the network model is provided in fig . .  using the extended channel notation of  ,the received vector during time slot can be written as here , is the diagonal matrix of channel coefficients from transmitter to receiver whereas ^t ] , and ^t ] , where each element is an vector .then , for each symbol time , the transmitter employs the interference alignment scheme , and maps to via , where we choose the interference alignment matrices as in  .we choose the secrecy and randomization rates as follows .  the above rates are inside the decodability region for each user , i.e. ,  , implying that each user can reliably decode its own streams as .hence , using the union bound argument , we can show that for a given there exists such that for any  for almost all codebooks in the ensemble .our second step is to show that can be made arbitrarily close to for any and for almost all codebooks in the ensemble . towards this end , it is sufficient to focus on the equivocation at an arbitrary receiver .furthermore , it is sufficient to establish perfect secrecy for the full message set since lemma [ thm : secrecy ] shows that perfect secrecy of the full message set implies secrecy for all subsets ( here , the full message set at the receiver refers to . ) . denoting the observation of the eavesdropper as , we write where the last equality follows from the fact that as the randomization ( i.e. , codeword ) indices are independent of the message ( i.e. , bin ) indices .we now bound each term of .first due to the markov chain relationship combining this with the fact that we obtain   second   to upper bound the last term , we use the following argument .assume that is transmitted .given these bin indices , the remaining randomness in at the eavesdropper can be resolved for almost all codebooks as the above choice of satisfies the multiple access channel achievability conditions ,  ( * ? ? ?* chapter 14 ) .then , by fano s inequality , we have , where as .then , defining , we have where as . using equations ( [ eq : bound1 ] ) , ( [ eq : bound2 ] ) , and ( [ eq : bound3 ] ) in ( [ eq : entropies ] ) and dividing both sides of by , we obtain  where we used the fact that and the rate assignment given by .it is already observed that as for almost all codebooks in the ensemble .the orthogonality of the intended message and interference at each respective receiver along with the full rank property of the gain matrices ( see lemma [ thm : lemmafullrank ] ) imply the followings .      where or depending on , and   using the observations , , , and in we see that for almost all codebooks in the ensemble .hence , for any given , we can make by letting grow . finally , due to , , and , we obtain which proves our result .first , it is easy to see that our previous results extend naturally when the eavesdropper csi is available _ a - priori _ at the different transmitters and receivers. intuitively , one can imagine the existence of a virtual transmitter associated with the external eavesdropper transforming our -user network into another one with -users .this way , one can achieve a secure dof of per frequency - time slot for each user using the scheme of the previous section .for example , for a two - user network with an external eavesdropper , it is possible to achieve secure dofs if the eavesdropper csi is available at the transmitters .more formally , we have the following result .  for the -user gaussian interference channel with an external eavesdropper ,a secure dof of per frequency - time slot is almost surely achievable for each user ( assuming the availability of the eavesdropper csi ) .more interestingly , it is still possible to achieve positive secure dof per user in the * absence * of the eavesdropper csi by exploiting the channel * ergodicity*. in the ergodic model , the channel gains are assumed to be fixed during a block of symbol times and then randomly change to another value in the next block for a total of blocks , where and .again , for illustration purposes , we use the case . here , the users of the network have total dof while the multiple access channel ( mac ) seen by the eavesdropper can have only one dof from its observations .hence , via an appropriate choice of secrecy codebooks , the additional dof can be _ evenly _ distributed among the network users _ on the average _ , allowing for a secure dof per user without any requirement on the eavesdropper csi . in the general case, we have the following result .[ thm : kuserext ] for the -user gaussian interference channel with an external eavesdropper , a secure dof of per frequency - time slot is achievable for each user in the ergodic setting ( in the absence of the eavesdropper csi ) .let and , where .we set and for .we generate all the permutations of length and denote this set by , where .then , for each fading block , we randomly pick , according to uniform distribution , a permutation from and denote it by . in order to ensure statistical symmetry , the interference alignment matrices in each fading block will be obtained according to a different user ordering induced by .more specifically , let and . using the newly ordered channel matrices , the interference alignment matrix for the user , i.e. , , is generated .for each secrecy codebook in the ensemble , we generate sequences each of length , where entries are chosen i.i.d . for some and that satisfies the long term average power constraint ( the existence of and follows from the argument of theorem [ thm : kuserconf ] ) .we independently assign each codeword to one of bins each having codewords .given , transmitter chooses the corresponding bin and independently ( according to uniform distribution ) chooses a codeword in that bin denoted by , where is the randomization index .this codeword is then divided into blocks , each with a length symbols .each block is then arranged in the following matrix  ] , in the following . to satisfy the achievability conditions and the secrecy requirements of the network , we choose and as follows - \max\limits_{p(\bar{{{\bf x}}}_{{{\cal k}}})}\mathbb{e } [ i(\bar{{{\bf x}}}_{{\cal k}};\bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e)]\right)\quad \text{and } \nonumber\\ r_k^x & = & \frac{1}{kf } \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal k}}};\bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e)],\end{aligned}\ ] ] where the maximization in the first equation is among all possible input distributions . with this choice of rates, we have the following - \frac{1}{kf}\max\limits_{p(\bar{{{\bf x}}}_{{{\cal k } } } ) } \mathbb{e}[i(\bar{{{\bf x}}}_{{\cal k}};\bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e ) ] + \frac{1}{kf } \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal k}}};\bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e)]\nonumber\\ & \leq & \frac{1}{f } \mathbb{e}[i(\tilde{{{\bf x}}};\bar{{{\bf y}}}|{{\bf h}})],\end{aligned}\ ] ] where the inequality is due to the maximization among _ all _ possible input distributions in the second term of the equation .hence , we have  ] , such that , if then , .  for a given , , and level of secrecy  ], we have ^t ] for any and for any .we note that the last property stated above is due to the symmetry between network users provided by the random choice of user ordering at each fading block .[ thm : randrates ] each user can set the randomization rates to be of the total dof per orthogonal time - frequency slot seen by the eavesdropper , i.e. , with a rate choice of ,\end{aligned}\ ] ] each randomization message ( codeword index ) , given the secrecy message ( bin index ) of each user , is decodable at the eavesdropper .let . from lemma [ thm :simplelemma2 ] , we have \leq \frac{1}{|{{\cal s}}| } \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bf x}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] .\ ] ] we continue as below . & \leq & \frac{1}{|{{\cal s}}| } \mathbb{e } [ i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bf x}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] \\ \leftrightarrow  & \leq & ( k-|{{\cal s}}| ) \mathbb{e } [ i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bfx}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] \\\leftrightarrow \frac{|{{\cal s}}|}{k } \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal s}}^c } ; \bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e ) ] & \leq & \frac{k-|{{\cal s}}|}{k } \mathbb{e } [ i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bfx}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] \\\leftrightarrow \frac{|{{\cal s}}|}{k } \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal k } } } ; \bar{{{\bf y}}}_e|{{\bf h}},{{\bf h}}_e ) ] & \leq & \mathbb{e}[i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bf x}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] , \end{aligned}\ ] ] from which we readily conclude that $ ] satisfies \leq \frac{1}{f}\mathbb{e } [ i(\tilde{{{\bf x}}}_{{{\cal s } } } ; \bar{{{\bf y}}}_e |\tilde{{{\bf x}}}_{{{\cal s}}^c},{{\bf h}},{{\bf h}}_e ) ] , \forall { { \cal s}}\subset{{\cal k}},\end{aligned}\ ] ] and hence randomization messages are decodable at the eavesdropper with this rate assignment .e. tekin and a. yener , `` the general gaussian multiple access and two - way wiretap channels : achievable rates and cooperative jamming , '' _ ieee trans .inf . theory _ ,54 , no . 6 , pp . 2735 - 2751 , june 2008 .maddah - ali , a. s. motahari , and a. k. khandani , `` communication over mimo x channels : interference alignment , decomposition , and performance analysis , '' _ ieee trans .inf . theory _ ,54 , no . 8 , pp . 3457 - 3470 , aug .r. liu , i. maric , p. spasojevic , and r. yates , `` discrete memoryless interference and broadcast channels with confidential messages , '' in _ proc .44th annual allerton conference on communication , control , and computing _ , monticello , il , sept .2006 .o. o. koyluoglu and h. el gamal `` on the secrecy rate region for the interference channel , '' in _ proc .2008 ieee international symposium on personal , indoor and mobile radio communications ( pimrc08 ) _ , cannes , france , sept . 2008 .      .the secret message of user results in the bin index and the randomization index for the corresponding user is .considering these two indices , each entry of the codebook is denoted as . ]

Abstract: this paper studies the frequency / time selective -user gaussian interference channel with secrecy constraints . two distinct models , namely the interference channel with confidential messages and the one with an external eavesdropper , are analyzed . the key difference between the two models is the lack of channel state information ( csi ) about the external eavesdropper . using interference alignment along with secrecy pre - coding , it is shown that each user can achieve non - zero secure degrees of freedom ( dof ) for both cases . more precisely , the proposed coding scheme achieves secure dof _ with probability one _ per user in the confidential messages model . for the external eavesdropper scenario , on the other hand , it is shown that each user can achieve secure dof _ in the ergodic setting_. remarkably , these results establish the _ positive impact _ of interference on the secrecy capacity region of wireless networks .


