Bernard Souny

Christophe Morlaas
email: morlaas@recherche.enac.fr

Usage of characteristic modes with specific boundary conditions for wide-band antenna design: application to a cage loaded monopole antenna

Keywords: compact antenna, wide bandwidth, characteristic modes, modal analysis, poles and zeros, rational fraction Goubau antenna, loaded antenna, monopole antenna

In this article, a technique to design a compact wideband antenna with the optimization of the input impedance is proposed. This technique is based on the use of two sets of characteristic modes. The first one is associated to an open circuit at the antenna input port while the second is associated to a short circuit at the antenna input port. The link of this analyze with a classical modal approach by a rational fraction approximation of the input impedance is demonstrated. This technique is firstly applied to a basic folded monopole antenna, then to a disc-loaded folded monopole antenna in the aim to optimize its bandwidth and radiation pattern while reducing its size. The disc-loaded folded monopole antenna is modeled and measured. Its VSWR performances are equivalent to thus of the Goubau antenna for equivalent dimensions. The radiation pattern is monopole-like on the full bandwidth.

I. INTRODUCTION

W ITH the development of the wireless telecommunica- tions, the use of wide bandwidth systems become an obligation. In many applications it is important to minimize the antenna size, which is however constraint by the physical laws [START_REF] Wheeler | Fundamental limitations of small antennas[END_REF].

The operating frequency band of the antenna is essentially linked to four notions: the maximal voltage standing wave ratio (VSWR) in the frequency band that is often fixed at 2, the radiation pattern, the polarization purity, and the efficiency.

In many applications, monopole-like pattern is required and compact monopole-like structures with wide-band properties are still extensively studied [START_REF] Akhoondzadeh-Asl | Novel low profile wideband monopole antenna for avionics applications[END_REF], [START_REF] Li | A compact, capacitively fed uwb antenna with monopole-like radiation characteristics[END_REF]. A kind of antenna that can be used is the disk loaded folded monopole and a well known type of such a wide bandwidth miniaturized monopole is the antenna proposed by G. Goubau [4] that presents one octave bandwidth in term of VWSR with a λ/10 height and a λ/10 radius at the lower frequency.

For small antennas the maximum usable bandwidth is frequently defined by the Chu limit [START_REF] Chu | Physical limitations of omni-directional antennas[END_REF], which links the quality factor to the antenna dimension. However, this quality factor is only linked to the bandwidth if the input reflection coefficient of the antenna only depends on one pole close to the useful frequency band [START_REF] Yaghjian | Impedance, bandwidth, and q of antennas[END_REF].

The poles of the reflection coefficient correspond to the eigenmodes of the structure loaded with a matched waveguide at the input. Generally they are complex. Some pure imaginary poles can be used to improve the matching [START_REF] Bode | Network analysis and feedback amplifier design[END_REF], [START_REF] Fano | Theoretical limitations on the broadband matching of arbitrary impedances[END_REF].

Additional poles that contribute to the radiation can also be used [START_REF] Stuart | Eigenmode analysis of small multielement spherical antennas[END_REF], [START_REF]Eigenmode analysis of a two element segmented capped monopole antenna[END_REF] to improve the bandwidth. However the radiation pattern and the polarization quality can be strongly affected by their radiation.

Characteristic modes have been initially introduced by Garbacz and Harrington [START_REF] Garbacz | A generalized expansion for radiated and scattered fields[END_REF], [START_REF] Harrington | Theory of characteristic modes for conducting bodies[END_REF], [START_REF]Computation of characteristic modes for conducting bodies[END_REF], [START_REF] Chang | A surface formulation for characteristic modes of material bodies[END_REF]. More recently, the mode characteristics have been used to facilitate the antenna design [START_REF] Inagaki | Eigenfunctions of composite hermitian operators with application to discrete and continuous radiating systems[END_REF], [START_REF] Cobos | A modified goubau-type antenna with two octaves of impedance bandwidth[END_REF], [START_REF] Cabedo-Fabres | Systematic design of antennas using the theory of characteristic modes[END_REF], [START_REF] Adams | Broadband equivalent circuit models for antenna impedances and fields using characteristic modes[END_REF]. In this article, we propose to link the characteristic mode to an poles analysis of an antenna input impedance or admittance also known as the modal analysis [START_REF] Schelkunoff | Antennas: theory and practice[END_REF]. An approximation by fractional ration is used in which, an open-circuit (OC) and a short-circuit (SC) at the input of the antenna are considered. Similarly, This analysis can be performed with the zeros of the impedance and the admittance. The link between these poles and the characteristic modes of the antenna in the two input configurations as mentioned above, i.e. OC and SC, is shown. We show how this analysis can be useful to optimize the input impedance bandwidth and the radiation of a wide-band antenna. As an example, the proposed method is applied to a simple folded monopole antenna and to a more complex case of a folded monopole loalded by a metallic disc already published in [START_REF] Morlaas | Cage-shaped wideband antenna based on folded monopoles[END_REF] which constitute an upgrade of the Goubau antenna [START_REF] Goubau | Multi-element monopole antennas[END_REF].

The section II presents the relations between classical modes analysis and characteristic modes of Garbacz/Harrington. Then, the usefulness of consider two input boundary conditions, i.e. an open circuit and a short circuit, at the input of a wide-band antenna is explained and illustrated for a folded monopole antenna. From this analysis the important points that affect the antenna design are reported and a new design is proposed.

The section III describes the classical mode and characteristic mode analysis of the improved antenna structure. The performances of this antenna as well as the ones of an optimized antenna structure are presented.

Finally, in section IV, the measurement performances of the optimized antenna version is presented.

The simulations has been done with FEKO and the rational fraction approximation is computed with OCTAVE.

II. RELATIONS BETWEEN CLASSICAL MODES ANALYSIS AND CHARACTERISTIC MODES OF GARBACZ/HARRINGTON

In this part, the link between a classical modal analysis and characteristic modes is shown. The usefulness of characteristic modes analysis with short-circuited or open-circuited boundary conditions according to the kind of antenna is depicted.

A. Modal analysis

The modal analysis based on the zeros and poles representation of the input impedance by a rational fraction is a classical technique for antennas study [START_REF] Schelkunoff | Antennas: theory and practice[END_REF]. The eigenmodes are generally associated to a complex pulsation due to the radiation phenomena. Poles of the impedance are linked to eigenmodes of the antenna structure with an open-circuited input and zeros of the impedance are linked to eigenmodes of the antenna with a short-circuited input and vice-versa for the admittance. Thus, the frequency behavior of the antenna can only be described in term of modes.

Theoretically each set of modes with SC and OC input conditions allows to describe the antenna performances through its input port, but as it will be seen, it is more useful to consider the two sets of modes together.

This analysis is suitable for the study of small antenna because only few poles and zeros takes action into the all bandwidth. indeed, complex poles and zeros of the input impedance are defined by pair of conjugate numbers but only the ones around the useful bandwidth are retained.

The process to obtain the poles and zeros is described as follows.

1) The input reflection coefficient of an antenna is measured or computed.

2) The input impedance is computed from the reflection coefficient (Γ) with z in = (1 + Γ)/(1 -Γ) and poles of impedance are computed by a classical method [START_REF] Gustavsen | Rational approximation of frequency domain responses by vector fitting[END_REF] [22] [23] [START_REF] Zeng | Modified rational function modeling technique for high speed circuits[END_REF].

