Yann Le Gal

Hengbo Cui

Pere Alemany

Enric Canadell

Reizo Kato

Thierry Roisnel

Vincent Dorcet

Marc Fourmigue

Dominique Lorcy

Marc Fourmigué

Mixed-valence gold bis(diselenolene) complex turning metallic under pressure

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Introduction

Among metal bis(dithiolene) complexes, gold complexes, 1,2 formulated as [Au(dt)2] -,• , are particularly investigated for their ability to form single component conductors 3 in their radical neutral form, [Au(dt)2] • . Their square planar geometry, together with extensively delocalized substituents, favors strong intermolecular interactions within stacks which are at the origin of the bulk conducting properties. [4][5][6][7][8][9][10][11][12][13][14][15][16][17][18] The main approach towards these highly conducting materials relies on the oxidation of the mono-anionic Au III [Au(dt)2] -complexes by electrocrystallization techniques. 19 Among the various neutral conducting gold bis(dithiolene) complexes reported to date, those involving the N-alkyl-1,3-dithiole-2-thione-4,5-dithiolate ligand (R-thiazdt) led to a variety of conducting materials, with different R groups such as R = Et, 20,21 Pr, 22 iPr, 23 cPr, 22 NMe2, 22 (CH2)2OH. 24 These neutral radical gold bis(R-thiazdt) complexes do not exhibit the usual tendency of radical species to dimerize but rather form uniform stacks in the solid state, leading to highly conducting semiconductors.

Application of pressure can in some instances favor the stabilization of a metallic state as in [Au(Et-thiazdt)2] • . 20,21 The most striking example in that respect is the [Au(Me-thiazdt)2] • complex, which exhibits a metallic state from room temperature to 4 K already at ambient pressure. 25 Along these lines, we have also shown that the introduction of selenium atoms in the metallacycles, i.e. in the corresponding gold bis(diselenolene) complexes, increases the room temperature conductivity by two orders of magnitude, as for instance between [Au(Et-thiazds)2] • and [Au(Et-thiazdt)2] • . 21 Such effect is due to stronger intermolecular interactions for selenium compounds that lead, consequently, to a notable increase of the band dispersion in the solid state. This overwhelming tendency of gold bis(dithiolene) complexes to form neutral radical species upon electrocrystallization of the [Au(dt)2] -species strongly contrasts with their Ni, Pt and Pd bis(dithiolene) congeners which give rise upon electrocrystallization to mixed-valence, multi-component conductors as the well-known 1:2 (TTF)[Ni(dmit)2]2 or C[Pd(dmit)2]2 salts with C + = Me4N + , Me4P + , Et2Me2P + , etc. 26 These compounds exhibit a variety of attractive physical properties, ranging from superconductivity in (TTF)[Ni(dmit)2]2 to Quantum Spin Liquid phases (QSL) in β'-EtMe3Sb[Pd(dmit)2]2. 27,28 Three decades ago, [Au(dmit)2] -and [Au(dsit)2] -(dmit : 4,5-dimercapto-1,3-dithiole-2-thione and dsit : 4,5-diselenolate-1,3dithiole-2-thione), [29][30][31][32][33][34] were also investigated as precursors of multi-component materials through oxidation of the monoanionic species. Among the different materials obtained, only two mixed-valence multi-component salts have been structurally characterized so far, namely (Me4N)[Au(dmit)2]2, 33 and (Et4N)[Au(dmit)2]2, 31 both of them exhibiting semi-conducting behavior in the whole temperature range at ambient pressure (σRT = 0.6-1 S.cm -1 ).

Considering the notably higher conductivities observed by replacing sulfur atoms in the metallacycles by selenium ones on the ethyl-substituted series, [Au(Et-thiazds)2] / [Au(Etthiazdt)2], we decided to investigate the diselenolene analogue in the methyl series, i.e.

[Au(Me-thiazds)2] • , since, as mentioned above, the dithiolene analog [Au(Me-thiazdt)2] exhibits a metallic behavior already at ambient pressure. 25 Herein, we describe the synthesis of the monoanionic diselenolene complex, [C + ][Au(Me-thiazds)2] (C + = Ph4P + , Et4N + ) with two different counterions in order to investigate their behavior upon electrocrystallization.

Contrary to our expectation with this class of complexes, we did not obtain the neutral radical single component material [Au(Me(thiazds)2] • but rather a rare 1:2 mixed-valence salt formulated as [Ph4P][Au(Me-thiazds)2]2. In this paper, we report our investigations on this multi-component conductor, which, as shown below, is the first mixed-valence gold complex with a metallic behavior.

Results and discussion

Syntheses.

As shown in Scheme 1, the synthesis of the monoanionic gold bis(diselenolene) complex starts with the cyanoethyl-protected diselenolene proligand 1. 35 ] both crystallize in the monoclinic system, space group P21/n for the Ph4P + salt and P21/c for the Et4N + salt. The square-planar monoanionic complex, [Au(Me-thiazds)2] -, adopts a cis configuration in the Et4N + salt (Figure 1a) while in the Ph4P + salt, it is localized in general position in the unit cell and is disordered on two positions (85:15) compatible with a trans configuration (Figure 1b). This disorder is reminiscent of that reported for its dithiolene analogue [Au(Me-thiazdt)2] -. 25 Indeed, as often encountered for such metal bis(dithiolene) complexes with dissymmetrical ligands, the cis and trans isomers can be formed. 36 With Rthiazdt dithiolene ligand, the trans configuration is the most often observed. The less common cis configuration was also observed together with the trans one in two cases only, namely in 24 and [Ph4P][Au(Pr-thiazdt)2.]. 22 The only monoanionic gold bis(diselenolene) complex crystallographically characterized so far within this ligand series, [Et4N][Au(Et-thiazds)2], was obtained as the trans isomer. 21 2 and Figure S1 in ESI), the reduction of the monoanionic complex into the dianionic one at E -2/-1 and the successive oxidation of the monoanionic complex into the neutral one at E -1/0 and then to the monocationic species at E 0/+1 . Both salts exhibit similar cyclic voltammograms with approximately the same redox potentials (Table 2). Actually, this is not surprising since in the conditions used the counterion of the supporting electrolyte Bu4N + will exchange with the counterion of the gold complex. Those values also compare with those reported with the analogous dithiolene compounds. PPh4 + for two gold bis(diselenolene) complexes. This 1:2 salt crystallizes in the monoclinic system, space group P2/c, with two crystallographically independent gold complexes, one (Au1) located in a two-fold axis and the other (Au2) on an inversion center. The Ph4P + cation is also on a two-fold axis, hence the 1:2 stoichiometry. The two crystallographic independent complexes cis-Au1 and trans-Au2 alternate and form uniform stacks along c with a unique Au1-Au2 interaction along the chain (Figure 3). The complexes adopt two different configurations, with cis geometry for Au1 and trans geometry for Au2 (Figure 4). A closer look at the intramolecular bond lengths indicates that the cis-Au1 complex is more oxidized than the trans-Au2 one (Cf Table 1). This can be determined by comparison of the bond lengths of the metallacycle with the monanionic species for the cis derivative and with the bond length of the Et analogue, [Au(Et-thiazds)2],

[Ph4P][Au(EtOH-thiazdt)2]
for the trans one. Indeed, for the cis isomer, the C=C bonds are clearly lengthened upon oxidation, while the C-Se bonds shorten. This tends to indicate that the cis-Au1 complex is closer to the neutral (and radical) state while the trans-Au2 one is closer to the unoxidized monoanionic state. This charge separation can have important consequences on the conductivity of such compounds as it is potentially associated with a charge localization and weak conductivity. In order to evaluate how this structure and its conductivity evolves under pressure, transport measurements were undertaken using a diamond anvil cell (DAC). Note that the pressure generated by a DAC tends to be anisotropic and its effect is often different from that of hydrostatic pressure. Some of us have developed a DAC technique that can generate highquality quasi-hydrostatic pressures, as reported earlier, 37,38 and used in the present work.

Under pressure, the room temperature conductivity was increased up to 7.14 S cm -1 as a pressure of 10.3 GPa was reached, and stays constant up to 11.6 GPa (Figure 6).

Furthermore, a change of regime from semiconducting to metallic occurs at 10.3 GPa. This

[Ph4P][Au(Me-thiazds)2]2 compound is therefore the first multi-component gold bis(diselenolene) (or dithiolene) complex that displays a metallic behavior.

Electronic Structure

The coexistence in the crystal structure of two isomers with apparently different charge is somewhat surprising since the cis-trans isomerism is not expected to provide the driving force for such a charge separation, yet the charge ordering may be at the origin of the semiconducting behavior. To gain some understanding about the origin of the charge separation and its possible evolution under pressure, we carried out a first-principles density where the interaction along the chains is considerably stronger, 20,25 the electronic structure around the Fermi level is almost fully determined by the SOMO and there is no internal electron transfer between the two levels. in Figure 7b, even without considering the role of pressure this is not the case. Under pressure, with the reasonable assumption that the space group is maintained, the inter-chain interaction should increase and the two-dimensional character of the Fermi surface should most likely be reinforced. Consequently, even if the system could undergo some structural or spin modulation destroying the nested parts of the Fermi surface, large portions would remain and a metallic state would be kept. For the time being we suggest that if the metallic state can be stabilized under pressure, even quite weak lateral interactions will most likely be enough to suppress the tendency of the stacks of gold complexes to undergo some kind of modulation along c and will preserve the metallic character.