this paper describes an advanced component separation technique and its application to the analysis of multi - spectral observations of the cosmic microwave background ( cmb ) .it is developed more specifically in view of the data processing pipeline for the forthcoming planck mission  of the european spatial agency ( esa ) .component separation is a critical part of cmb data processing because no frequency channel exists that would be sensitive only to cmb emission . as an illustration , figure [ fig : inputs ] shows ( simulated ) maps of the microwave emission of the sky observed in the frequency channels of planck _i.e. _ in spectral bands centered around frequencies . the red ( in an arbitrary color map ) horizontal strips along the galactic equator , in the middle of the maps , are due to galactic emissions ( that is , emissions from _ our _ galaxy ) .in contrast , towards the poles where the galactic emissions are weaker , one can see in the center channels ( 100 , 143 and 217 ghz ) , an homogeneous texture which is the signal of interest : the ( spatial ) fluctuations ( or ` anisotropies ' ) of the cmb . unfortunately , even in the polar areas , the contamination of the cmb by various astrophysical emissions ( or foregrounds , as opposed to the cosmic _ background _ ) is still significant .there , especially at the smallest angular scales , contamination is dominated by the emission of remote extragalactic objects galaxies ( or ` point sources ' ) of various types ( which emit radiation through similar physical processes as our own galaxy ) , and galaxy clusters ( which emit via the sunyaev - zeldovich effect , hereafter : the ` sz component ' ) .there is no significant occlusion between the various physical components so that the signal measured at a frequency in direction is a linear superposition : where is the contribution due to a particular component and accounts for instrumental noise at the map level .it must be stressed that , even though the lowest ( 30 , 44 , 70 ghz ) and highest ( 353 , 545 , 857 ghz ) frequency channels are more contaminated by galactic emissions , they still are very informative because they can be used to predict and remove the galactic emission in the center channels , leaving only cmb and noise  in an ideal data processing scenario .successful scientific exploitation of the cmb measurements for cosmological studies critically depends on accurate cleaning of the cmb and on quantifying as well as possible any remaining contamination .figure [ fig : reccmb ] illustrates on simulated data what can be achieved with our technique on a very large fraction of the sky .even though cmb recovery is the main goal , recovering the other components also is of scientific interest .for instance , planck should also yield a very rich catalog of galaxy clusters , and provide a lot of information about the emission of the interstellar medium in our own galaxy .hence , we are actually facing a problem of _ component separation _ from multiple observations .the component separation problem is not new to the astrophysics community . in the past ten years however , component separation for cmb observations specifically has motivated the development of a large variety of methods with various degrees of sophistication ( see for a review ) .the most direct methods rely on modelling and subtracting foreground emission .such methods , typically , involve a thorough understanding of the emission processes , necessary to build the foreground model .other methods use simple decorrelation , or make use of differences between properly weighted maps .the processing of data from the wmap satellite mission makes uses of both kinds of methods .the large majority of existing component separation methods in the context of cmb observations rely on a ` mixing model ' as follows : for any available direction in the sky , the channels at frequencies provide data points which can be collected in an vector which is modelled as where each entry of contains the sky - pattern for the emission of a given component .the emissivity of a given component depends on the frequency ; hence , each column of matrix reflects the emission law of the corresponding component .this equation , then , forms the basis for many component separation algorithms but it forces a special structure on each component , namely that it ` scales rigidly with frequency ' .some methods are developed to solve this component separation problem when the mixing matrix is known _a priori_. this is the case , for instance , of the work described in .component separation is a well studied problem in signal processing , but the nature of the available data and the scientific objectives call for specific techniques. in particular , one may think of resorting to independent component analysis ( ica ) methods since these ` blind ' techniques may be useful to deal with some of the uncertainties in the data structure .in addition , columns of the mixing matrix corresponding to some of the astrophysical components ( e.g. components modelling the emission of our own galaxy ) are known a priori up to significant uncertainty , and more precise knowledge is an objective of the forementioned astrophysical observations .implementations of ica ideas in the context of cmb observations can be found in .  