3) The input admittance is computed from the reflection coefficient (Γ) with y in = (1 -Γ)/(1 + Γ) and poles of admittance that are zeros of impedance are computed by the same classical method. Let us take the case of a dipole antenna, The reflection coefficient obtained with a simulation between 400 MHz and 2 GHz and the rational fraction results are plotted in the complex plane in Fig. 1.

The rational fraction gives a good approximation of the simulated points with the numerator and denominator order equal to 3. The frequency and quality factor of useful poles and zeros close to the analysis band are given in (1)

f z 1 = 0.894 GHz, Qz 1 = 5.3 (1) f p 1 = 1.74 GHz, Qp 1 = 3
with Q defined in appendix VI

B. Influence of the kind of excitation on characteristic modes

The electrical and magnetic impress currents J s , M s defined in the Maxwell equations [START_REF] Harrington | Time-harmonic electromagnetic fields[END_REF] which contribute to the incident field [START_REF] Harrington | Theory of characteristic modes for conducting bodies[END_REF] are introduced in the characteristic modes theory without difficulties. It is well established that the characteristic modes associated to a fixed structure defined by materials and boundary conditions do not depend on the sources J s and M s . Then, the modification of their amplitude and position does not change characteristic modes. Besides, the excitation of an antenna implies the definition of a port associated to a boundary condition before to apply the source. Well, characteristic modes depend on boundary conditions and altough a superposition of characteristic modes always allow to obtain the excited fields whatever the source because they defined a complete base, the choice of the appropriate boundary conditions allow to use the minimum number of characteristic modes to describe the excited field.

In theoretical work current or voltage sources are very often used [START_REF] Harrington | Time-harmonic electromagnetic fields[END_REF]. An example of sources compliant with a dipole antenna is presented in Fig. 2. These sources are connected to the antenna by the areas Σ1 and Σ2.

Boundary conditions associated with these sources are a short circuit (Perfect electrical conductor) for a voltage source (Fig. 2-a) and an open circuit (Perfect magnetic conductor) for a current source (Fig. 2-b). In practice this latter is often approached by no conductor, obtained by neglecting the displacement currents (Fig. 2-c). But in practice antennas are generally fed through a port on which a matched source with a finite internal resistance is put, for example 50Ω. This configuration is depicted in Fig. 3-a. In this case, the use of the equivalent theorem allows to replace the real source by equivalent (impress) sources J eq and M eq with the internal media replaced by an arbitrary media as shown in Fig. 3-b. If the media is replaced by a perfect electrical conductor (PEC), that corresponds to a voltage source. If the media is replaced by a perfect magnetic conductor (PMC), that corresponds to a current source.

Fig. 4 shows the λ/2 dipole with a feeding port at its center using sources defined in Fig. 2. Fig. 4-a We observe that the excitation by mean of a voltage source corresponds to the half-wavelength dipole fed with a loop of a magnetic current located at its center but the excitation by mean of a current source corresponds to a structure completely different due to the difference of the boundary conditions: two dipole of half length.

For instance, consider a dipole of l = 150 mm length and a d = 2 mm diameter. Frequencies obtained with a characteristic mode analysis with an OC input (poles of impedance) and a SC input (zeros of impedance) are given in [START_REF] Akhoondzadeh-Asl | Novel low profile wideband monopole antenna for avionics applications[END_REF].

f cz 1 = 0.935 GHz (2) f cp 1 = 1.830 GHz f cz 2 = 1.927 GHz
We observed that frequencies extracted from the rational fraction approximation given in (1) are close to the ones of the characteristic modes analysis given in [START_REF] Akhoondzadeh-Asl | Novel low profile wideband monopole antenna for avionics applications[END_REF]. Nevertheless, a slight difference exists because the associated quality factors have low values.

The characteristic modes for the two source configurations (SC/OC) are very different as it can be seen in Fig. 5. The currents of the first characteristic mode with a SC input as boundary condition correspond to the ones of the fundamental mode of a l = 150mm length dipole (Fig. 5-a). The currents of the first characteristic mode with an OC input as boundary condition correspond to the ones of the fundamental mode of two dipoles of l = 75mm length with a odd symmetry (Fig. 5b). While the currents of the second characteristic mode with an SC input as boundary condition correspond to the ones of the fundamental mode of two dipoles of l = 75mm length with an even symmetry (Fig. 5-c). In the appendix VI, the link between eigenmodes and characteristic modes with OC and SC input for isolated modes is demonstrated. The relevance of the isolation of a mode for an isomorphic operation associated to a pole p n = α n + jω n depends on the proximity of the others modes and of the associated quality factor.

In the following we assume that modes can be considered as isolated. Therefore, when the operating frequency is close to the one of an isolated characteristic mode with a SC input the actually used source can be replaced by an equivalent voltage source without any fields modification. The characteristic modes are then the ones of the structure with a SC input as boundary condition and the isolated state of the mode close to the operating frequency induces that the excited field is essentially the one of the considered characteristic mode.

Dualistically, when the operating frequency is close to the one of an isolated characteristic mode with an OC input the actually used source can be replaced by an equivalent current source without any fields modification. The characteristic modes are then the ones of the structure with an OC input as boundary condition and the isolated state of the mode close to the operating frequency induces that the excited field is essentially the one of the considered characteristic mode.

In practice, to observe the currents on the antenna structure, an impedance matching source is used.

The currents of the dipole excited by a resistive source are given in the Fig. 6-a, b, c at the frequency f cz 1 , f cp 1 and f cz 2 respectively. At f cz 1 , the currents are very similar to the ones of the characteristic mode with SC as boundary condition at this frequency (Fig. 5-a). At f cp 1 and f cz 2 , the currents are very similar to the ones of the characteristic mode with OC as boundary condition at f cp 1 (Fig. 5-b). The mode corresponding to the characteristic mode at f cz 2 is not excited because it presents an even symmetry and cannot be excited by a source at the center of the dipole.

Because characteristic modes constitute a complete base, they always allow to represent the excited field and close to f cp1 we can represent fields as a superposition of characteristic modes with SC boundary condition but it will be far simpler to represent it with only one characteristic mode with OC boundary. So it will be very easy to predict the dipole operation close to f cp1 using only this characteristic mode. For narrow operating bandwidth, two kinds of antennas (resonant antennas) can be distinguished. The ones defined by only one pole of the impedance (a half-wavelength slot antenna for instance) for which the corresponding characteristic modes with OC boundary condition help the antenna analysis along. The ones defined by only one pole of the admittance (a half-wavelength dipole antenna for instance) for which the characteristic modes with SC boundary condition help the antenna analysis along.

Concerning the wide-band antennas, for which poles and zeros of the impedance act together, the characteristic modes analysis with SC and OC boundary conditions help the analysis along as shown in the following study of the folded monople and the suggested antenna.

C. Illustration on a basic folded-monopole

The analysis of a basic folded-monopole (Fig. 7) is given first because the modes of the proposed antenna are an evolution of the basic folded-monopole modes.