ACCEPTED MANUSCRIPT -CLEAN COPY B. Localized states. Looking at the band structure of Figure 7a it is clear that the two upper SOMO bands exhibit a relatively modest dispersion. Such dispersion is of the same order or even a bit smaller than that calculated in the same way for the single component gold complex 15 (dm-dddt:5,6-dimethyl-5,6-dihydro-1,4-dithiin-2,3-dithiolate), which is already a localized system. The weak dispersion is easy to understand when looking at the four different types of interactions (Figures 8b-e) and the nature of the SOMO of the present cis and trans gold complexes (Figure 8a). Interactions along the transverse b direction (Figures 8b andc) are of the usually not quite strong lateral -type and are associated with long Se•••Se contacts thus leading to relatively weak interactions. The interaction along the uniform chains (Figure 8d) is associated with Au•••Se contacts but the Au participation in the SOMO is quite small (Figure 8a) so that this interaction cannot be very strong. Finally, the other interaction between chains (Figure 8e) implicates three Se•••Se contacts but because of the antisymmetric nature of the SOMO, the interactions partially compensate and the interaction can neither be strong. Thus, it seems that the different SOMO•••SOMO interactions occurring in this lattice must be relatively modest. In order to put this observation in a more quantitative basis we have calculated the SOMO-SOMO interaction energies 39 associated with each interaction with the usual extended Hückel approach. The absolute values of the SOMO-SOMO for the interactions in Figures 8b,c The calculated band structure of the ferromagnetic state (FM) is shown in Figure 9a.

[Au(R,R-dm-dddt)2]
The spin-up (red) and spin-down (blue) bands are practically rigidly shifted (~ 0.2 eV) and have almost the same shape as in the metallic state. A very small band gap separates the filled from the empty bands. This FM state is calculated to be 18.4 meV per [Ph4P][Au(Methiazds)2]2 formula unit more stable than the metallic state. Thus, localization is favored although the energy difference with the metallic state is weak and comparable to those found for other single component gold complexes which become metallic under 5-15 kbar pressure. 20,21 Analysis of the contribution of the cis and trans complexes to the DOS led to the same result (within numerical errors) as for the metallic state: the electrons are located ~ 40% on the cis Au1 complexes and ~ 60% on the trans Au2 complexes.

The band structure of the antiferromagnetic state (AF1) is shown in Figure 9b. Every band in this figure is really a pair of identical bands located on two equivalent parts of the lattice, one associated with spin-up and the other with spin-down. A band gap of approximately 0.1 eV opens as a result of the antiferromagnetic interactions. This state is found to be 3.1 meV per formula unit below the FM state. Note that both FM and AF1 have been calculated using the crystallographic unit cell so that ferromagnetic interactions along the b transverse direction are forced. When carrying out calculations using a double cell along b so as to also allow antiferromagnetic interactions along this direction we located four different states: FM, AF1 as well as the antiferromagnetic counterparts along b of the FM (labeled AF2) and AF1 (labeled AF3) states. The AF2 and AF3 states are slightly more stable than the FM by 1.9 and 1.3 meV per formula unit, respectively, and thus have energies in between those of the FM and AF1 states. In other words, the four different localized states are found to be within a very narrow energy range of 3 meV per formula unit and not that far from the metallic state. Let us note that because of the large size of the unit cells we are almost reaching the accuracy limit of the calculations so that the energy differences between the four localized states should not be overemphasized. However, since the four states are associated with different spin orderings it is understandable that [Ph4P][Au(Me-thiazds)2]2

does not exhibit any magnetic ordering in our measurements. complexes is now nil because of the delocalized nature of the transferred electrons and the change of the spin sign along the chain. This picture of the transferred electrons centered on the trans-Au2 complexes but "dressed" with a smaller but significant contribution on the two adjacent cis-Au1 complexes provides a simple but accurate description of our calculations and seems to be common to both the metallic and all localized states.

Origin of the charge order and possible effect of pressure. At this point it is important to understand why the two isomers prefer to exhibit different charges. In principle, it seems unlikely that the preference can be an intrinsic effect associated to the different location of the two N-CH3 groups. To test this idea, we have carried out DFT structural optimizations for the two isolated isomers and found that the optimized structures are practically identical (see Figure S3 in Supplementary Information). For these optimized structures, the larger bond distance differences between the two isomers amount to about 0.2%, although most of the bonds are identical. Note that, in contrast, the bond distance differences in the crystal structure reach values of up to 5-6% for some bonds. Under such circumstances, the preference should be an extrinsic effect due to the environment.

Considering the nature of the cation-anion interface and that the positive charge in the Ph4P + cation is found in our calculations to be very uniformly distributed over all C-H units of this large cation, the electrostatic interaction between the two partners should mostly originate from local C-H•••S interactions between the sulfur atoms of the gold complexes and the hydrogen atoms of the Ph4P + cations. Shown in Figure 3 interactions with the trans-Au2 complex (Figure 10a). In contrast, the cis-Au1 complex does not find a good match to create strong C-H•••S interactions with the cations, and the shortest contacts of this type are noticeably longer, ~3.20 Å (Figure 10b). of electronic density around these sulfur atoms. When the structural details make these interactions possible, the sulfur atom will tend to reinforce these interactions by acquiring as much electronic density as possible from the rest of the molecule. This results in a polarization of the electron density which ultimately leads to the buildup of an electronic hole in the inner part of the molecule, which thus will try to keep the electrons that had been transferred. In other words, the strength of the C-H•••S interactions in the outer part of the Au complex is intimately related with the amount of additional electron density transferred to the SOMO and the Au complexes located in the cavities allowing the better C-H•••S interactions will be those more negatively charged, i.e. the trans-Au2 complexes in this case. This mechanism is not unrelated to that proposed [40][41][42] for the charge ordering transition of systems such as -(BEDT-TTF)2I3 and -(EDT-TTF-CONMe2)2Br, as well as the anion ordering transitions in the (TMTTF/TMTSF)2X Fabre/Bechgaard salts.

Since the SOMOs of nearest neighbor cis-Au1 and trans-Au2 complexes are not isolated but somewhat linked (i.e. the associated transfer integral is not nil), the fraction of electrons transferred to both complexes is interrelated and will subtly depend on the C-H•••S contacts around the two complexes. This feature provides a very effective mechanism to control the partitioning of the transferred electrons among the two different Au complexes which certainly must be at the origin of the insulator to metal conversion under applied pressure (see below). Note that the charge order described here is different in origin from most charge order states described in the literature, 43 where optimization of the electronic interactions within the conducting sublattice is the driving force for the ordering. In the present case it is induced by the environment and is almost independent of the electronic state of the gold complexes.

The effect of pressure on a crystal structure containing flexible Ph4P + cations, with many structural degrees of freedom, is not easy to predict. However, with the previous discussion in mind it is not difficult to suggest a plausible mechanism for the pressure induced metallicity of the system. To begin with, according to our calculations the metallic state for the ambient pressure structure seems to be quite close to the localized states. Pressure certainly will compress the lattice and increase the strength of the SOMO•••SOMO interactions, thus favoring the metallic state. However, just strengthening some of these interactions is most probably not enough to induce the metallic state. As we have discussed, Alternatively, the present system could be described in a more physically oriented language as being built from a series of three-quarter filled uniform chains with moderate intra-chain interactions, which is thus prone to electronic localization and charge ordering because of the different site potentials associated with the two complexes. However, such description, based on an simplified model which takes only the anions into account, does not provide (i) a simple chemical understanding of the origin of the different site potentials, (ii) the importance of the inter-layer interactions to stabilize a possible metallic state and (iii) a plausible mechanism of the observed pressure induced metallization. First-principles DFT approach adopted in this work includes all the interactions between the different species (anionic and cationic) in the system, treating them on an equal foot. Transport measurements.

Conclusions

Single crystals with an average size of 0.13  0.04  0.01 mm 3 were used for the highpressure measurements. The sample was mounted in the DAC by using the same technique as that used for [Ni(ptdt)2]. 46 The sample was encapsulated with a mixture of epoxy and alumina.

The diamond culet size was 0.56 mm. Tension annealed SUS301 was used. Electrical contacts were obtained by attaching four 10 μm gold wires with gold paint, and the four-probe DC method was used for all measurements. Daphne Oil 7373 was used as the pressure transmitting medium. The pressure was determined by the shift in the ruby fluorescence R1 lines at room temperature.

Electronic structure calculations

The first-principles calculations for the solid were carried out using a numeric atomic orbitals density functional theory (DFT) approach 47,48 developed for efficient calculations in large systems and implemented in the SIESTA code. [49][50][51][52] We used the generalized gradient approximation (GGA) to DFT and, in particular, the functional of Perdew, Burke, and Ernzerhof. 53 To study the relative energies of states with localized electrons, spin polarized band calculations for appropriate supercells have been undertaken. All calculations included a Hubbard correction term Ueff = U -J = 6.0 eV for the S 3p and Se 4p states. 54 In previous work 55 we have found that this U term on the chalcogen atoms is needed for appropriately describing the electronic structure of molecular conductors were accurate experimental information on the bandwidth and charge transfer is available. Only the valence electrons are considered in the calculation, with the core being replaced by norm-conserving scalar relativistic pseudopotentials 56 factorized in the Kleinman-Bylander form. 57 We have used a split-valence double- basis set including polarization orbitals with an energy shift of 10 meV 58 for S, Se, C, N, P and H atoms. For gold atoms we have used a split-valence basis set of double-ξ plus polarization quality, where the 5d electrons of Au were treated also as valence electrons. The basis functions used for Au have been optimized in order to reproduce the geometry and the bulk modulus for the ccp crystal structure of metallic gold. 59 The energy cutoff of the real space integration mesh was 300 Ry. The Brillouin zone was sampled using grids 60 of (3×9×5) and (20×20×20) k-points for building the charge density and calculation of the Fermi Surface, respectively. The crystal structure at 150 K was used for the computations.

The DFT calculations for isolated molecular systems have been carried out with the Gaussian 09 program 61 using the PBE functional with the double-ξ quality LANL2DZ basis set and the Los Alamos effective core potentials. 62 The tight-binding band structure calculations were of the extended Hückel type 63 using a modified Wolfsberg-Helmholtz formula to calculate the non-diagonal H values. 64 All valence electrons were taken into account in the calculations and the basis set consisted of Slater-type orbitals of double-ζ quality for Au 5d and of single-ζ quality for Au 6s and 6p, C 2s and 2p, N 2s and 2p, S 3s and 3p, Se 4s and 4p and H 1s.The ionization potentials, contraction coefficients and exponents were taken from previous work. 15

Conflicts of interest

There are no conflicts to declare.

  Scheme 1. Synthetic route toward monoanionic gold diselenolene complexes, [Ph4P][Au(Methiazds)2] and [Et4N][Au(Me-thiazds)2.