compared with all the above methods ,our framework allows the inclusion of components with arbitrary structure ( at least in terms of second order correlations ) .this structure , as will be detailed later on , can be matched to prior knowledge of the nature of the astrophysical emission .along the same spirit of matching a parametric model of the emissions to the data , although quite different in implementation , are the work described in .although there is substantial motivation for letting the data alone indicate the appropriate model of the emissions , a fully blind approach would leave out a lot of valuable prior information in the cmb context .the approach described in this paper is rooted in ica but allows for the inclusion an arbitrary amount of prior information , in particular by allowing components with an arbitrary structure .again , the approach to component separation described herein is not specific to this particular problem and may be considered for application to any situation where ` expensive ' data deserve special care and have to be fitted by a complex component model .the paper is organized as follows : section [ sec : model ] describes our approach , section [ sec : discussion ] focuses in specific aspects ; implementation is discussed in section [ sec : impl ] in terms of a library of components ; section [ sec : results - simul - planck ] illustrates an application of the method on data from the planck sky model .this section introduces a key ingredient of our approach : the development of a flexible model of _ additive components _ in which the notion of ` mixing matrix ' , which is central to standard ica , is disposed of . not only the mixing matrix is not needed but it is ` considered harmful ' .we start the exposition by considering the most general component model for an random vector : we only postulate that is made of the linear superposition of components : where each component is an random vector . for simplicity, we shall assume that all components have zero mean .denote ( _ resp . _ ) the covariance matrix of vector ( _ resp ._ of its -th component ) .a key assumption of our model is mutual decorrelation between all components , which implies that the covariance matrix of decomposes as consider now recovering linearly a particular component from the superposition ( [ eq : compmodel ] ) .denote the matrix which best predicts based on in the sense that this problem is easily solved under our assumptions and one readily finds : at this stage , we wish to deliver a simple but important message : eq .( [ eq : wiener ] ) means that component separation , understood as computing , requires only knowing and .of course , covariance matrix may ( but need not to ) be simply estimated from the empirical covariance matrix of .therefore , component separation is solved if we can uniquely solve the _ covariance separation problem _ , in the sense of identifying the component terms in decomposition ( [ eq : deccov ] ) .  in order to achieve covariance separation ,some additional information or assumptions are of course needed .our method jointly exploits two possibilities : considering _ localized matrices _ ( for more information ) and considering _ structured matrices _ ( for more constraints ) .this section defines the ` localized statistics ' on which our method is based .consider the component separation problem based on samples on detectors .we shall not compress this data set into a single sample covariance matrix but rather ` localize ' the statistics and form a _ set _ of _ localized covariance matrices_. by ` localization ' , we typically mean localization in time , space , frequency , wavelet space , depending on the problem at hand .the simplest example of localization is suggested by the separation of non - stationary time series as considered in where the data are time - indexed and one would divide the observation interval into sub - intervals and compute an estimate of the covariance of over each of these intervals . more generally , we assume that the data are available as vectors , each of size where the -th entry of is either the -th sample of the -th input signal or its -th coefficient in some basis ( _ e.g. _ the fourier basis ) . in our application to cmb analysis , we use the spherical harmonic basis ( see section [ sec : results - simul - planck ] ) .basis choice is discussed below ; it does have an impact on separation but it does not change the nature of the problem in the sense that , if the component model ( [ eq : compmodel ] ) holds in some basis , it also holds in any other basis and that if a component is reconstructed in a given basis , it becomes available in any basis .localized statistics are formed by dividing the index set  ] and localized statistics are computed ( defined ) as where is the number of coefficients in domain .  the expected value of the sample covariance matrix for the domain is defined / denoted as and we shall use the same notation for each component so that , these being mutually uncorrelated by assumption , one has the decomposition we can then perform a localized separation in the sense that the localized wiener filter for recovering component in domain is a key point is that , by definition , this filter is specialized to operate on domain , hence it is adapted to the local correlation conditions .   implementing component separation by ( [ eq : locwiener ] ) requires knowning the local covariance matrices .these are usually unknown and must be estimated , based on , the sample estimate ( [ eq : empscm ] ) of their sum .it goes without saying that , without constraining the possible values of , it is not possible in general to uniquely resolve ( an estimate of ) this sum into its components . constraining a given component achieved by parametrizing the matrix set by a parameter vector _ i.e. _ a parametric model for the -th component is a function .some examples of these parametric models are given in section [ sec : mylib ] .a parametric model follows by assuming component decorrelation ( [ eq : decrq ] ) and taking the global parameter as the concatenation of the parameter vectors of each component :   the parametric model is identified by fitting it to data , that is , by minimizing a measure of mismatch between and as : where is measure of mismatch between two positive matrices and are positive weights .our most common choice is to use \ ] ] because this is the form which stems from the maximum likelihood principle .specifically , if a coefficient belonging to domain is modeled as and as independent from for , then is the negative log - likelihood of the model  .   