The reflection coefficient obtained with a simulation between 400 MHz and 1.7 GHz and the rational fraction results are plotted in the complex plane in Fig. 8.

The rational fraction gives a good approximation of the simulated points with the numerator and denominator order equal to 3. The simulated point number used to build the approximation is not critical but they must be well spread over the curve.

The frequency and quality factor of useful poles and zeros close to the analysis band are given in (3)

f p 1 = 0.58 GHz, Qp 1 = 18.3 (3) f z 1 = 0.847 GHz, Qz 1 = 4.0 f p 2 = 1.2 GHz, Qp 2 = 4.3
Besides, frequencies obtained with a characteristic mode analysis with an open-circuited input (poles) and a shortcircuited input (zeros) are given in [START_REF] Goubau | Multi-element monopole antennas[END_REF].

f cp 1 = 0.562 GHz (4) f cz 1 = 0.883 GHz f cp 2 = 1.24 GHz
Wire currents obtained with a characteristic mode analysis are plotted in Fig. 9,

The characteristic mode at f cp1 = 0.562 GHz (Fig. 9-a) corresponds to the λ/4 resonant mode of the loop with an OC input. The characteristic mode at f cp2 = 1.24 GHz (Fig. 9-c) corresponds to the 3λ/4 resonant mode of the loop with an OC input and the characteristic mode at f cz1 = 0.883 GHz (Fig. 9-b) corresponds to the λ/2 resonant mode of the loop with a SC input which is the λ/4 mode of the monopole antenna.

D. Evolution of the antenna

From some simulations it is easy to observe that f cz1 mode have few dependence with the distance between the two vertical wires contrary to the modes at f cp1 and f cp2. After having identified the mode at f cz1 = 0.883 GHz, this frequency can be reduced by increasing the size of the upper conductor by applying a top disc for example. This top disc allows also to reduce f cp1 and f cp2 as maintaining the bandwidth. Because the currents associated to the f cp1 mode are strong onto the wire without the port (outer wire), this frequency can be decreased relatively to the one of the f cp2 mode by lengthen this wire with a meander shape for example.

Besides, the input impedance can be reduced by increasing the diameter of the wire containing the port.

Then, from this first analysis on the folded monopole, it is easy to optimize the distance between these wires to maximize the bandwidth and to reduce the antenna high. The first evolution of the folded monopole antenna is shown in Fig. 10 where the central conductor is a stepped design to allow to increase the diameter of the wire containing the port, to reduce the input impedance, and to simply connect it to a standard coaxial connector. To close the loop between the top disc to the ground plane four outer conductors are used to avoid an undulation of the radiation pattern at high frequency as explained in [START_REF] Morlaas | Cage-shaped wideband antenna based on folded monopoles[END_REF].

The input reflection coefficient is given in Fig. 11. This is typically a two poles and one zero inserted system.

Frequencies obtained with a modal analysis (left side) and a characteristic mode analysis (right side) with an open-circuited input (poles) and a short-circuited input (zeros) are given in [START_REF] Chu | Physical limitations of omni-directional antennas[END_REF].

f p 1 = 0.50 GHz f cp 1 = 0.50 GHz (5) f z 1 = .60 GHz f cz 1 = .68 GHz f p 2 = .775 GHz f cp 2 = 1.20 GHz f z 2 = 1.06 GHz f cz 2 = 1.375 GHz
f cp 1 and f cz 1 are too close to f cp 2 to obtain a wideband operation. Then, to decrease f cp 1 and f cz 1 a solution consists of lengthen the outer wires. To lengthen the wires different solutions can be used; a portion of helix, a meander line section, a solenoid section.

The simplest is the first one and it is the solution used in Fig. 12.

III. MONOPOLE ANTENNA LOADED WITH A QUADRIFILAR CAGE

A. Antenna description

This antenna is an evolution of the basic folded monopole, it is a cage loaded monopole antenna and is called the cage antenna.

A picture of this antenna is given in Fig. 12. The antenna is intended to be used on a metallic roof. Thus, for simulation purpose the antenna is over an infinite perfect electrical (IPEC) conductor. It consists of two circular discs linked each-other with four inner vertical wires (IW) of same diameters (2 mm). The bottom disc (BD) of D2 = 7 cm diameter is linked to the IPEC by mean of the coaxial feed wire (FW) and the top metallic disc (TD) of D1 = 12 cm diameter is linked to the IPEC via four outer helical shape wires (OW). The total height is h1 = 5.5 cm.

During the design process, an idealized model is used to accelerate simulation: perfect conductor discs are infinitely thin.

B. Modal analysis

The antenna input reflection coefficient (Gamma) is computed via a simulation between 300 MHz and 1.8 GHz. It is plotted in Fig. 13.

The obtained rational fraction gives a good approximation of poles and zeros close to this frequency band with the numerator and denominator order equal to 4. The frequencies and quality factor of useful poles and zeros close to the analysis band are given in ( 6) with corresponding frequencies of associated characteristic modes. The analysis is extended to frequencies higher than the useful band to explain certain pattern deformation at high frequency.

f p 1 = 0.377 GHz, Qp 1 = 15.5 f cp 1 = 0.386 GHz (6) f z 1 = 0.539 GHz, Qz 1 = 3.2 f cz 1 = 0.581 GHz f p 2 = 0.806 GHz, Qp 2 = 2.1 f cp 2 = 1.01 GHz f z 2 = 1.126 GHz, Qz 2 = 5.5 f cz 2 = 1.272 GHz f p 3 = 1.55 GHz, Qp 3 = 35 f cp 3 = 1.554 GHz 1)
Mode p 1 : 377 MHz: The antenna is excited at 377 MHz, the wire currents are plotted in part (a) of Fig. 14 where only one snapshot is given at phases 90°because all pieces of current are in phase and maximal at this phase. Currents on the disc are essentially the continuation of the wire current and for readability, they are not plotted. We can see that the currents along the helical cage (OW) are strong while they are weak along the inner vertical wires (IW). In fact this is an evolution of the p 1 mode identified in the basic folded monopole (unfed wire). This mode is strongly dependent to the helical cage parameters: the length of the wires (this parameter depends on the turn number N turn ) and the upper disc (TD) diameter. The corresponding radiating mode is principally a T M 01 .

The corresponding characteristic mode with an opencircuited input is find at 387 MHz. These currents are given in Fig. 14 (b). We can see that they are very similar to those obtained with the rational fraction process given in Fig. 14 (a).

2) Mode z 1 : 539 MHz: The antenna is excited at 539 MHz, the wire currents are plotted in part (a) and (b) of Fig. 15 where two snapshots are given at phases 0°and 90°because all pieces of current are not in phase. The corresponding radiating mode is principally a T M 01 .

The corresponding characteristic mode with a shortcircuited input is find at 581 MHz. These currents are given in Fig. 15 (c). We can see that they are similar but not identical to wire currents in antenna excited at 539 MHz. However we can see that this mode corresponds to an evolution of the z 1 mode identified on the basic folded monopole.

3) Mode p 2 : 806 MHz: The antenna is excited at 806 MHz, the wire currents are plotted in part (a) and (b) of Fig. 16. The corresponding radiating mode is principally a T M 01 .

The corresponding characteristic mode with an opencircuited input is found at 1014 MHz. These currents are given in Fig. 16 (c). We can see that they are similar to the wire current in the antenna excited at 806 MHz.