  From the obtained results and what is found in the literature, there is no clear pattern on which factors favor one ACCEPTED MANUSCRIPT -CLEAN COPY 6 configuration over the other, as even for the same ligand, and the same metal, the two configurations are obtained. It is therefore more the overall result of crystal packing forces, which might favor one configuration over the other.

Figure 1 .

 1 Figure 1. Molecular view of the monoanionic species [Au(Me-thiazds)2] -in (a) [Et4N][Au(Me-thiazds)2] and (b) [Ph4P][Au(Me-thiazds)2]. The two disordered moieties in [Ph4P][Au(Me-thiazds)2] are highlighted with light grey and black bonds.

Figure 2 .

 2 Figure 2. Cyclic voltammogram of [Et4N][Au(Me-thiazds)2] recorded in CH2Cl2-Bu4NPF6 0.1 M at a platinum electrode, scan rate 100 mVs -1 .

Figure 3 .

 3 Figure 3. Projection view of the unit cell of [Ph4P][Au(Me-thiazds)2]2 along b. C-H•••S interactions shorter than 3.1 Å are highlighted as dashed red lines.

Figure 4 .

 4 Figure 4. Details of both crystallographically independent gold complexes (cis-Au1, trans-Au2) in [Ph4P][Au(Me-thiazds)2]2, with selected intramolecular bond distances.

Figure 5 .

 5 Figure 5. Intermolecular interactions between diselenolene complexes in [Ph4P][Au(Methiazds)2]2.

Figure 6 .

 6 Figure 6. (a)Temperature dependence of the resistivity of [PPh4][Au(Me-thiazds)2]2 at different pressures, (b) Pressure dependence of room temperature conductivity and activation energy

  functional theory (DFT) study of the electronic structure of [Ph4P][Au(Me-thiazds)2]2.A. Metallic State. A good starting point in describing the DFT electronic structure of [Ph4P][Au(Me-thiazds)2]2 is the hypothetical metallic state of this salt (Figure7a) calculated without spin-polarization using the ambient pressure crystallographic cell (i.e. containing four gold complexes). The four upper bands are based on the SOMO (singly occupied molecular orbital) of the gold complexes whereas the four lower bands are based on the SOMO-1 (i.e. the orbital immediately below the SOMO). The two sets of bands overlap only well below the Fermi level so that, in contrast with other single component gold bis(dithiolene) conductors

Figure 7 .

 7 Figure 7. DFT Band Structure (a) Fermi Surface (b) and Density of States (b) calculated for the hypothetical metallic state of [Ph4P][Au(Me-thiazds)2]2 based on the ambient pressure crystal structure. The dashed line refers to the Fermi level and = (0, 0, 0), X= (a*/2, 0, 0), Y= (0, b*/2, 0), Z= (0, 0, c*/2) and M= (0, b*/2, c*/2) in units of the monoclinic reciprocal lattice vectors. In (c) the projected densities of states (PDOS in units of electrons/eV unit cell) associated with the cis (Au1, blue) and trans (Au2, red) gold complexes are shown.

13 An

 13 important aspect to consider is the different degree of localization of the charge on the two types of gold complexes. Shown in Figure7cis the total density of states (DOS) for [Ph4P][Au(Me-thiazds)2]2 as well as the contributions (PDOS) of the cis-Au1 (blue line) and trans-Au2 (red line) complexes. The distribution of the transferred electrons among the two different complexes can be obtained from the associated PDOS curves. This leads to the values -0.39 and -0.61 for the cis-Au1 and trans-Au2 complexes, respectively. Note that the cis and trans contributions overlap considerably all along the energy range of the SOMO orbitals (from ~ -0.2 to +0.1 eV) suggesting a sizable delocalization of the transferred electrons. Thus, even in the hypothetical metallic state with this structure, the transferred electrons should be more strongly located on the trans-Au2 isomer, although with an important delocalization toward the adjacent cis-Au1 isomers.

  , d and e are (in eV) 0.0003, 0.0177, 0.1603 and 0.0097, respectively. These values together with the different energies of the SOMOs of the two complexes in the uniform chains (= 0.151 eV, which effectively decreases the last two interactions) lead to: (a) a non-negligible inter-chain coupling and, (b) the modest band dispersion of the SOMO bands in Figure 7. These results prompted us to examine also different localized states of the [Ph4P][Au(Me-thiazds)2]2 solid. We carried out spin-polarized DFT calculations for different ferromagnetic and antiferromagnetic states of [Ph4P][Au(Me-thiazds)2]2 using both the crystallographic unit cell as well as a double cell along b so as to also allow for antiferromagnetic interactions along this transverse direction.

Figure 8 .

 8 Figure 8. (a) SOMO orbital for the cis and trans gold complexes. (b) and (c) Intermolecular interactions along the transverse b-direction. (d) Intermolecular interaction along the chain cdirection. (e) Intermolecular interaction between chains. See also Figure 5.

Figure 9 .

 9 Figure 9. DFT Band Structure for the ferromagnetic (a) and antiferromagnetic AF1 (b) states calculated using the crystallographic cell of [Ph4P][Au(Me-thiazds)2]2. The dashed line refers to the highest occupied level and = (0, 0, 0), X= (a*/2, 0, 0), Y= (0, b*/2, 0), Z= (0, 0, c*/2) and M= (0, b*/2, c*/2) in units of the monoclinic reciprocal lattice vectors. Spin-up and spindown bands are shown in blue and red, respectively. The spin-up and spin-down bands are identical although located in spatially different but equivalent sites in (b) so that only the blue bands are visible.

  is a view of the cation-anion interface where the S•••H distances shorter than 3.1 Å are highlighted in red. It is quite clear that only the trans-Au2 complexes undergo strong C-H•••S interactions with the cations. Three phenyl groups on each side of the molecule are well oriented to establish C-H•••S

Figure 10 .

 10 Figure 10. (a) Environment of the trans-Au2 complex where the C-H•••S interactions shorter than 3.1 Å are shown as red dashed lines. (b) Environment of the cis-Au1 complex where the shorter but relatively long C-H•••S interactions are shown as black dashed lines.

  the charge separation is related to the neighboring cations. The more effective way to favor the metallic state is by increasing the delocalization of the holes which are already somewhat delocalized in the semiconducting localized states. Of course, increasing the SOMO•••SOMO interactions will favor delocalization. But there is another subtle and easy way to do it, just by decreasing the difference in strength of the C-H•••S interactions for the cis and trans complexes. Since establishment of these localized electrostatic interactions increases the stiffness of the lattice around them, pressure should have a larger effect on the interaction of the closer-to-neutral cis complex with the environment. Thus, the cis complex might tend to acquire some extra electron density from the neighboring trans complexes toward a hypothetical regular cis/trans 0.5/0.5 charge distribution. Thus, pressure will certainly provide a strong driving force for the metallization through (i) intrinsic effects (i.e. increase of the transfer integrals within the layers of gold complexes) and (ii) extrinsic effects (i.e. decreasing the difference in the C-H•••S interactions of the environment with the two types of Au complexes. With a metallic state nearby and these two effects at work, quite affordable pressures should be able to induce the metallic behavior. As noted above, this metallic state should be quite stable because of the non nil inter-chain interactions. The question about which of the two effects, intrinsic or extrinsic, are dominant in the mechanism of the observed pressure metallization is something that must await structural work under pressure.

  Based on the observation of a substantial increase of the conductivity by replacement of the sulfur atoms by selenium ones in the metallacycles of the ethyl-substituted series [Au(Etthiazds)2] / [Au(Et-thiazdt)2], we have tried to use the same strategy to prepare highly conducting solids based on the methyl-substituted analogues. We have been able to prepare a new gold bis(diselenolene) which upon electrocrystalization affords an original mixedvalence 1:2 salt, [Ph4P][Au(Me-thiazds)2]2. The crystal structure of this salt contains unusual layers where the gold complexes exhibit two different configurations with cis and trans geometries and different degree of oxidation. Analysis of the correlation between the crystal and electronic structures reveals that the charge differentiation is induced by the different strength of the C-H•••S interactions at the cation-anion interface. DFT type calculations show that the ground state of this salt is an electronically localized semiconducting state although the metallic state lies not far in energy. In contrast with other 1:2 gold bis(dithiolene) salts, the present system turns metallic under pressure (> 10 GPa) and is thus the first 1:2 mixedvalence gold complex metallic salt. These results open a new pathway in the search for highly conducting gold complex salts.

Table 1

 1 Intramolecular bond distances within the complexes [Au(R-thiazds)2] -1,0 , R = Me, Et

	Monanions:	a	a'	b	b'	c	Ref
	cis-[Au(Me-thiazds)2] -(a)	2.435(2) 2.448(2)	2.441(2) 2.423(2)	1.896(19) 1.82(2)	1.897(17) 1.96(2)	1.25 (2) 1.251(17)	this work
	trans-[Au(Et-thiazds)2] -	2.4405(6)	2.4456(6)	1.874(6)	1.910(6)	1.342(8)	21
	In [Ph4P][Au(Me-thiazds)2]2:	a	a'	b	b'	c
	cis-[Au(Me-thiazds)2] -x	2.437(3)	2.439(2)	1.82(2)	1.86(2)	1.38(3)	this work
	trans-[Au(Me-thiazds)2] -1+x	2.454(2)	2.437(2)	1.88(2)	1.87(2)	1.31(3)	this work
	(a) In the Et4N + salt,