in summary ,our method is based on the following steps and design choices :  [ it : basis ] choice of a basis to obtain coefficients ( original space , fourier or spherical - harmonic space , wavelet space , ) .[ it : domain ] choice of domains to localize their second - order statistics ,  [ it : models ] choice of a model , for the contribution of each component to the covariance matrix over each domain .minimization of the matching criterion ( [ eq : idcrit ] ) to obtain the estimate , hence estimates for the parameters of each component .estimation of the coefficients of each component by , that is the version of ( [ eq : locwiener ] ) based on estimated parameters .reconstruction of the components from their coefficients .  at this stage ,most of the statistical framework is in place but our method is not completely specified yet because of the flexibility in design choices in items [ it : basis ] , [ it : domain ] and [ it : models ] above among other issues .next section discusses some of them .we wish to contrast the standard form of the noisy linear component model of eq .( [ eq : multmod ] ) with the unconstrained model of eq .( [ eq : compmodel ] ) .obviously , the standard model is included in the unconstrained model since one may always write with for and where denotes the -th column of and denotes the entry of .  a component which can be written as is fully coherent in the sense that any two entries of vector are 100% correlated . of course , this is because all the randomness in comes from a single scalar random variable .a fully coherent component contributes exactly the same pattern on all sensors ( or channels ) up to a fixed proportionality coefficient .it has a rank - one covariance matrix : ; for this reason , it could also be called a one - dimensional since it leaves in the 1d space spanned by vector or in the 1d range space of .  in our application ,the cmb is a fully coherent component : from one frequency channel to another , only its overall intensity changes ( ignoring the frequency - dependent resolution ) .this is in contrast with , for instance , the galactic emission , which can not be represented over the sky as a fully coherent component ( see fig . [fig : decinputs ] ) .assume now that , in the model , two sources , say and , are statistically _ dependent _ while all other pairs of sources in are mutually independent . these two sources contribute to and we decide to lump them into a single component denoted for some index : . then can still be written as in eq .( [ eq : compmodel ] ) and all the _ components _ are independent , _again_. in other words , we represent here the contributions of two correlated sources as the contribution of a single 2-dimensional component : we now have one less component but it is 2-dimensional . clearly , this component is not fully coherent : it contributes a pattern on sensor which is not proportional to the pattern contributed to another sensor .one can obviously define multidimensional components of any dimension , each one possible representing the contribution of several correlated sources . in section [ sec : results - simul - planck ] , we use a 4-dimensional component to model galactic emission .  note that , in eq .( [ eq : compmodel ] ) , the noise term is included as one of the components .if the noise is uncorrelated from channel to channel , as is often the case , then the noise component is -dimensional ( the largest possible dimension ) .more generally , our model does not require any component to be low dimensional .rather , our model is a plain superposition of components as in eq .( [ eq : compmodel ] ) .none of these components is required to have any special structure , one - dimensional or otherwise .we call a collection of parametric models a _ library _ of components . in practice, each member of the library must not only specify a parametrization but also its gradient and related quantities ( see sec .[ sec : comp - deriv ] ) .typical examples of component models are now listed .the ` classic ' ica component is fully coherent ( one dimensional ) . denoting average variance of over the domain , the contribution of this component to is the rank - one matrix this component can be described by an vector of parameters containing the entries of and the variance values .such a parametrization is redundant , but we leave this issue aside for the moment .a -dimensional component can be modeled as where is an matrix and is an positive matrix varying freely over all domains. this can be parametrized by a vector of scalar parameters ( the entries of and of ) . again , this is redundant , but we ignore this issue for the time being.  noise component . a simple noise model is given by that is , uncorrelated noise from channel to channel , with the same level in all domains but not in all channels . this component is described by a vector of only parameters .a more general model is meaning that the noise changes from domain to domain ; then parameter vector has size .  as a final example , for modeling ` point sources ' in spectral domain , one may use .such a component contributes identically in all domains corresponding to a flat spectrum .if , for instance , we assume that this contribution is known , then the parameter vector is void .if is known up to a scale factor , then is just a scalar , etc in the demonstration test discussed in [ sec : results - simul - planck ] , however , we consider instead the sum of galactic and point source emission as one single 4-dimensional component a choice which the flexibility of the present model allows us to do .there are two strong , very different motivations for localizing the statistics .