The central conductor current is much stronger than the cage current and thus, this mode corresponds to an evolution of the p 2 mode identified on the basic folded monopole.

4) Mode z2: 1.128 GHz: The antenna is excited at 1.128 GHz, the wire currents are plotted in part (a) and (b) of Fig. 17.

The corresponding characteristic mode with a short- circuited input is find at 1260 MHz. These currents are given in Fig. 17 circuited input is find at 1554 MHz. These currents are given in Fig. 18 (c). The radiation is no longer a T M 01 mode and the radiation pattern is given in Fig. 19. If during the design process this mode is too close to the useful frequency band, this modal current will be strong and it will produce some pattern distortions.

C. Antenna performances

In Fig. 20 the gain curves versus the azimuth angles are plotted in the H plane for three frequencies corresponding respectively to the bottom, the center, and the top of the frequency band. We can observe an undulating gain increasing with the frequency but it remains less than 0.1 dB at the upper part of the bandwidth.

These undulations come from the wire currents of the helical cage as explained in [START_REF] Morlaas | Cage-shaped wideband antenna based on folded monopoles[END_REF]. - In Fig. 21 the gain curves versus the polar angles are plotted in the E plane for three frequencies corresponding respectively to the bottom, the center, and the top of the frequency band. we can verify that the gain is constant with the frequency.

The main polarization is vertical with a maximal crosspolarization at θ = 40 • equal to -25dB whatever the frequency is.

In Fig. 22 the S 11 magnitude is plotted. The shape of the obtained curve is typically the one of a two order system (two pairs of conjugated poles).

D. Meander-line and solenoid wires antennas

The outer wires winding around the top disc presents some drawback on the polarization quality. This solution introduces a coupling between the co and the cross-polarization modes (T M 01 and T E 01 modes). For the four wire antenna (in Sec.

III-C) the cross-polarization ratio is lower than -25dB

To avoid this problem, each wire is replaced by a small diameter helix (Fig. 23) or a meander-line wire (Fig. 24). In these cases, the simulation results are given in Fig. 25 where the gain is plotted in the E plane. The obtained cross-polarization ratio is lower than -33dB and -47dB for the helix and the meander-line wire respectively. The co-polarization gain is not affected by the kind of wires.

IV. MEASUREMENTS

To test the antenna, a prototype has been constructed with wires and plates. The meander-line wires are chosen. Plates are 0.8 mm thick and the meander-line are manufactured by bending wires. Thus, a new more realistic model for simulation is now used and is shown in Fig. 26. In this model h1 is the total height (including thickness of plates), the other dimensions are the same as in Fig. 12.

A picture of the prototype antenna with a small ground plane 20 cm × 20 cm is given in Fig. 27. It is important to note that during measurements, a larger ground plane screwed to the small one is used (1m × 1m).

The measured reflection coefficient is compared to the one modeled with the infinite ground plane in Fig. 28. We obtain very closed results between measurement and modeled antenna. Noise in measured data are due to parasitic reflections in the measurement environment: we don't have access to an anechoic chamber operating at these relatively low frequencies and measurements are not done in very clean environment. For this reason no pattern measurement have been performed.

The obtained bandwidth at V SW R = 2 is about 1.8 : 1 (0.48 MHz/0.9 MHz) and 1.9 : 1 (0.47 MHz/0.9 MHz) for modeling and measurement respectively. The efficiency has been estimated using simulation with a finite 1m × 1m ground plane and all pieces made with brass. The efficiency is assessed greater than 99% over all bandwidth.

The simulated H plane gain pattern of the antenna on an infinite ground plane is given in Fig. 29 at the lower and upper frequency in the bandwidth. As we can see the radiation pattern is quite omnidirectional in the H plane.

V. CONCLUSION

The presented design method is based on a modal analysis using a rational fraction approximation of the input reflection coefficient. To complete this analysis it is interesting to compute characteristic modes with an open-circuited input associated with poles and characteristic modes with a short-circuited input associated with zeros. This technique allows an easy and complete understanding of the antenna operating mode, thus it is easy to identify pertinent geometric dimensions in order to obtain specific performances. This method can be applied to every antenna structure having a finite number of (useful) poles and zeros, which is generally the case for small antennas. An antenna structure (meanderline cage antenna) having VSWR performances very similar to the ones of the Goubau antenna with similar dimensions has been designed. Unlike Goubau antenna whose radiation pattern lack uniformity in the high part of the band the radiation pattern of the new antenna is very stable in all the bandwidth. Manufacturing of this antenna is easy and can be done economically with plates and wires thanks to the low sensitivity to the dimension accuracy.

VI. APPENDIX A

A. Input impedance calculation from a current source

The source of the Fig. 2-c is connected to a metallic structure through the two areas Σ1 and Σ2. The Harrington formulation [START_REF] Harrington | Theory of characteristic modes for conducting bodies[END_REF], (Eq. 33) is used in this part.

On the Σ3 area, the impressed current density is fixed by assumption and noticed Js u with u, the unitary vector associated to the current. Then, the current source is

Is = C1 Js dl (7)
Besides, the modal coefficient V n is given by

V n = Σ3 E n • Js u dΣ (8)
with E n , the electrical field of the characteristic mode n into the source volume.

For a lumped source , E n can be considered constant over Σ3. Then,

V n = h E n • u Is. (9)
Consider also the electrical field E defined everywhere in space and particularly over the source, it can be written as

E = Σ n
V n E n 1 + jλ n [START_REF]Eigenmode analysis of a two element segmented capped monopole antenna[END_REF] with λ n , the eigenvalue associated to the characteristic mode n as defined in [START_REF] Harrington | Theory of characteristic modes for conducting bodies[END_REF] (Eq. 31).

If the characteristic mode n is isolated, i.e. for ω ≃ ω n with λ(ω n ) = 0, if the electromagnetic fields are the ones of the characteristic mode n alone, then this mode is dominant and

E ≃ V n E n 1 + jλ n . (11)
In that conditions, the real function λ n of the real variable ω can be expended on

λ n (ω) ≃ λ ′ n (ω n )(ω -ω n ). (12)
Also, E can be considered as a function of jω such as

E(jω) ≃ V n E n 1 + jλ ′ n (ω n )(ω -ω n ) . ( 13
)
Now, inserting the function of the complex variable p defined as

p = jω ω ∈ [-∞ + ∞] (14)
p n = jω n -1 λ ′ n and substituting (9) into [START_REF]Computation of characteristic modes for conducting bodies[END_REF], this equation becomes

E(p) = h E n Is λ ′ n 1 p -p n E n . ( 15
)
Over the p domain defined by ( 14), E(p) = E(jω). Then, E(p) is an analytic extension of E(jω).