Table 2

 2 Redox

	Me, Et) complexes.
		E -2/-1	E -1/0	E 0/+1	Ref
	[Et4N][Au(Et-thiazdt)2]	-0.90*	+0.52	+0.65	21
	[Et4N][Au(Et-thiazds)2]	-0.89*	+0.52*	-	21
	[Et4N][Au(Me-thiazdt)2]	-0.85*	+0.48	0.73/0.58	25
	[Ph4P][Au(Me-thiazds)2]	-0.87/-0.75	+0.51/0.40	0.76/0.54	this work
	[Et4N][Au(Me-thiazds)2]	-0.87/-0.73	+0.52/0.40	0.74/0.54	this work
	E = (Epa+Epc)/2, E is given when the kinetics of the electron transfer is fast, Epa-Epc close to 60 mV,
	while for large Epa-Epc, Epa/Epc, anodic and cathodic peak potentials, are given. * irreversible process

potentials (in V vs. SCE) of the [Au(R-thiazdt)2] and [Au(R-thiazds)2] (R =

Table 3

 3 Crystallographic data

		[PPh4]	[NEt4]	[PPh4]
		[Au(Me-thiazds)2]	[cisAu(Me-thiazds)2],	[Au(Me-thiazds)2]2,
	Formulae	C32H26AuN2PS4Se4	C16H26N3AuS4Se4	C40H32Au2N4PS8Se8
	FW (g.mol -1 )	1110.56	901.44	1881.76
	System	monoclinic	monoclinic	monoclinic
	Space group	P21/n	P21/c	P2/c
	a (Å)	10.7225(3)	7.5529(11)	21.436(2)
	b (Å)	19.7086(5)	27.483(5)	7.4127(7)
	c (Å)	16.5545(6)	12.3490(19)	16.0209(18)
	α (deg)	90	90	90
	β (deg)	99.7380(10)	94.562(6)	104.331(4)
	γ (deg)	90	90	90
	V (Å 3 )	3447.98(18)	2555.2(7)	2466.5(5)
	T (K)	150(2)	150(2)	150(2)
	Z	4	4	2
	Dcalc (g.cm -3 )	2.139	2.343	2.534
	µ (mm -1 )	8.805	11.792	12.253
	Total refls	51928	22161	5750
	Abs corr	multi-scan	multi-scan	multiscan
	Uniq refls (Rint)	8147(0.0460)	5819 (0.0511)	5750
	Uniq refls (I > 2σ(I))	7090	4716	5093
	R1, wR2	0.0518, 01115	0.0891, 0.1946	0.0769, 0.1977
	R1, wR2 (all data)	0.0612, 0.1158	0.1059, 0.2020	0.0935, 0.2155
	GOF	1.240	1.175	1.101

Acknowledgements

Work in Japan was supported by the JSPS KAKENHI (Grant Number JP16H06346). Work in Spain was supported by MINECO through Grants PGC2018-096955-B-C44 and PGC2018-093863-B-C22, Generalitat de Catalunya (2017SGR1506 and 2017SGR1289). E. C. acknowledges support of the Spanish MINECO through the Severo Ochoa FUNFUTURE (CEX2019-000917-S) Excellence Centre distinction and P. A. from the Maria de Maeztu Units of Excellence Program (MDM-2017-0767). Their support is gratefully acknowledged.

Notes and references

EXPERIMENTAL SECTION

General. Chemicals and materials from commercial sources were used without further purification. All the reactions were performed under an argon atmosphere. Melting points were measured on a Kofler hot-stage apparatus and are uncorrected. Mass spectra were recorded by the Centre Régional de Mesures Physiques de l'Ouest, Rennes. Methanol, acetonitrile and dichloromethane were dried using an Inert pure solvent column device. CVs were carried out on a 10 -3 M solution of complex in CH2Cl2 with NBu4PF6 0.1 M. Potentials were measured versus Saturated Calomel Electrode (SCE). The proligands 1 was prepared as previously reported. 35 [Et4N][Au(Me-thiazds)2]. Under inert atmosphere, a solution of NaOMe (4.55 mmol, prepared from 105 mg of Na in 10 mL of dry MeOH) was added to the proligand 1 (225 mg, 0.57 mmol). After complete dissolution, the solution was stirred at room temperature for 1 hour. Then a solution of of KAuCl4 (129 mg, 0.34 mmol) in 15 mL of dry MeOH was added, followed 5 hours later by the addition NEt4Br (131 mg, 0.63 mmol). The reaction mixture was stirred at room temperature for 24 hours and the precipitate was filtered and recrystallized in Me3CN to afford the monoanion complex [NEt4][Au(Me-thiazds)2] as dark crystals. Yield: 84%; Mp = 225°C. 1 H NMR (300 MHz, CD3CN) δ 1.22 (m, 12H, CH3), 3.18 (q, 8H, CH2, J = 7.2 Hz), 3.58 (s, 6H, CH3); 13 (10 mg) was placed in the anodic compartment, and (PPh4)(PF6) (100 mg) in both compartments. Crystals suitable for X-ray diffraction studies, were obtained on the anode upon application of a constant current of 0.5 μA for 10 days.

X-Ray Crystallography

Suitable crystals for X-ray diffraction single crystal experiment were selected and mounted with a cryoloop on the goniometer head of a D8 Venture diffractometer equipped with a (CMOS) PHOTON 100 detector, using Mo-K radiation (λ = 0.71073 Å, multilayer monochromator). Structures were solved by a dual-space algorithm using the SHELXT program, 44 and then refined with full-matrix least-square methods based on F 2 (SHELXL program). 45 All non-hydrogen atoms were refined with anisotropic atomic displacement parameters. H atoms were finally included in their calculated positions and treated as riding on their parent atom with constrained thermal parameters. Details of the final refinements are summarized in Table 3.


Member, IEEE Wei Gao
email: weigao@ujs.edu.cn

Senior Member, IEEE Jie Chen
email: dr.jie.chen@ieee.org

Senior Member, IEEE Cédric Richard
email: cedric.richard@unice.fr

Member, IEEE Wentao Shi

Member, IEEE Qunfei Zhang
email: zhangqf@nwpu.edu.cn

Transient Performance Analysis of the 1 -RLS

Keywords: Transient analysis, sparse system, online identification, 1-RLS

The recursive least-squares algorithm with 1-norm regularization ( 1-RLS) exhibits excellent performance in terms of convergence rate and steady-state error in identification of sparse systems. Nevertheless few works have studied its stochastic behavior, in particular its transient performance. In this letter, we derive analytical models of the transient behavior of the 1-RLS in the mean and mean-square sense. Simulation results illustrate the accuracy of these models.

I. INTRODUCTION

S PARSITY aware adaptive filters have been successfully applied to a wide range of applications, e.g., echo cancellation [START_REF] Elisei-Iliescu | Recursive least-squares algorithms for the identification of low-rank systems[END_REF], channel estimation [START_REF] Pelekanakis | Adaptive sparse channel estimation under symmetric alpha-stable noise[END_REF], and system identification [START_REF] Zakharov | DCD-RLS adaptive filters with penalties for sparse identification[END_REF]. As the recursive least-squares (RLS) algorithm achieves better performance than the least-mean-square (LMS) algorithm for time-invariant system identification [START_REF] Haykin | Adaptive Filter Theory[END_REF], [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF], sparse RLS-type algorithms have attracted considerable attention.

The recursive 1 -regularized least-squares (SPARLS) algorithm was introduced using the expectation-maximization scheme [START_REF] Babadi | SPARLS: The sparse RLS algorithm[END_REF]. The greedy sparse RLS algorithm with exponential window exploiting the orthogonal matching pursuit was devised in [START_REF] Dumitrescu | Greedy sparse RLS[END_REF]. The 1 -norm regularized RLS ( 1 -RLS) algorithm was proposed based on the modified least-squares cost function with sparsity promoting regularization [START_REF] Eks ¸ioglu | RLS adaptive filtering with sparsity regularization[END_REF]. By considering two possible weighted 1 -norm sparsity constraints with the cost function, two weighted 1 -RLS algorithms were presented in [START_REF] Eksioglu | Sparsity regularised recursive least squares adaptive filtering[END_REF]. They can be viewed as particular cases of the convex regularized RLS (CR-RLS) algorithm, that was derived by considering any convex function in the regularizer [START_REF] Eks | RLS algorithm with convex regularization[END_REF]. In [START_REF] Hong | Zero-attracting recursive least squares algorithms[END_REF], the cost function of the original RLS was modified by adding an adaptively weighted 2 -norm penalty resulting in the proposal of two zero-attracting RLS (ZA-RLS) algorithms. A distributed sparse RLS algorithm was also proposed for decentralized scenarios over networks in [START_REF] Liu | Distributed sparse recursive least-squares over networks[END_REF]- [START_REF] Djuric | Cooperative and Graph Signal Processing: Principles and Applications[END_REF].

The zero-attracting LMS (ZA-LMS) was extensively studied in [START_REF] Shi | Convergence analysis of sparse LMS algorithms with l 1 -norm penalty based on white input signal[END_REF]- [START_REF] Chen | Transient performance analysis of zero-attracting LMS[END_REF], with a focus on its transient behavior. In [START_REF] Das | Robust adaptive filtering via convex combination of l 0 -RLS adaptive filters[END_REF], the authors analyzed the mean and mean-square deviations at steady-state of the 0 -norm regularized RLS ( 0 -RLS) algorithm. In contrast, despite the superiority of the 1 -RLS algorithm, no theoretical analysis of its transient behavior has been reported so far. This may be due to the fact that there are few analyses of the transient behavior of the RLS algorithm in the literature. In addition, the analysis of the update equation in the 1 -RLS is critical from a statistical perspective. To address this concern, we start this letter by reformulating the update equation of the 1 -RLS algorithm into a concise form. This makes the analysis of its transient behavior tractable both in the mean and mean-square sense. Simulation results validate the theoretical findings.

Notation: [x] i and [X] ij denote the i-th entry of column vector x and the (i, j)-th entry of matrix X, respectively. The superscript (•) represents the transpose of vector or matrix. The matrix trace is denoted by tr{•}. The operator sgn{•} takes the sign of the entries of its argument. All-zero vector of length N is denoted by 0 N , and all-one vector of length N is denoted by 1 N . The Gaussian distribution with mean µ and variance σ 2 is denoted by N (µ, σ 2 ). The multivariate Gaussian distribution with mean µ and covariance matrix Σ is denoted by N (µ, Σ). The cumulative distribution function (CDF) of the standard Gaussian distribution is denoted by φ(x). The CDF of the multivariate Gaussian distribution is denoted by Φ(x, µ, Σ).