* localization for accuracy . *the first motivation is separation accuracy .if the strength of the various components ( including noise ) varies significantly across the domains , reconstruction is improved by the localized filter .indeed , the best linear reconstruction of based on would be , as seen above , obtained as .this requires knowing both and . in practice, it seems difficult to obtain estimates for these matrices for all samples , _i.e. _ for each value of .however , if they do not vary too much for all values of across a domain , then a good approximation to the best reconstruction is ( [ eq : locwiener ] ) _ i.e. _ the reconstruction filter also is localized , taking advantage of the ` local snr conditions ' on domain .  * localization for diversity / identifiability . *second , the diversity of the statistics of the components over several domains is precisely what may make this model blindly identifiable .for instance , in the basic ica model ( all components are one - dimensional , no noise ) , if are fourier coefficients and are spectral bands , it is known that spectral diversity ( no two components have identical spectrum ) is a sufficient condition for blind identifiability .is this a _ blind _ component separation method ?it all depends on the component model . if all components are modeled as ` classic ' ica components ( see sec .[ sec : mylib ] ) , then the method is as blind as regular ica .our approach , however , leaves open the possibility of tuning the blindness level at will by specifying more or less stringent models for some or all of the components .  of course, it may be difficult to predict if a given parametrization ensures the identifiability of the model : this is to be discussed on a case - by - case basis .however , over - parametrization can be tested numerically because the fisher information matrix ( fim ) is available in our framework .it has a natural block structure where the block related to a pair of components is the matrix of size : {cc ' }  =  \frac12 \sum_q p_q\ \mathrm{trace }  \left (   \frac{\partial\r_q^ c  ( \theta^c  ) } { \partial\theta^c  } \r_q\inv(\theta )    \frac{\partial\r_q^{c'}(\theta^{c'})}{\partial\theta^{c ' } } \r_q\inv(\theta )   \right)\ ] ] under the assumptions which make the mismatch measure proportional to the log - likelihood ( see sec . [sec : constr - comp - model ] ) .the fim is also used for computing ( approximate ) error bars .we discuss the practical issue of actually minimizing the mismatch using an arbitrary library of components .  note that for a noise - free model containing only ` classic ica ' components ( no other constraints than being one - dimensional ) , criterion boils down to a joint diagonalization criterion which can be very efficiently minimized by a specialized algorithm  . for a modelincluding only unconstrained multi - dimensional components and noise , it is possible to use the em algorithm  .em , however , is not convenient for general component models and , in addition , it appears too slow for our purposes .therefore , we had to consider more efficient and more general optimization procedures .we found the conjugate gradient ( cg ) algorithm well suited for minimizing .its implementation requires computing the gradient and ( possibly an approximation of ) the hessian for pre - conditioning .since actually is a negative log - likelihood in disguise , its hessian can classically be approximated by , the fisher information matrix ( fim ) of eq .( [ eq : cm : deffimblock ] ) .these computations offer no particular difficulty in theory but we aim at an implementation it in the framework of a library of components _i.e. _ computations should be organized in such a way that each component model works as a ` plug - in ' .   * computing the gradient . *the partial derivative of with respect to takes the form where matrix is defined as hence the computation of at a given point can be organized as follows .a first loop through all components computes by adding up the contribution of each component .then , a second loop over all domains computes matrices which are stored in a common work space .finally , a third loop over all components concatenates all partial gradients , each component implementing the computation of the right hand side of ( [ eq : cm : parderphi ] ) in the best possible way , using the available matrices .* computing an ( approximate ) hessian . *the fisher information matrix can be partitioned component - wise and each block be computed according to eq .( [ eq : cm : deffimblock ] ) .therefore its computation can be organized with a double nested loop over and as soon as the code implementing a given component is able to return the matrix set .a straightforward implementation of this idea may be impractical , though , because this is a set of matrices , possibly very large .this problem can be alleviated in the frequent case where components have ` local ' variables .* local variables . *consider the case when can be partitioned into blocks : where , for , the sub - vector affects only while collects all the remaining parameters _i.e. _ those which affect the covariance matrix over two or more domains .we then say that this component model has ` local variables ' .this is a fairly frequent situation which occurs for instance when the power of a component can be freely adjusted in each domain ( the simplest example is the ` classic ' ica component : , for which and for .the global parameter vector inherits this structure by being partitioned accordingly into a ` global part ' and local parts .a first and major benefit of such a local / global partitioning is that it introduces many zero blocks in the fim since then {qq'}=\mathbf{0} ] .note that the angular power spectrum of the cmb is obtained as a by - product of the global fitting procedure .we show and comment some results from our best - fit model .