Besides, the voltage at the current source terminals is given by U = -hE(p) • u [START_REF] Cobos | A modified goubau-type antenna with two octaves of impedance bandwidth[END_REF] and the input impedance is defined as

Z in = U Is . (17)
Then, when substituting (15) into ( 16) and ( 16) into [START_REF] Cabedo-Fabres | Systematic design of antennas using the theory of characteristic modes[END_REF] it follows

Z in ≃ - (hE n • u) 2 λ ′ n 1 p -p n . ( 18
)
The relation [START_REF] Adams | Broadband equivalent circuit models for antenna impedances and fields using characteristic modes[END_REF] shows that if a characteristic mode with an OC input is isolated, namely that, close to the associated frequency the field in the excited structure is quite the field of this mode, it is associated to a complex pole of the input impedance Z in (p) which is also a zero of the input admittance Y in (p) and the zeros of the admittance correspond to the eigenmodes with a OC input [START_REF] Bode | Network analysis and feedback amplifier design[END_REF]. Dualistically, if a characteristic mode with a SC input is isolated, then it is associated to a complex pole of the input admittance Y in (p) which is also a zero of the input impedance Z in (p) and the zeros of the impedance correspond to the eigenmodes with a SC input [START_REF] Bode | Network analysis and feedback amplifier design[END_REF].

Otherwise, from the classical study of RLC circuits in the complex domain [START_REF] Bode | Network analysis and feedback amplifier design[END_REF], the eigenmode associated to a pole p n of the impedance is all the more dominant so the quality factor given in 19 presents strong values for a harmonic state.

P n = α n + jω n Q = ω n 2|α n | = ω n |λ ′ n | 2 (19)
The Q value of pole given by ( 19) is the same as the Q value of characteristic mode given by Harrington in [START_REF] Harrington | Control of radar scattering by reactive loading[END_REF]. Then, the quality factor Q is a good characteristic to identify a dominant (isolated) mode in the operation of an antenna.

Fig. 1 .

 1 Fig. 1. Real versus imaginary part of the input reflection coefficient of the dipole antenna between 400 MHz and 2 GHz. The square points correspond to the simulation values. The full line corresponds to the interpolation with the rational fraction.

Fig. 2 .

 2 Fig. 2. Classical lumped sources

Fig. 3 .

 3 Fig. 3. Equivalent real source.

Fig. 4 .

 4 Fig. 4. Half-wave dipole with voltage and current excitation.

Fig. 5 .

 5 Fig. 5. First characteristic modes with a short-circuited input and an opencircuited input for the half-wave dipole.

  Fig. 6. Currents of the half-Wave dipole exited with a resistive source.

Fig. 7 .

 7 Fig. 7. Basic folded-monopole. H=74mm, W=10mm, D=2mm.

Fig. 8 .

 8 Fig. 8. Real versus imaginary part of the input reflection coefficient of the basic folded monopole antenna between 400 MHz and 1.7 GHz. The square points correspond to the simulation values. The full line corresponds to the interpolation with the rational fraction.

Fig. 9 .

 9 Fig. 9. Characteristic mode currents for a basic folded monopole antenna.

Fig. 10 .

 10 Fig. 10. Evolution N°1 of the folded monopole.

Fig. 11 .

 11 Fig. 11. Real versus imaginary part of the input reflection coefficient (Gamma) of the evolution N°1 between 300 MHz and 1.5 GHz. The square points correspond to the simulation values. The full line corresponds to the interpolation with the rational fraction.

Fig. 12 .

 12 Fig. 12. Monopole antenna with quadrifilar cage (section III): h1 = 5.5 cm, h2 = 1.5 cm, D1 = 12 cm, D2 = 7 cm, Wire diameter 2 mm, N turn = .25.

Fig. 13 .

 13 Fig.[START_REF]Computation of characteristic modes for conducting bodies[END_REF]. Real versus imaginary part of the input reflection coefficient (Gamma) of the cage antenna between 300 MHz and 1.8 GHz. The square points correspond to the simulation values. The full line corresponds to the interpolation with the rational fraction.

(a) p 1

 1 Fig. 14. p 1 mode and characteristic mode currents with OC input for the cage antenna.

(a) p 2 Fig. 17 .

 217 Fig.[START_REF] Cobos | A modified goubau-type antenna with two octaves of impedance bandwidth[END_REF]. p 2 mode and characteristic mode currents with OC input for the cage antenna.

  Fig. 18. p 3 mode and characteristic mode currents with OC input for the cage antenna.

Fig. 19 .

 19 Fig. 19. Radiation pattern of the characteristic mode with an open-circuited input at 1554 MHz (f cp3).

Fig. 20 .

 20 Fig. 20. Cage antenna gain (dBi) in the H plane ( for low, central and high operating frequency.

Fig. 21 .

 21 Fig. 21. Cage antenna gain (dBi) in the E plane for low, central and high operating frequency.

Fig. 22 .

 22 Fig. 22. Magnitude of the reflection coefficient (S 11 ) of the cage antenna for a VSWR<2 bandwidth.

Fig. 23 .

 23 Fig. 23. Four helix wires Antenna. Dimension (see Fig.12) h1 = 6 cm, h2 = 1.5 cm, D1 = 12 cm, D2 = 7 cm, Wire diameter 2 mm, helix diameter: 10 mm, N turn = 2.5.

Fig. 24 .

 24 Fig.[START_REF] Zeng | Modified rational function modeling technique for high speed circuits[END_REF]. Four meander lines Antenna. Dimension (see Fig.12) h1 = 6 cm, h2 = 1.5 cm, D1 = 12 cm, D2 = 7 cm, input wire diameter 1.1 mm, All other wires diameter 1.8 mm, meander-line width: 60 cm.

Fig. 25 .Fig. 26 .

 2526 Fig.[START_REF] Harrington | Time-harmonic electromagnetic fields[END_REF]. Co (vertical) and cross (horizontal) polarization gain (dBi) in the E plane for: Four wires antenna, four helix wires antenna, four meander lines antenna.

Fig. 28 .

 28 Fig. 28. Measured and Modeled reflection coefficient.

Fig. 29 .

 29 Fig. 29. Simulated meander line antenna gain in the H plane at lower and upper frequency in bandwidth.


Anil Goyal

Emilie Morvant

Massih-Reza Amini

Multiview Learning of Weighted Majority Vote by Bregman Divergence Minimization

We tackle the issue of classifier combinations when observations have multiple views. Our method jointly learns view-specific weighted majority vote classifiers (i.e. for each view) over a set of base voters, and a second weighted majority vote classifier over the set of these view-specific weighted majority vote classifiers. We show that the empirical risk minimization of the final majority vote given a multiview training set can be cast as the minimization of Bregman divergences. This allows us to derive a parallel-update optimization algorithm for learning our multiview model. We empirically study our algorithm with a particular focus on the impact of the training set size on the multiview learning results. The experiments show that our approach is able to overcome the lack of labeled information.

Introduction

In many real-life applications, observations are produced by more than one source and are so-called multiview [START_REF] Sun | A survey of multi-view machine learning[END_REF]. For example, in multilingual regions of the world, including many regions of Europe or in Canada, documents are available in more than one language. The aim of multiview learning is to use this multimodal information by combining the predictions of each classifier (or the models themselves) operating over each view (called view-specific classifier) in order to improve the overall performance beyond that of predictors trained on each view separately, or by combining directly the views [START_REF] Snoek | Early versus late fusion in semantic video analysis[END_REF].