II. THE PROBLEM AND 1 -RLS ALGORITHM

Assume that the input-output sequences are generated by an unknown time-invariant system with sparse impulse response:

y n = x n w + z n (1)
where x n ∈ R L is the regression vector at time instant n with positive definite correlation matrix R x = E{x n x n }, and w ∈ R L is the sparse optimal weight vector to be estimated. The modeling error z n is assumed to be stationary, white and Gaussian with zero-mean and variance σ 2 z , and statistically independent of any other signal. Consider the batch least-absolute shrinkage and selection operator (LASSO) estimator usually considered for sparse system identification problems [START_REF] Tibshirani | Regression shrinkage and selection via the LASSO[END_REF]:

w o = arg min w∈R L 1 2 n i=0 λ n-i |y i -w x i | 2 + δ w 1 (2)
with 0 λ < 1 a forgetting factor, and δ > 0 a regularization parameter that controls the trade-off between the estimation error and the sparsity of the weight vector.

Based on the modified deterministic normal equation resulting from the subgradient vector of ( 2), the 1 -RLS algorithm arXiv:2109.06749v1 [eess.SP] 14 Sep 2021

proposed in [START_REF] Eks ¸ioglu | RLS adaptive filtering with sparsity regularization[END_REF]- [START_REF] Eks | RLS algorithm with convex regularization[END_REF] is given by:

k n = P n-1 x n λ + x n P n-1 x n , (3)
w n = w n-1 + e n k n + δ λ -1 λ I L -k n x n (4)
× P n-1 sgn{w n-1 },

P n = λ -1 (P n-1 -k n x n P n-1 ), (5)
with the instantaneous estimation error e n = y n -x n w n-1 .

Here, matrix P n is defined as the inverse of the time-averaged correlation matrix Φ n for the input vector [START_REF] Haykin | Adaptive Filter Theory[END_REF], [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF], given by:

Φ n = n i=0 λ n-i x i x i + δλ n+1 I L = λΦ n-1 + x n x n . (6)
Recursion ( 4) is too complex to use in convergence analysis. By inserting

k n = P n x n and I L -k n x n = λP n P -1 n-1
from ( 3) and ( 5), respectively, into (4), it can be simplified as follows:

w n = w n-1 + e n P n x n + γP n sgn{w n-1 } (7)
with γ = δ(λ -1). This alternative recursion of the 1 -RLS algorithm makes the following analysis easier to handle. It can be seen from ( 7) that the only difference between the RLS and the 1 -RLS lies in the rightmost zero attractor term.

III. TRANSIENT PERFORMANCE ANALYSIS OF 1 -RLS

We shall now study the transient behavior of the -RLS algorithm in the mean and mean-square sense. We define the weight error vector w n as the difference between the weight vector w n and w , i.e.,

w n = w n -w . (8)
In essence, the analysis of the 1 -RLS consists of studying the evolution over time of the expectation of w n and its correlation matrix

K n = E{ w n w n }.
Before proceeding, we introduce the following statistical assumptions for mathematical tractability.

A1: The weight error vector w n-1 is statistically independent of the regression vector x n .

A2: Any pair of entries [ w n ] i and [ w n ] j with i = j is jointly Gaussian.

The so-called independence assumption (IA) A1 is widely used in the convergence analysis of adaptive filters [START_REF] Haykin | Adaptive Filter Theory[END_REF], [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF]. Assumption A2 has been used successfully in the analysis of the ZA-LMS algorithm [START_REF] Chen | Transient performance analysis of zero-attracting LMS[END_REF] as it makes the analysis of the nonlinear sign term in [START_REF] Dumitrescu | Greedy sparse RLS[END_REF] tractable. We shall illustrate the validity of A2 with histograms of [ w n ] i versus [ w n ] j .

A. Mean Weight Error Behavior Model

We focus on the mean weight error analysis of the 1 -RLS. Taking the expectation of both sides of (6) yields:

E{Φ n } = λE{Φ n-1 } + R x ( 9
)
with Φ 0 = ε -1 I L and ε a positive number. This recursion will be used in the following analysis. From ( 1) and ( 8), the instantaneous estimation error e n can be rewritten as follows:

e n = z n -x n w n-1 . (10)
Subtracting w from both sides of (7), using ( 8) and ( 10), we have:

w n = w n-1 -P n x n x n w n-1 + z n P n x n + γP n sgn{w + w n-1 }. (11)
Pre-multiplying both sides of (11) by P -1 n , then applying (6) and definition Φ n = P -1 n , leads to

Φ n w n = λΦ n-1 w n-1 + z n x n + γsgn{w + w n-1 }. (12)
Taking the expectation of both sides of [START_REF] Liu | Distributed sparse recursive least-squares over networks[END_REF], considering that noise z n is statistically independent of any other signal and it is zero mean, we have:

E{Φ n w n } = λE{Φ n-1 w n-1 } + γE sgn{w + w n-1 } .
(13) Using the following approximation presented in detail in [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF], and valid for small values of λ:

E{Φ n w n } ≈ E{Φ n }E{ w n } (14)
equation ( 13) becomes:

E{Φ n }E{ w n } = λE{Φ n-1 }E{ w n-1 } + γE sgn{w + w n-1 } . (15)
Let u i ∼ N (µ i , σ 2 i ) a Gaussian random variable. Lemma 1 presented in [START_REF] Chen | Transient performance analysis of zero-attracting LMS[END_REF] shows that:

E sgn{u i } = 1 -2φ(-µ i /σ i ). (16)
In order to evaluate the last term on the r.h.s. of (15), we set the i-th entry of E sgn{w + w n-1 } as follows:

[w + w n-1 ] i → u i (17)
with

[w ] i + E [ w n-1 ] i → µ i , (18)
E [ w n-1 ] 2 i -E [ w n-1 ] i 2 → σ 2 i , (19)
where

E [ w n-1 ] 2 i
can be extracted from the main diagonal entries of matrix K n-1 that will be determined in the next subsection. With the above definitions, it then follows that:

E{Φ n }E{ w n } = λE{Φ n-1 }E{ w n-1 } + γ(1 L -2ϕ) (20)
with ϕ = φ(-µ 1 /σ 1 ), φ(-µ 2 /σ 2 ), . . . , φ(-µ L /σ L ) . By pre-multiplying both sides of (20) by E{Φ n } -1 , we arrive at the mean weight error behavior for the 1 -RLS algorithm:

E{ w n } = λE{Φ n } -1 E{Φ n-1 }E{ w n-1 } + γE{Φ n } -1 (1 L -2ϕ) (21)
where ( 9) is used to perform the above recursion.

B. Mean-Square Error Behavior Model

We shall now analyze the 1 -RLS algorithm in the meansquare sense. Squaring [START_REF] Eks | RLS algorithm with convex regularization[END_REF] and taking its expected value, then considering assumption A1 and the statistical property of z n , the mean-square error (MSE) can be expressed as

E{e 2 n } ≈ σ 2 z + tr{R x K n-1 } (22)
where the second term on the r.h.s. of the above equation denotes the excess mean-square error (EMSE) [START_REF] Haykin | Adaptive Filter Theory[END_REF], [START_REF] Sayed | Fundamentals of Adaptive Filtering[END_REF]. Moreover, the instantaneous mean-square-deviation (MSD) is defined by:

MSD n = E{ w n 2 } = tr{K n }. (23)
In order to evaluate the EMSE or the MSE, and the MSD, we have to determine a recursion to calculate the correlation matrix K n . Post-multiplying ( 12) by its transpose, taking the expectation of both sides, and using the statistical property of noise z n , yields

E{Φ n w n w n Φ n } = λ 2 E{Φ n-1 w n-1 w n-1 Φ n-1 } + σ 2 z R x + γ 2 E sgn{w n + w n-1 }sgn {w n + w n-1 } + λγE Φ n-1 w n-1 sgn {w n + w n-1 } (24) + λγE sgn{w n + w n-1 } w n-1 Φ n-1 .
Evaluating ( 24) is difficult without using some approximations. The following approximations are used in the sequel:

E{Φ n w n w n Φ n } ≈ E{Φ n }K n E{Φ n }, (25)
E Φ n-1 w n-1 sgn {w n + w n-1 } ≈ E{Φ n-1 }E w n-1 sgn {w n + w n-1 } , (26)
E sgn{w n + w n-1 } w n-1 Φ n-1 ≈ E sgn{w n + w n-1 } w n-1 E{Φ n-1 }. ( 27
)
Similar approximations are presented in detail in [START_REF] Eweda | Stochastic analysis of the recursive least squares algorithm for cyclostationary colored inputs[END_REF]. To understand these approximations, we denote by ∆ n the random fluctuation of matrix Φ n around E{Φ n }, namely,

Φ n = E{Φ n } + ∆ n . (28)
Then, we have:

E{Φ n w n w n Φ n } = E{Φ n }K n E{Φ n } + E{Φ n }E{ w n w n ∆ n } + E{∆ n w n w n }E{Φ n } + E{∆ n w n w n ∆ n }, (29)
E Φ n-1 w n-1 sgn {w n + w n-1 } = E{Φ n-1 }E w n-1 sgn {w n + w n-1 } + E ∆ n-1 w n-1 sgn {w n + w n-1 } , (30)
E sgn{w n + w n-1 } w n-1 Φ n-1 = E sgn{w n + w n-1 } w n-1 E{Φ n-1 } + E sgn{w n + w n-1 } w n-1 ∆ n-1 . ( 31
)
We assume that the entries of ∆ n are small with respect to those of E{Φ n } based on the fact that [START_REF] Babadi | SPARLS: The sparse RLS algorithm[END_REF] shows that Φ n is a low-pass filtering of x n x n . The first term on the r.h.s. of each one of the above three equations then dominates the remaining ones, which leads to (25)-( 27). Simulation results in the sequel will confirm the validity of these approximations.