* goodness of fit*. the top panel of figure [ fig : mismatch ] displays the values of versus the spectral domain index ( actually : versus the average angular frequency for this domain ) .the overall mismatch measure is just the sum of these quantities over all domains : see eq .( [ eq : idcrit ] ) .it is possible to predict the average mismatch value ( in the asymptotic regime of many fourier modes per domain ) when the model holds .this provides at least a reference value in terms of goodness of fit .this value ( and twice this value ) are displayed as a horizontal line on fig .[ fig : mismatch ] .the fit appears good for but not so good for .this reflects the difficulty of modelling the complex galactic structure ( since this component is important at this range scale ) and also possibly the fact that the assumptions required for predicting the average mismatch are not met in this domain .the bottom panel of the figure shows how the mismatch increases when the contribution of any one of the components ( except noise ) is removed from the model . in this case , the mismatch explodes , showing that all fitted components are significant here .* power decomposition at sensor level . * the decomposition ( [ eq : decrq ] ) allows to find the contribution of each component on each detector by domain , _i.e. _ as a function of angular frequency in the present case. figure [ fig : specsmic ] displays for all detectors ( ) the values of {jj} ] . actually the plot of the former quantity is almost entierly masked by the plot of the latter , evidencing a near perfect fit of the global angular spectrum .  ) .the line labelled _ stats _ shows the values of {jj} ] ; the latter is hidden by the former almost everywhere.,title="fig:",width=264 ] ) .the line labelled _ stats _ shows the values of {jj} ] ; the latter is hidden by the former almost everywhere.,title="fig:",width=264 ] + ) .the line labelled _ stats _ shows the values of {jj} ] ; the latter is hidden by the former almost everywhere.,title="fig:",width=264 ] ) .the line labelled _ stats _ shows the values of {jj} ] ; the latter is hidden by the former almost everywhere.,title="fig:",width=264 ]  figure [ fig : specsmiccross ] shows the cross - spectra between channels , that is the values of {ij}$ ] for selected pairs of channels .  ) .same conventions as in figure [ fig : specsmic].,title="fig:",width=264 ] ) .same conventions as in figure [ fig : specsmic].,title="fig:",width=264 ] + ) .same conventions as in figure [ fig : specsmic].,title="fig:",width=264 ] ) .same conventions as in figure [ fig : specsmic].,title="fig:",width=264 ]  * cmb angular spectra . * left panel of figure [ fig : cl_est+res ] shows the estimated angular spectrum with error bars computed from the fisher information matrix . the angular spectrum of the cmb process is known in this simulated data set and displayed as a solid line . an excellent recovery is observed up to angular frequency , together with plausible error bars . *cmb map*. the cmb map recovered through the localized wiener filter ( [ eq : locwiener ] ) is shown in figure [ fig : reccmb ] .it has no visible contamination but reconstruction is not over the whole sky : the small galactic region ( shown on fig .[ fig : hitcount+mask ] ) is not reconstructed because the galactic emission is so strong that avoiding any contamination is very difficult .the bottom map of fig .[ fig : reccmb ] is the residual _i.e. _ the error between true and reconstructed cmb maps. some galactic residuals are now visible indicating that the localization strategy should be improved ( the galactic emission is known to vary a lot across the sky , and especially near the galactic plane ) .the angular spectrum of the _ residual _ map ( cmb error map ) is shown on the right panel of fig . [fig : cl_est+res ] . the linear ( on this log scale )trend indicates that the error is noise - dominated ( as also seen from the residual map ) .   with error bars given by the fisher information matrix .we ( conventionally ) plot rather than the plain angular spectrum .right : angular spectrum of the reconstructed cmb and of the residual in mkcmb .the red and the black curves are the same reference input cmb spectrum which has been used to generate the cmb component.,title="fig:",height=226 ]  with error bars given by the fisher information matrix . we ( conventionally ) plot rather than the plain angular spectrum .right : angular spectrum of the reconstructed cmb and of the residual in mkcmb .the red and the black curves are the same reference input cmb spectrum which has been used to generate the cmb component.,title="fig:",height=226 ]  mk cmb ) .bottom : difference map ( color scale :  mk cmb).,title="fig : " ] mk cmb ) .bottom : difference map ( color scale :  mk cmb).,title="fig : " ] + mk cmb ) .bottom : difference map ( color scale :  mk cmb).,title="fig : " ]  * galactic and point source emission reconstruction*. the most complex component is due to the radiation from the interstellar medium in our galaxy ( galactic emission ) as well as other galaxies ( which contribute most of the emission of extragalactic ` point sources ' ) .the present work focuses on cmb reconstruction , and no attempt has been made to separate the galactic emissions on the basis of the astrophysical radiation emission process ( synchrotron emission , ` free - free ' emission , or grey body emission from dust particles ) , nor on the basis of their origin ( within our own galaxy , or in other galaxies ) . for this reason, the sum of these emissions _ outside our galactic + point source mask _ has been tentatively represented by a 4-dimensional ` catch all ' model as explained above .the result of the separation for this component is shown in fig .