Related works. The main idea here follows the conclusion of the seminal work of [START_REF] Blum | Combining Labeled and Unlabeled Data with Co-Training[END_REF] which states that correlated yet not completely redundant views contain valuable information for learning. Based on this idea, many studies on multiview learning have been conducted and they can be grouped in three main categories. These approaches exploit the redundancy in different representations of data, either by projecting the view-specific representations in a common canonical space [START_REF] Gönen | Multiple kernel learning algorithms[END_REF], [START_REF] Zhang | A novel ensemble construction method for multi-view data using random cross-view correlation between within-class examples[END_REF], [START_REF] Xu | Large-margin multi-viewinformation bottleneck[END_REF], or by constraining the classifiers to have similar outputs on the same observations; for example by adding a disagreement term in their objective functions [START_REF] Sindhwani | An RKHS for multi-view learning and manifold co-regularization[END_REF], or lastly by exploiting diversity in the views in order to learn the final classifier defined as the majority vote over the set of view-specific classifiers [START_REF] Peng | Shareboost: Boosting for multi-view learning with performance guarantees[END_REF][START_REF] Peng | Multiview boosting with information propagation for classification[END_REF], [START_REF] Xiao | Multi-view adaboost for multilingual subjectivity analysis[END_REF]. While the two first families of approaches were designed for learning with labeled and unlabeled training data, the last one, were developed in the context of supervised learning. In this line, most of the supervised multiview learning algorithms dealt with the particular case of two view learning [START_REF] Farquhar | Two view learning: Svm-2k, theory and practice[END_REF], [START_REF] Janodet | Boosting Classifiers built from Different Subsets of Features[END_REF], [START_REF] Xu | An algorithm on multi-view adaboost[END_REF], and some recent works studied the general case of multiview learning with more than two views under the majority vote setting. Amini et al. [START_REF] Amini | Learning from Multiple Partially Observed Views -an Application to Multilingual Text Categorization[END_REF] derived a generalization error bound for classifiers learned on multiview examples and identified situations where it is more interesting to use all views to learn a uniformly weighted majority vote classifier instead of single view learning. [START_REF] Koc | A boosting approach to multiview classification with cooperation[END_REF] proposed a Boosting-based strategy that maintains a different distribution of examples with respect to each view. For a given view, the corresponding distribution is updated based on view-specific weak classifiers from that view and all the other views with the idea of using all the view-specific distributions to weight hard examples for the next iteration. Peng et al. [START_REF] Peng | Shareboost: Boosting for multi-view learning with performance guarantees[END_REF][START_REF] Peng | Multiview boosting with information propagation for classification[END_REF] enhanced this idea by maintaining a single weight distribution among the multiple views in order to ensure consistency between them. [START_REF] Xiao | Multi-view adaboost for multilingual subjectivity analysis[END_REF] proposed a multiview learning algorithm where they boost the performance of view-specific classifiers by combining multiview learning with Adaboost.

all views v = 1 h 1,1 h 1,2 h 1,3 h 1,4 v = 2 h 2,1 h 2,2 h 2,3 h 2,4 h 2,5 v = 3 h 3,1 h 3,2 h 3,3
Figure 1: Illustration of MωMvC 2 with V =3. For all views v ∈ {1, 2, 3}, we have a set of view-specific weak classifiers (H v ) 1≤v≤V that are learned over a multiview training set. The objective is then to learn the weights Π (black histograms) associated to (H v ) 1≤v≤V ; and the weights ρ (hatched histograms) associated to weighted majority vote classifiers such that the ρΠ-weighted majority vote classifier B ρΠ (Equation 1) will have the smallest possible generalization error.

Contribution. In this work, we propose a multiview Boosting-based algorithm, called MωMvC 2 , for the general case where observations are described by more than two views. Our algorithm combines previously learned view-specific classifiers as in [START_REF] Amini | Learning from Multiple Partially Observed Views -an Application to Multilingual Text Categorization[END_REF] but with the difference that it jointly learns two sets of weights for, first, combining view-specific weak classifiers; and then combining the obtained view-specific weighted majority vote classifiers to get a final weighted majority vote classifier. We show that the minimization of the classification error over a multiview training set can be cast as the minimization of Bregman divergences allowing the development of an efficient parallel update scheme to learn the weights. Using a large publicly available corpus of multilingual documents extracted from the Reuters RCV1 and RCV2 corpora as well as MNIST 1 and MNIST 2 collections, we show that our approach consistently improves over other methods, in the particular when there are only few training examples available for learning. This is a particularly interesting setting when resources are limited, and corresponds, for example, to the common situation of multilingual data.

Organization of the paper. In the next section, we present the double weighted majority vote classifier for multiview learning. Section 3 shows that the learning problem is equivalent to a Bregman-divergence minimization and describes the Boosting-based algorithm we developed to learn the classifier. In Section 4, we present experimental results obtained with our approach. Finally, in Section 5 we discuss the outcomes of this study and give some pointers to further research.

Notations and Setting

For any positive integer N , [N ] denotes the set [N ] . ={1, . . . , N }. We consider binary classification problems with V ≥2 input spaces X v ⊂ R dv ; ∀v ∈ [V ], and an output space Y={-1, +1}. Each multiview observation

x ∈ X 1 × • • • ×X V is a sequence x . =(x 1 , • • • , x V )
where each view x v provides a representation of the same observation in a different vector space X v (each vector space are not necessarily of the same dimension). We further assume that we have a finite set of weak classifiers

H v . ={h v,j : X v → {-1, +1} | j ∈ [n v ]} of size n v .
We aim at learning a two-level encompassed weighted majority vote classifier where at the first level a weighted majority vote is build for each view v∈[V ] over the associated set of weak classifiers H v , and the final classifier, referred to as the Multiview double ωeighted Majority vote Classifier (MωMvC 2 ), is a weighted majority vote over the previous view-specific majority vote classifiers (see Figure 1 for an illustration). Given a training set S=(x i , y i ) 1≤i≤m of size m drawn i.i.d. with respect to a fixed, yet unknown, distribution D over (X 1 × • • • ×X V )×Y, the learning objective is to train the weak view-specific classifiers (H v ) 1≤v≤V and to choose two sets of weights;

Π = (π v ) 1≤v≤V , where ∀v ∈ [V ], π v =(π v,j ) 1≤j≤nv , and ρ=(ρ v ) 1≤v≤V , such that the ρΠ-weighted majority vote classifier B ρΠ B ρΠ (x) = V v=1 ρ v nv j=1 π v,j h v,j (x v ) (1)
has the smallest possible generalization error on D. We follow the Empirical Risk Minimization principle [START_REF] Vapnik | The Nature of Statistical Learning Theory[END_REF], and aim at minimizing the 0/1-loss over S:

L0/1 m (B ρΠ , S) = 1 m m i=1 1 yiB ρΠ (xi)≤0 ,
where 1 p is equal to 1 if the predicate p is true, and 0 otherwise. As this loss function is non-continuous and non-differentiable, it is typically replaced by an appropriate convex and differentiable proxy. Here, we replace 1 z≤0 by the logistic upper bound a log(1 + e -z ), with a=(log 2) -1 . The misclassification cost becomes

Lm (B ρΠ , S) = a m m i=1 ln 1 + exp -y i B ρΠ (x i ) , (2)
and the objective would be then to find the optimal combination weights Π and ρ that minimize this surrogate logistic loss.