Using ( 25)-( 27), expression (24) can then be written as:

E{Φ n }K n E{Φ n } = λ 2 E{Φ n-1 }K n-1 E{Φ n-1 } + σ 2 z R x + γ 2 Q 1 + λγ E Φ n-1 }Q 2 + Q 2 E{Φ n-1 } (32) with Q 1 = E sgn{w n + w n-1 }sgn {w n + w n-1 } , (33)
Q 2 = E w n-1 sgn {w n + w n-1 } . ( 34
)
Pre-multiplying and post-multiplying (32) by E{Φ n } -1 simultaneously, it results that:

K n = λ 2 E{Φ n } -1 E{Φ n-1 }K n-1 E{Φ n-1 }E{Φ n } -1 + σ 2 z E{Φ n } -1 R x E{Φ n } -1 + γ 2 E{Φ n } -1 Q 1 E{Φ n } -1 + λγE{Φ n } -1 E Φ n-1 }Q 2 + Q 2 E{Φ n-1 } E{Φ n } -1 . (35)
In order to perform recursion (35), we shall now calculate Q 1 and Q 2 . Before proceeding by using assumption A2, consider two jointly Gaussian random variables u and v defined by:

u v ∼ N µ := µ u µ v , Σ uv := σ 2 u ρ uv ρ uv σ 2 v (36)
where µ and Σ uv denote the mean vector and the covariance matrix, respectively. According to Lemma 2 presented in [START_REF] Chen | Transient performance analysis of zero-attracting LMS[END_REF], it holds that:

E sgn{u}sgn{v} = Φ 0 2 , [µ u , µ v ] , Σ uv + Φ 0 2 , -[µ u , µ v ] , Σ uv -Φ 0 2 , [µ u , -µ v ] , Σ uv -Φ 0 2 , [-µ u , µ v ] , Σ uv (37)
with

Σ uv = σ 2 u -ρ uv -ρ uv σ 2 v . (38)
On the one hand, observe that the main diagonal entries [Q 1 ] ii for 1 ≤ i ≤ L are equal to 1. On the other hand, considering assumption A2 and (37), the off-diagonal entries [Q 1 ] ij for 1 ≤ i = j ≤ L can be obtained by making the identifications:

[w + w n-1 ] i → u, (39)
[w + w n-1 ] j → v, (40)
with

E [w + w n-1 ] i → µ u , (41)
E [w + w n-1 ] j → µ v , (42)
E [ w n-1 ] 2 i -E [ w n-1 ] i 2 → σ 2 u , (43)
E [ w n-1 ] 2 j -E [ w n-1 ] j 2 → σ 2 v , (44)
E [ w n-1 ] i [ w n-1 ] j -E [ w n-1 ] i E [ w n-1 ] j → ρ uv , (45)
where the expectations E [ w n-1 ] i [ w n-1 ] j are available from the off-diagonal entries [K n-1 ] ij . Furthermore, based on definition (36) and Lemma 3 proved in [START_REF] Chen | Transient performance analysis of zero-attracting LMS[END_REF], it holds that: where θ = b -c 2 /a > 0 with

E u sgn{v} (46)
Σ -1 uv = a c c b . ( 47
)
Likewise, in order to evaluate matrix Q 2 , we can make the following identifications:

[ w n-1 ] i → u, ( 48
) [w + w n-1 ] j → v, (49)
with

E [ w n-1 ] i → µ u , (50)
E [w + w ] j → µ v , (51)
for 1 ≤ i, j ≤ L. Consequently, all the entries of matrix Q 2 can be determined according to (46) under assumption A2. Using ( 9), (33), and (34), we can finally perform recursion (35), which allows us to characterize the transient mean-square error behavior of the 1 -RLS algorithm.

IV. NUMERICAL TESTS

The good performance of the 1 -RLS algorithm has already been illustrated in the literature by comparing it with related adaptive filters for sparse system identification [8]- [START_REF] Eks | RLS algorithm with convex regularization[END_REF]. We shall now examine the accuracy of the analytical models derived in this paper, via simulation results. All empirical curves were obtained by averaging over 500 Monte Carlo runs.

The input signal was generated with a first-order AR model, namely, x n = 0.6 x n-1 + s n , with s n a zero-mean white Gaussian random sequence. Its variance was set to σ 2 s = 0.64, in order that the variance of the input signal x n was σ 2

x = 1. The noise z n was zero-mean white and Gaussian with variance σ 2 z = 0.09. The optimal weight vector in (1) was set to: w = [0.9, 0.7, 0.5, 0.3, 0.1, 0 22 , -0.1, -0.3, -0.5, -0.7, -0.9] ∈ R 32 .

The exponential forgetting factor was set to λ = 0.995, and the regularization parameter was set to δ = 0.25. The initial parameter ε was set to 0.1. The weight vector was initialized to zero, that is, w 0 = 0 32 . Four histograms are depicted in Fig. 1 for two arbitrarily selected pairs of entries of the weight error vector w n , i.e., [ w n ] i , [ w n ] j have the required Gaussian-like profiles, which validates the feasibility of assumption A2. The mean weight behavior is shown in Fig. 2(a). One can observe that all the theoretical curves of weight coefficients w i predicted by (21) are generally consistent with those empirical curves including the zero coefficients. Fig. 2(b) shows that the empirical learning curves of the MSE and EMSE coincide with their theoretical curves obtained from ( 22) and (35), respectively. Fig. 2(c) shows the good agreement between the empirical learning curve of MSD and its theoretical prediction obtained from (23) and (35). Notice that there is a mismatch between the simulated and theoretical results during the initial transient stage due to the approximations (25)-(27). Fig. 2 illustrates the correctness and accuracy of our analytical models and the necessary approximations used in the analysis. These models offer an effective means of profoundly understanding the convergence behavior of the 1 -RLS algorithm in the context of online sparse system identification.

V. CONCLUSION In this letter, the transient behavior of the 1 -RLS algorithm was theoretically studied in the mean and mean-square sense. Simulation results illustrated their accuracy.

  (a) Empirical vs. theoretical evolution for w i . MSE and EMSE curves.

Fig. 2 .

 2 Fig. 2. Comparisons of empirical and theoretical simulation results for the 1 -RLS algorithm.

[ w n ] 2 ,

 2 [ w n ] 10 and [ w n ] 13 , [ w n ] 25 from 5 × 10 3 simulated samples, at time instant n = 200 and n = 1500, respectively. As shown in Fig. 1, the four histograms of bivariate vector(a) [ wn] 2 , [ wn] 10 at n = 200. (b) [ wn] 2 , [ wn] 10 at n = 1500. (c) [ wn] 13 , [ wn] 25 at n = 200. (d) [ wn] 13 , [ wn] 25 at n = 1500.

Fig. 1 .

 1 Fig. 1. Histograms of bivariate vector [ wn] i , [ wn] j with 5 × 10 3 samples.

Manuscript received MM DD, 2021; revised MM DD, 2021. This work was supported in part by the National NSFC under Grants 62171205 and 62171380.


Danqi Jin
email: danqijin@mail.nwpu.edu.cndr.jie.chen@ieee.org

Jie Chen

Cédric Richard
email: cedric.richard@unice.frjingdongchen@ieee.org

Jingdong Chen

PROXIMAL MULTITASK LEARNING OVER DISTRIBUTED NETWORKS WITH JOINTLY SPARSE STRUCTURE

Keywords: Distributed optimization, diffusion strategy, proximal operator, joint sparsity, ∞, 1-norm regularization

Modeling relations between local optimum parameter vectors in multitask networks has attracted much attention over the last years. This work considers a distributed optimization problem for parameter vectors with a jointly sparse structure among nodes, that is, the parameter vectors share the same support set. By introducing an ∞,1-norm penalty at each node, and using a proximal gradient method to minimize the regularized cost, we devise a proximal multitask diffusion LMS algorithm which promotes the joint-sparsity to enhance the estimation performance. Analyses are provided to ensure the stability. Simulation results are presented to highlight the performance.

INTRODUCTION

Diffusion adaptation has been widely used in multi-agent networks to address estimation problems in a distributed and online manner due to their superior performance and wide stability range [START_REF] Tu | Diffusion strategies outperform consensus strategies for distributed estimation over adaptive netowrks[END_REF]. Several diffusion algorithms have been devised under specific settings in the literature [START_REF] Lopes | Diffusion least-mean squares over adaptive networks: Formulation and performance analysis[END_REF][START_REF] Cattivelli | Diffusion recursive least-squares for distributed estimation over adaptive networks[END_REF][START_REF] Li | Distributed adaptive estimation based on the APA algorithm over diffusion networks with changing topology[END_REF][START_REF] Liu | Diffusion sparse least-mean squares over networks[END_REF][START_REF] Lorenzo | Sparse distributed learning based on diffusion adaptation[END_REF].

According to the relations between the optimal parameter vectors over the entire network, diffusion networks are divided into single-task and multitask networks. In single-task networks, all nodes estimate the same parameter vector. Typical works include [START_REF] Sayed | Diffusion adaptation over networks[END_REF][START_REF] Sayed | Adaptive networks[END_REF][START_REF] Sayed | Adaptation, Learning, and Optimization over Networks[END_REF]. In multitask networks, multiple different but related parameter vectors are inferred simultaneously in a cooperative manner, so as to improve the estimation accuracy by exploiting the similarities between tasks. One of the efficient ways to leverage these similarities is to introduce appropriate regularization terms. Related works include [START_REF] Chen | Multitask diffusion adaptation over networks[END_REF][START_REF] Nassif | Proximal multitask learning over networks with sparsity-inducing coregularization[END_REF][START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF][START_REF] Nassif | Diffusion LMS for multitask problems with local linear equality constraints[END_REF].

Multitask learning considerably enriches the modelling capacity of diffusion networks. Beyond the multitask models appeared in the aforementioned references, there are also applications where the optimal parameter vectors have a jointly sparse structure. Applications include, for instance, distributed dictionary learning and distributed spectrum sensing [START_REF] Lorenzo | Distributed spectrum estimation for small cell networks based on sparse diffusion adaptation[END_REF][START_REF] Liang | Distributed dictionary learning for sparse representation in sensor networks[END_REF]. Several works have been proposed to address problems with jointly sparse structure over diffusion networks.

The work of Jie Chen was supported in part by NSFC grants 61671382 and 61811530283. The work of Jingdong Chen was supported in part by NSFC grants 61761146001 and 61811530283. The work of C. Richard was funded in part by ANR under grant ANR-19-CE48-0002 and by the CoopIn-tEER program CNRS-NSFC (DIALOG project). Corresponding author: Jie Chen.

In [START_REF] Gu | Learning distributed jointly sparse systems by collaborative LMS[END_REF], the authors propose to use the mixed 2,0-norm. In [START_REF] Li | Distributed jointly sparse multitask learning over networks[END_REF], the authors devise an algorithm with 2,0, 2,1 and reweighted 2,1 regularizers. However, these works use the subgradient method because the objective functions involve non-differentiable regularization terms, which is unfavorable for an accurate and fast convergence.