[ fig : galrec ] . in spite of small discrepancies in the reconstruction on large angular scales ,the result is quite satisfactory .angular power spectra for this model , displayed in fig .[ fig : specsmic ] clearly correspond to the sum of the steep galactic power spectrum , and a flat plateau at large angular frequency , due to the contribution of emissions from the large number of extragalactic sources not masked by our point source mask .+  +  +  +the component separation technique discussed in this paper offers great modeling flexibility from the realization that separation can start with _ covariance matrix separation _ _i.e . _ the identification of individual component terms in the domain - wise decomposition ( [ eq : decrq]) followed by _ data separation _ according to ( [ eq : wiener ] ) .too much flexibility may also introduce difficulties , though . indeed , whether or not minimizing the covariance matching criterion leads to _ uniquely _ identified components depends on the particular choice of component models . in our approach , however , the amount of constraints imposed on any component is fully tunable . by using more or less constrained components ,the method ranges from totally blind to semi - blind , to non - blind .  some other good points are the following .* speed * : the method is potentially fast because large data sets are compressed into a much smaller set of covariance matrices . *accuracy * : the method is potentially accurate because it can model complex components and then recover separated data via local wiener filters which are naturally adapted to the local snr conditions . * noise * : the method can take noise into account without increased complexity since noise is not processed differently from any other component .* prior * : the implementation also allows for easy inclusion of prior information about a component if it can be cast in the form of a prior probability distribution in which case one only need to subtracting from and the related changes can be delegated to the component code . * varying resolution * : in our application , and possibly others , the input channels are acquired by sensors with channel - dependent resolution . accurate component separation requires to take this effect into account .this can be achieved with relative simplicity if the data coefficients entering in are fourier ( or spherical harmonic ) coefficients .* built - in goodness of fit * via the mismatch measure ( [ eq : idcrit ] ) .this paper combines several ideas already known in the ica literature : lumping together correlated components into a single multidimensional component is in ; minimization of a covariance - matching contrast derived from the log - likelihood of a simple gaussian model is found for instance in  ; the extension to noisy models is already explained in  .the current paper goes one step further by showing how arbitrarily structured components can be separated and how the related complexity can be managed at the software level by a library of components .  * about gaussianity*. the specific choice of the matching criterion ( [ eq : idcrit ] ) stems from a gaussian model for the signal coefficients and the linear wiener filter ( [ eq : wiener ] ) is optimal only for gaussian signals .hence , there is no doubt that an improved statistical efficiency could be gained by resorting to non gaussian models and to non - linear filtering for strongly non gaussian data .however , this is more easily said than done , non gaussian modeling / processing being often more difficult and costly to implement . by sticking to simple gaussian assumptions, we can afford to model a non trivial correlation structure ( through domains and through sensors ) so it is not clear yet what the good trade - off is .it may depend very much on the scientific objectives ( recovery of cmb versus recovery of other components , ability to predict estimation errors , ) and technical constraints ( _ e.g. _ fast codes are important for assessment via monte - carlo runs ) .intensive work is in progress within the planck collaboration to assess the performance of various approaches for cmb analysis .we acknowledge the use of healpix  for all sphere - based computations .our code is implemented in octave ( ` octave.org ` ) .synthetic data are made available thanks to the planck collaboration .maude martin was partially supported by astromap and cosmostat grants , two aci programs of cnrs .    c. l. bennett , r. s. hill , g. hinshaw , m. r. nolta , n. odegard , l. page , d. n. spergel , j. l. weiland , e. l. wright , m. halpern , n. jarosik , a. kogut , m. limon , s. s. meyer , g. s. tucker , and e. wollack . ., 148:97117 , september 2003 .jacques delabrouille and jean - franois cardoso ., chapter diffuse source separation in cmb observations .ecture notes in physics .springer , 2007 .editors : vicent j. martinez , enn saar , enrique martinez - gonzalez , maria jesus pons - borderia .jacques delabrouille , jean - franois cardoso , and guillaume patanchon .detector multi component spectral matching and applications for cmb data analysis ., 346(4):10891102 , december 2003 . also available as http://arxiv.org/abs/astro-ph/0211504.           h. snoussi , g. patanchon , j. f. macas - prez , a. mohammad - djafari , and j. delabrouille . .in r. l. fry , editor , _ bayesian inference and maximum entropy methods in science and engineering _ , volume 617 of _ american institute of physics conference series _, pages 125140 , may 2002 .

Abstract: this paper offers a new point of view on component separation , based on a model of additive components which enjoys a much greater flexibility than more traditional linear component models . this flexibility is needed to process the complex full - sky observations expected from the planck space mission , for which it was developed , but it may also be useful in any context where accurate component separation is needed .