3 An iterative parallel update algorithm to learn MωMvC 2

In this section, we first show how the minimization of the surrogate loss of Equation 2 is equivalent to the minimization of a given Bregman divergence. Then, this equivalence allows us to employ a parallel-update optimization algorithm to learn the weights Π=(π v ) 1≤v≤V and ρ leading to this minimization.

Bregman-divergence optimization

We first recall the definition of a Bregman divergence [START_REF] Bregman | The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming[END_REF], [START_REF] Lafferty | Additive models, boosting, and inference for generalized divergences[END_REF].

Definition 1 (Bregman divergence) Let Ω ⊆ R m and F : Ω → R be a continuously differentiable and strictly convex real-valued function. The Bregman divergence D F associated to F is defined for all (p, q) ∈ Ω × Ω as

D F (p||q) . = F (p) -F (q) -∇F (q), (p -q) , (3)
where ∇F (q) is the gradient of F estimated at q, and the operator •, • is the dot product function.

The optimization problem arising from this definition that we are interested in, is to find a vector p ∈ Ω-that is the closest to a given vector q 0 ∈ Ω-under the set P of V linear constraints

P . = {p ∈ Ω|∀v ∈ [V ], ρ v p M v = ρ v p M v },
where p∈Ω is a specified vector, and M v is a m×n v matrix with n v =|H v | the number of weak classifiers for view v∈[V ]. Defining the Legendre transform as

L F q, V v=1 ρ v M v π v . = arg min p∈Ω D F (p||q) + V v=1 ρ v M v π v , p .
the dual optimization problem can be stated as finding a vector q in Q, the closure of the set

Q . = q = L F q 0 , V v=1 ρ v M v π v ρ ∈ R V ; ∀v, π v ∈ R nv ,
for which D F (p||q ) is the lowest. It can be shown that both of these optimization problems have the same unique solution Della Pietra et al. [1997], [START_REF] Lafferty | Additive models, boosting, and inference for generalized divergences[END_REF], with the advantage of having parallel-update optimization algorithms to find the solution of the dual form in the mono-view case [START_REF] Darroch | Generalized iterative scaling for log-linear models[END_REF], Della Pietra et al.

[1997], [START_REF] Collins | Logistic regression, adaboost and bregman distances[END_REF], making the use of the latter more appealing. According to our multiview setting and to optimize Equation (2) through a Bregman divergence, we consider the function F defined for all p∈Ω=[0, 1] m as

F (p) . = m i=1 p i ln(p i ) + (1 -p i ) ln(1 -p i ),
which from Definition 1 and the definition of the Legendre transform, yields that for all (p, q) ∈ Ω × Ω and r ∈ Ω

D F (p||q) = m i=1 p i ln p i q i + (1 -p i ) ln 1 -p i 1 -q i , (4)
and ∀i ∈ [m], L F (q, r) i = q i e -ri 1 -q i + q i e -ri , (5)
Algorithm 1 Learning MωMvC 2 Input: Training set S = (x i , y i ) 1≤i≤m , where ∀i, x i = (x 1 i , . . . , x V i ) and y i ∈ {-1, 1}; and a maximal number of iterations T .

Initialization: ρ (1) ← 1 V 1 V and ∀v, π (1) v ← 1 n V 1 nv Train the weak classifiers (H v ) 1≤v≤V over S For v ∈ [V ] set the m × n v matrix M v such that ∀i ∈ [m], ∀j ∈ [n v ], (M v ) ij = y i h v,j (x v i ) 1: for t = 1, . . . , T do 2: for i = 1, . . . , m do 3: q (t) i = σ   y i V v=1 ρ (t) v nv j=1 π (t) v,j h v,j (x v i )   4:
for v = 1, . . . , V do 5:

for j = 1, . . . , n v do 6:

W (t)+ v,j = i:sign((Mv)ij )=+1 q (t) i |(M v ) ij | 7: W (t)- v,j = i:sign((Mv)ij )=-1 q (t) i |(M v ) ij | 8: δ (t) v,j = 1 2 ln W (t)+ v,j W (t)- v,j 9: π (t+1) v = π (t) v + δ (t) v 10:
Set ρ (t+1) , as the solution of :

min ρ - V v=1 ρ v nv j=1 W (t)+ v,j -W (t)- v,j 2 (8) s.t. V v=1 ρ v = 1, ρ v ≥ 0 ∀v ∈ [V ]
Return: Weights ρ (T ) and Π (T ) .

with a i the i th characteristic of a=(a i ) 1≤i≤m (a being p, q, r or L F (q, r)). Now, let q 0 = 1 2 1 m be the vector with all its components set to 1 2 . For all i ∈ [m], we define L F (q 0 , v) i = σ(v i ) with σ(z) = (1 + e z ) -1 , ∀z ∈ R. We set the matrix M v as for all (i, j)

∈ [m] × [n v ], (M v ) ij = y i h v,j (x v i ).
Then using Equations ( 4) and ( 5), it comes

D F 0 L F q 0 , V v=1 ρ v M v π v = m i=1 ln   1+exp   -y i V v=1 ρ v nv j=1 π v,j h v,j (x v i )     . (6)
As a consequence, minimizing Equation ( 2) is equivalent to minimizing D F (0||q) over q ∈ Q0 , where for

Ω = [0, 1] m Q 0 =    q ∈ Ω q i = σ   y i V v=1 ρ v nv j=1 π v,j h v,j (x v i )   ; ρ, Π    . ( 7
)
For a set of weak-classifiers (H v ) 1≤v≤V learned over a training set S; this equivalence allows us to adapt the parallel-update optimization algorithm described in [START_REF] Collins | Logistic regression, adaboost and bregman distances[END_REF] to find the optimal weights Π and ρ defining MωMvC 2 of Equation (1), as described in Algorithm 1.

A multiview parallel update algorithm

Once all view-specific weak classifiers (H v ) 1≤v≤V have been trained, we start from an initial point q (1) ∈ Q 0 (Eq. 7) corresponding to uniform values of weights ρ

(1) = 1 V 1 V and ∀v ∈ [V ], π (1)
v = 1 nv 1 nv . Then, we iteratively update the weights such that at each iteration t, using the current parameters ρ (t) , Π (t) and q (t) ∈ Q 0 , we seek new parameters ρ (t+1) and δ (t) v such that for

q (t+1) = L F (q 0 , V v=1 ρ (t+1) v M v (π (t) v + δ (t) v )), (9)
we get D F (0||q (t+1) ) ≤ D F (0||q (t) ).