Compared to subgradient-based methods, it is known that using proximal operators is a more efficient way to solve optimization problems with non-differentiable regularizers. It often results in iterations with subproblems that admit closed-form solutions or can be solved with simple specialized methods [START_REF] Parikh | Proximal algorithms[END_REF]. For distributed estimation over networks, proximal algorithms have been used to estimate sparse parameter vectors in [START_REF] Wee | A proximal splitting approach to regularized distributed adaptive estimation in diffusion networks[END_REF]. They are further used to optimize general stochastic costs with non-differentiable regularizers, and non-smooth regularizers in [START_REF] Vlaski | Proximal diffusion for stochastic costs with non-differentiable regularizers[END_REF] and [START_REF] Vlaski | Diffusion stochastic optimization with non-smooth regularizers[END_REF], respectively. Under the multitask assumption, the authors in [START_REF] Nassif | Proximal multitask learning over networks with sparsity-inducing coregularization[END_REF] derive a closed-form solution to the proximal operator of the 1-distance between parameter vectors. In this paper, we propose a proximal diffusion LMS strategy for multitask networks with jointly sparse structure. In contrast to existing works [START_REF] Gu | Learning distributed jointly sparse systems by collaborative LMS[END_REF][START_REF] Li | Distributed jointly sparse multitask learning over networks[END_REF], the optimization problem is formulated with the ∞,1-norm regularization on the parameter matrices, and the closed-form solution for the proximal operator under this case is derived. Conditions to ensure the stability in the mean and mean-square sense are also provided. The superiority of the proposed method is validated with numerical experiments.

Notation. Normal font x, boldface small letters x and capital letters X denote scalars, column vectors and matrices, respectively. Symbol [ • ]m denotes the m-th entry of its vector argument. The superscript (•) denotes the transpose operator. The mathematical expectation is denoted by E{•}. The Gaussian distribution with mean µ and variance σ 2 is denoted by N (µ, σ 2 ). Operator | • | takes the absolute value of its scalar or vector argument. Operator max{•,•} extracts the maximum value of its two arguments. Symbol denotes a component-wise inequality. The set N k denotes the neighbors of node k, including k itself, and |N k | denotes its cardinality. The N - k denotes the neighbors of node k, excluding node k. Vector 1L is the all-one vector of dimension L × 1.

PROBLEM FORMULATION

Consider a connected network consisting of N nodes. Each node k has access to streaming data {d k,n , u k,n } at time instant n, where u k,n ∈ IR L×1 is the regression vector and d k,n denotes the observed signal. We assume that the data at time instant n are related via the linear model:

d k,n = u k,n w k + z k,n , (1)
where w k ∈ IR L×1 is the unknown system vector to estimate, and z k,n is a zero-mean additive noise. We assume that z k,n is independent of any other signal. Further, we assume that vectors w k over the entire network are jointly sparse. This means not only each w k is a sparse vector but, in addition, they all share the same support, namely,

supp(w 1 ) = • • • = supp(w k ) = • • • = supp(w N ) (2)
where supp(w k ) {j : [w k ] j = 0} is the support of w k [START_REF] Huang | The benefit of group sparsity[END_REF].

In this work, we focus on distributed processing where only local information exchange is authorized. We thus collect w over the neighborhood of node k into an L × |N k | matrix, and replace the k-th column w k by the optimization variable w k . This leads us to the local parameter matrix:

W k w k , w with ∈ N - k ∈ IR L×|N k | . (3)
Without loss of generality, we suppose that the columns of W k are sorted in increasing order according to the values of k and . Several mixed-norms have been introduced in the literature to promote the jointly sparse structure of a matrix, including the mixed 2,1-norm and ∞,1-norm. Evaluating the mixed p,1-norm of matrix W k with p = 2 or ∞ results in the following two steps:

Step 1: Evaluate the p-norm of each row of W k , and stack the results into an L × 1 intermediate vector;

Step 2: Evaluate the 1-norm of the obtained intermediate vector to promote sparsity.

Though 2,1-norm can be more efficient in some cases [START_REF] Negahban | Simultaneous support recovery in high dimensions: Benefits and perils of block 1/ ∞-regularization[END_REF], we shall consider the ∞,1-norm to promote the joint-sparsity. It is element-wise separable and facilitates the derivation of the proximal operator.

PROXIMAL MULTITASK DIFFUSION LMS

Before proceeding, to facilitate the following derivation we also denote W k by

W k = w k,1 • • • w k,m • • • w k,L , (4)
where wk,m is the m-th row of matrix W k . To determine the unknown vectors w k with jointly sparse structure, we consider the regularized cost at node k:

J k (w k ) = J k (w k ) + λ k g(w k ) (5)
with

J k (w k ) 1 2 E |d k,n -u k,n w k | 2 . The nonnegative pa- rameter λ k is used to control the regularization strength, g(w k ) L m=1
wk,m ∞ evaluates the ∞,1-norm of W k . At each node k, we then consider the convex optimization problem [START_REF] Tropp | Algorithms for simultaneous sparse approximation. Part II: Convex relaxation[END_REF]:

w † k = argmin w k J k (w k ). (6)
Within the context of online learning, such optimization problem is usually solved via subgradient-based methods. In this paper, we propose to devise a proximal algorithm since it is more stable than subgradient iterations [START_REF] Wee | A proximal splitting approach to regularized distributed adaptive estimation in diffusion networks[END_REF][START_REF] Bertsekas | Incremental proximal methods for large scale convex optimization[END_REF]. Proximal gradient methods generate a sequence of estimates by the following iterations [START_REF] Parikh | Proximal algorithms[END_REF]:

w k,n+1 = prox µ k λ k g w k,n -µ k ∇J k (w k,n ) , (7)
where µ k is a positive small step-size, and the proximal operator is defined by

prox λg (v) argmin w k g(w k ) + 1 2λ w k -v 2 2 . (8)
By introducing the intermediate quantity ψ k,n+1 , calculating the gradient of J k (w k ) at w k,n and using instantaneous approximation for unknown statistical quantities, we obtain from (7) the proximal multitask diffusion LMS algorithm for jointly sparse networks reported in Algorithm 1. Different from regular diffusion algorithms where the intermediate estimates are fused by weighted average, in this algorithm, estimates from neighboring nodes are fused via the proximal operator of g(w k,n ).

ψ k,n+1 = w k,n +µ k u k,n d k,n -u k,n w k,n w k,n+1 = prox µ k λ k g ψ k,n+1 (9)

PROXIMAL OPERATOR EVALUATION

To perform Algorithm 1, we need to derive a closed-form expression for the following proximal operator:

w k,n+1 = prox µ k λ k g (ψ k,n+1 ) = argmin w k g(w k )+ 1 2µ k λ k w k -ψ k,n+1 2 2 .
(

) 10
As g(w k ) is separable over its all entries, its proximal operator can be evaluated in an element-wise manner as [START_REF] Parikh | Proximal algorithms[END_REF]:

[prox µ k λ k g (ψ k,n+1 )] m = prox µ k λ k gm ([ψ k,n+1 ] m ) (11)
with gm([w k ] m ) wk,m ∞, [w k ] m is the m-th entry of w k , and wk,m is the m-th row of matrix W k in (3). This leads us to:

[w k,n+1 ]m = argmin [w k ] m max{|[w k ] m |, |[w ] m | with ∈ N - k } + 1 2µ k λ k [w k ] m -[ψ k,n+1 ] m 2 . ( 12
)
For ease of presentation, we shall denote [w k,n+1 ]m by ŵ as long as there is no ambiguity, and denote the maximal value of

|[w ] m | for ∈ N - k as [w o k ] m . According to the relation between |[w k ] m | and [w o
k ] m , we further split the problem into the following two cases:

• Case 1: |[w k ] m | < [w o k ] m .
In this case, [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF] becomes:

ŵ = argmin [w k ] m |[w k ] m |<[w o k ] m [w o k ] m + 1 2µ k λ k [w k ] m -[ψ k,n+1 ] m 2 . ( 13
)
The solution is directly given by:

ŵ =      [ψ k,n+1 ] m , if |[ψ k,n+1 ] m | < [w o k ] m [w o k ] m , if [ψ k,n+1 ] m ≥ [w o k ] m -[w o k ] m , if [ψ k,n+1 ] m ≤ -[w o k ] m . (14)
• Case 2: 12) becomes:

|[w k ] m | ≥ [w o k ] m . Equation (
ŵ = argmin [w k ] m |[w k ] m |≥[w o k ] m |[w k ] m |+ 1 2µ k λ k [w k ] m -[ψ k,n+1 ] m 2 (15)
We shall first discard the constraint

|[w k ] m | ≥ [w o k ] m
, and denote by ŵo the solution of the unconstrained problem. As the cost function in [START_REF] Liang | Distributed dictionary learning for sparse representation in sensor networks[END_REF] is convex on the real domain, this constraint will be taken into account in the course of the calculation. Consider first:

ŵo = argmin [w k ] m |[w k ] m |+ 1 2µ k λ k [w k ] m -[ψ k,n+1 ] m 2 ( 16
) the solution is given by the soft thresholding operator defined as [START_REF] Boyd | Distributed optimization and statistical learning via the alternating direction method of multipliers[END_REF]: [START_REF] Liang | Distributed dictionary learning for sparse representation in sensor networks[END_REF] becomes unconstrained and we have:

ŵo = S µ k λ k [ψ k,n+1 ] m = (17)      [ψ k,n+1 ] m +µ k λ k , if [ψ k,n+1 ] m < -µ k λ k [ψ k,n+1 ] m -µ k λ k , if [ψ k,n+1 ] m > µ k λ k 0 otherwise. If [w o k ] m = 0, problem
ŵ = ŵo (18)
Otherwise, since problem ( 15) is convex, considering constraint

|[w k ] m | ≥ [w o k ] m > 0 with (17) leads to: ŵ = (19)
             [ψ k,n+1 ] m +µ k λ k , if [ψ k,n+1 ] m ≤ -[w o k ] m -µ k λ k -[w o k ] m , if -[w o k ] m -µ k λ k < [ψ k,n+1 ] m <0 -[w o k ] m or [w o k ] m , if [ψ k,n+1 ] m = 0 [w o k ] m , if 0 < [ψ k,n+1 ] m < [w o k ] m +µ k λ k [ψ k,n+1 ] m -µ k λ k , if [ψ k,n+1 ] m ≥ [w o k ] m +µ k λ k
To evaluate the proximal operator [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF], several issues have to be addressed.

1. One of the main issues is that we first need to know which of ( 14), [START_REF] Li | Distributed jointly sparse multitask learning over networks[END_REF] or [START_REF] Wee | A proximal splitting approach to regularized distributed adaptive estimation in diffusion networks[END_REF] has to be applied as the proximal operator of [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF]. We now consider the following two cases:

[w o k ] m = 0 and [w o k ] m > 0. • Case A: [w o k ] m = 0. Since condition |[w k ] m | < [w o k ] m of Case 1 cannot
hold, we only consider Case 2. The proximal operator is given by ( 17) directly.

Let us now consider the second case:

• Case B: [w o k ] m > 0.
Proximal operators ( 14) and ( 19) hold simultaneously. We shall choose the solution that minimizes the cost [START_REF] Chen | Multitask diffusion adaptation over networks with common latent representations[END_REF]. We arrive at the following expression:

ŵ = (20)                [ψ k,n+1 ] m +µ k λ k , if [ψ k,n+1 ] m ≤ -[w o k ] m -µ k λ k -[w o k ] m , if -[w o k ] m -µ k λ k <[ψ k,n+1 ] m ≤-[w o k ] m [ψ k,n+1 ] m , if [ψ k,n+1 ] m < [w o k ] m [w o k ] m , if [w o k ] m ≤ [ψ k,n+1 ] m < [w o k ] m +µ k λ k [ψ k,n+1 ] m -µ k λ k , if [ψ k,n+1 ] m ≥ [w o k ] m +µ k λ k 2.
Another issue is that ŵ cannot be evaluated with ( 17) and ( 20) since [w o k ] m is unknown. To bypass this problem, we follow a strategy adopted in the literature [START_REF] Chen | Diffusion LMS over multitask networks[END_REF] by using ψ ,n+1 as an approximation of w . An approximation of

[w o k ] m is then given by max ∈N - k { [ψ ,n+1 ]m }.

Examining expressions in Case

A and Case B, we notice that only the proximal operator [START_REF] Li | Distributed jointly sparse multitask learning over networks[END_REF] in Case A has the capability to drive [w k ] m to zero and promote sparsity. Condition [w o k ] m = 0 has to be satisfied to trigger Case A, otherwise Case B is considered. Within the context of online learning with stochastic gradient descent algorithms, due to the existence of gradient noise, the estimates of [w o k ] m for zero-valued entries often fluctuate around zero rather than being exact null, so that condition [w o k ] m = 0 of Case A is seldom satisfied. To promote the sparsity of the estimates, we introduce a small positive threshold value τ instead of zero to make a distinguish between zero-valued and nonzero-valued entries. As a consequence, we arrive at conditions [w o k ] m ≤ τ to trigger Case A and [w o k ] m > τ to select Case B. The value of τ needs to be fine-tuned to ensure the performance.

We summarize the proximal operator of ∞,1-norm in Algorithm 2.

Algorithm 2: Proximal operator of ∞,1-norm 1 Initialization: Choose threshold value τ > 0.

2 Proximal operator: At each instant n ≥ 0, for each node k, utilize ψ k,n+1 to evaluate w k,n+1 in an elementwise manner:

1. Calculate [w o k ] m as the maximal value of [ψ ,n+1 ]m for all ∈ N - k ; 2. If [w o k ] m ≤ τ , then calculate [w k,n+1 ]m as ŵo via (17); 3. If [w o k ] m > τ , then calculate [w k,n+1
]m as ŵ via (20).

CONVERGENCE ANALYSIS

The proximal operator of ∞,1-norm can be expressed as:

prox µ k λ k g ψ k,n+1 = ψ k,n+1 -γ k,n+1 , (21)
where γ k,n+1 is a vector of dimension L×1. The explicit expression of γ k,n+1 is omitted here, which can be derived from ( 17) and ( 20) in an element-wise manner. Define

wk,n+1 w k,n+1 -w k . (22)
By collecting w k , w k,n+1 , wk,n+1 and γ k,n+1 over the entire network into block column vectors, we obtain quantities w , wn+1, wn+1 and γn+1, respectively. To facilitate theoretical analysis, we introduce the following assumptions: A1 (Independent Regressors): The regression vector u k,n , generated from a zero-mean random process, is temporally stationary, white (over n) and spatially independent (over k)

with covariance matrix R u,k = E{u k,n u k,n } > 0.
A2 (Small step-sizes): The step-sizes µ k of the network are small enough, so that terms depending of higher-order powers of the step-sizes can be ignored.

Mean behavior analysis

Subtracting w k from (9), using signal model (1) and block notations, and taking expectation under assumption A1, we arrive at the mean behavior given by:

E{ wn+1} = BE{ wn} -E{γn+1}, (23)
where

B I -U M (24)
M diag Ru,1, Ru,2, • • • , Ru,N (25)
E{γn+1} = col E{γ1,n+1}, • • • , E{γN,n+1} . (26)
We point out that vector E{γ k,n+1 } is absolutely bounded with |E{γ k,n+1 }| µ k λ k 1L at all time instant n. This can be derived from the explicit expression of E{γ k,n+1 }. We provide the following condition on the step-size to ensure the mean stability, without proof due to the limited space. Theorem 1. (Mean stability) Assume data model (1) and assumption A1 hold. Then for any initial conditions, the distributed networks with proximal multitask diffusion LMS algorithm ( 9) is stable in the mean, if the step-sizes µ k of the network satisfies:

0 < µ k < 2 λmax{R u,k } , k = 1, • • • , N, (27)
where λmax{•} denotes the maximal eigenvalue of its matrix argument.

Mean-square behavior analysis

Under assumptions A1, A2, and ignoring terms containing higherorder powers of the step-size, then for any semi-positive definite matrix Σ of compatible dimension, the weighted mean-square behavior of wn+1 is given by: E{ wn+1

2 σ } = E{ wn 2 F σ }+[vec{H}] σ + E{ γn+1 2 σ } -2E{ w n B Σγn+1}, (28)
where we use the notations E{ wn+1 x Σx, and we have defined the following quantities:

σ vec{Σ} (29) F B ⊗ B ( 30
)
H U diag σ 2 z,1 Ru,1, • • • , σ 2 z,N Ru,N U . ( 31
)
It is noted that (30) holds only for sufficiently small step-sizes. We provide the following condition on the step-size to ensure the meansquare stability, without proof due to the limited space. Theorem 2. (Mean-square stability) Assume data model (1) and assumptions A1, A2 hold. Further assume that approximation (30) is reasonable for sufficiently small step-sizes. Then for any initial conditions, the distributed networks with proximal multitask diffusion LMS algorithm [START_REF] Sayed | Adaptation, Learning, and Optimization over Networks[END_REF] is stable in the mean-square sense, if the stepsizes µ k of the network are sufficiently small and satisfy [START_REF] Chen | Diffusion LMS over multitask networks[END_REF].

SIMULATION RESULTS

In this section, we present simulation results to validate the effectiveness of the proposed algorithm. All curves were obtained by averaging over 100 Monte-Carlo runs. We considered a nonstationary jointly sparse system identification scenario with w k varying over time. The evolution of w k was divided into three stationary stages and two transient stages. During stationary stages, sparse vectors w k were set to sparsity degree of 1/30, 5/30 and 3/30, respectively. Each nonzero element of w k was generated independently from standard Gaussian distribution. The transient stages were designed by using linear interpolation over 500 time instants. The regressors u k,n were generated from a zero-mean Gaussian distribution with covariance matrices R u,k = σ 2 u,k I30 for white inputs, and with R u,k = σ 2 u,k R † for colored inputs, where R † is an 30 × 30 Hermite matrix with eigenvalue spread matrix argument. For comparison purpose, non-cooperative diffusion LMS algorithm, sparse diffusion LMS [START_REF] Lorenzo | Sparse distributed learning based on diffusion adaptation[END_REF] with zero-attracting (ZA) regularizer and reweighted zero-attracting (RZA) regularizer, multitask diffusion LMS with adaptive combiner [START_REF] Chen | Diffusion LMS over multitask networks[END_REF] and jointly sparse multitask diffusion LMS [START_REF] Li | Distributed jointly sparse multitask learning over networks[END_REF] with 2,1-regularization were taken into consideration. We adopted a uniform step-size 0.01 for all algorithms. For proximal multitask LMS with ∞,1-regularization, we set λ k to 0.08 and τ to 0.1. For the other algorithms, we adjusted the parameters to reach the best performance. The results are illustrated in Fig. 1 for white inputs. We observe that multitask LMS with adaptive combiner is the worst one due to inappropriate cooperation between nodes. Since jointly sparse systems can be regarded as special cases of general sparse systems, sparse diffusion LMS with ZA regularizer and RZA regularizer work better than non-cooperative LMS. Similarly, two jointly sparse multitask LMS algorithms have better performance than the non-cooperative LMS. The proposed proximal multitask LMS with ∞,1-regularization has the best performance among all competing algorithms, especially when systems are more sparse. Similar conclusions can be achieved from Fig. 2 for colored inputs.

CONCLUSIONS

Many practical problems of interest happen to have the jointly-sparse structure. In this paper, by evaluating the proximal operator of ∞,1- norm, we obtained a proximal multitask diffusion LMS algorithm for networks with jointly-sparse structure. We derived conditions to ensure the stabilities in the mean and mean-square sense. Simulation results illustrated the effectiveness of the proposed algorithm.

Algorithm 1 :

 1 Proximal multitask diffusion LMS1 Initialize w k,0 for all k = 1, 2, • • • , N , and repeat:

2 σ } and E{ wn+1 2 Σ

 22 } interchangeably with x 2 Σ

Fig. 1 .Fig. 2 .

 12 Fig. 1. Simulation results with white inputs.