Following [Collins et al., 2002, Theorem 3], it is straightforward to show that in this case, the following inequality holds:

D F (0||q (t+1) ) -D F (0||q (t) ) ≤ A (t) , (10)
where

A (t) = - V v=1 ρ (t+1) v nv j=1 W (t)+ v,j (e -δ (t) v,j -1) -W (t)- v,j (e δ (t) v,j -1) 2 , with ∀j ∈ [n v ]; W (t)± v,j = i:sign((Mv)ij )=±1 q (t) i |(M v ) ij |.
By fixing the set of parameters ρ (t+1) ; the parameters δ (t) v that minimize A (t) are defined as ∀v ∈

[V ], ∀j ∈ [n v ]; δ (t) v,j = 1 2 ln W (t)+ v,j W (t)- v,j
. Plugging back these values into the above equation gives

A (t) = - V v=1 ρ (t+1) v nv j=1 W (t)+ v,j -W (t)- v,j 2 . ( 11
)
Now by fixing the set of parameters (W (t)± v,j ) v,j , the weights ρ (t+1) are found by minimizing Equation ( 11) under the linear constraints ∀v ∈ [V ], ρ v ≥ 0 and V v=1 ρ v = 1. This alternating optimization of A (t) bears similarity with the block-coordinate descent technique [START_REF] Bertsekas | Nonlinear Programming[END_REF], where at each iteration, variables are split into two subsets-the set of the active variables, and the set of the inactive ones-and the objective function is minimized along active dimensions while inactive variables are fixed at current values.

Convergence of Algorithm. The sequences of weights (Π (t) ) t∈N and (ρ (t) ) t∈N found by Algorithm 1 converge to the minimizers of the multiview classification loss (Equation 2), as with the resulting sequence (q (t) ) t∈N (Equation 9), the sequence (D F (0||q (t) )) t∈N is decreasing and since it is lower-bounded (Equation 6), it converges to the minimum of Equation ( 2).

Experimental Results

We present below the results of the experiments we have performed to evaluate the efficiency of Algorithm 1 to learn the set of weights Π and ρ involved in the definition of the ρΠ-weighted majority vote classifier B ρΠ (Equation ( 1)).

Datasets

MNIST is a publicly available dataset consisting of 70, 000 images of handwritten digits distributed over 10 classes [START_REF] Lecun | Gradient-based learning applied to document recognition[END_REF]. For our experiments, we created 2 multiview collections from the initial dataset. Following [START_REF] Chen | Multi-view generative adversarial networks[END_REF], the first dataset (MNIST 1 ) was created by extracting 4 no-overlapping quarters of each image considered as its 4 views. The second dataset (MNIST 2 ) was made by extracting 4 overlapping quarters from each image as its 4 views. We randomly splitted each collection by keeping 10, 000 images for testing and the remaining images for training.

Reuters RCV1/RCV2 is a multilingual text classification data extracted from Reuters RCV1 and RCV2 corpus 1 . It consists of more than 110, 000 documents written in five different languages (English, French, German, Italian and Spanish) distributed over six classes. In this paper we consider each language as a view. We reserved 30% of documents for testing and the remaining for training.

Experimental Protocol

In our experiments, we set up binary classification tasks by using all multiview observations from one class as positive examples and all the others as negative examples. We reduced the imbalance between positive and negative examples by subsampling the latter in the training sets, and used decision trees as view specific weak classifiers. We compare our approach to the following seven algorithms.

• Mono is the best performing decision tree model operating on a single view. • Concat is an early fusion approach, where a mono-view decision tree operates over the concatenation of all views of multiview observations.

• Fusion is a late fusion approach, sometimes referred to as stacking, where view-specific classifiers are trained independently over different views using 60% of the training examples. A final multiview model is then trained over the predictions of the view-specific classifiers using the rest of the training examples.

• MVMLsp [START_REF] Huusari | Multi-view Metric Learning in Vector-valued Kernel Spaces[END_REF] is a multiview metric learning approach, where multiview kernels are learned to capture the view-specific information and relation between the views. We kept the experimental setup of [START_REF] Huusari | Multi-view Metric Learning in Vector-valued Kernel Spaces[END_REF] with Nyström parameter 0.24.2

• MV-MV [START_REF] Amini | Learning from Multiple Partially Observed Views -an Application to Multilingual Text Categorization[END_REF] is a multiview algorithm where view-specific classifiers are trained over the views using all the training examples. The final model is the uniformly weighted majority vote.

• MVWAB Xiao and Guo [2012] is a Multiview Weighted Voting AdaBoost algorithm, where multiview learning and ababoost techniques are combined to learn a weighted majority vote over view-specific classifiers but without any notion of learning weights over views.

• rBoost. SH Peng et al. [2011, 2017] is a multiview boosting approach where a single distribution over different views of training examples is maintained and, the distribution over the views are updated using the multiarmed bandit framework. For the tuning of parameters, we followed the experimental setup of [START_REF] Peng | Multiview boosting with information propagation for classification[END_REF].

Fusion, MV-MV, MVWAB, and rBoost.SH make decision based on some majority vote strategies, as the proposed MωMvC 2 classifier. The difference relies on how the view-specific classifiers are combined. For MVWAB and rBoost.SH, we used decision tree model to learn view-specific weak classifiers at each iteration of algorithm and fixed the maximum number of iterations to T = 100. To learn MωMvC 2 , we generated the matrix M v by considering a set of weak decision tree classifiers with different depths (from 1 to max d -2, where max d is maximum possible depth of a decision tree). We tuned the maximum number of iterations by cross-validation which came out to be T = 2 in most of the cases and that we fixed throughout all of the experiments. To solve the optimization problem for finding the weights ρ (Equation 8), we used the Sequential Least SQuares Programming (SLSQP) implementation of scikit-learn [START_REF] Pedregosa | Scikit-learn: Machine learning in Python[END_REF], that we also used to learn the decision trees. Results are computed over the test set using the accuracy and the standard F 1 -score [START_REF] Powers | Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation[END_REF], which is the harmonic average of precision and recall. Experiments are repeated 20 times by each time splitting the training and the test sets at random over the initial datasets. other approaches and compared to the second best strategy the gain in accuracy (resp. F 1 -score) varies between 0.2% and 2.2% (resp. 2.2% and 3.8%) across the collections. These results provide evidence that majority voting for multiview learning is an effective way to overcome the lack of labeled information and that all the views do not have the same strength (or do not bring information in the same way) as the learning of weights, as it is done in MωMvC 2 , is much more effective than the uniform combination of view-specific classifiers as it is done in MV-MV.

Results

We also analyze the behavior of the algorithms for growing initial amounts of labeled data. Figure 2 illustrates this by showing the evolution of the accuracy and the F 1 -score with respect to the number of labeled examples in the initial labeled training sets on MNIST 1 , MNIST 2 and Reuters datasets. As expected, all performance curves increase monotonically w.r.t the additional labeled data. When there are sufficient labeled examples, the performance increase of all algorithms actually begins to slow, suggesting that the labeled data carries sufficient information and that the different views do not bring additional information.

An important point here is that rBoost.SH-which takes into account both view-consistency and diversity between views-provides the worst results on MNIST 1 where there is no overlapping between the views, while the weighted majority vote as it is performed in MωMvC 2 still provides an efficient model. Furthermore, MVMLsp-which learns multiview kernels to capture views-specific informations and relation between views-performs worst on all the datasets. We believe that the superior performance of our method stands in our two-level framework. Indeed, thanks to this trick, we are able to consider the view-specific information by learning weights over view-specific classifiers, and to capture the importance of each view in the final ensemble by learning weights over the views.

A note on the Complexity of Algorithm

For each view v, the complexity of learning decision tree classifiers is O(d v mlog(m)). We learn the weights over the views by optimizing Equation (11) (Step 10 of our algorithm) using SLSQP method which has time complexity of O(V 3 ). Therefore, the overall complexity is O(V d v m.log(m) + T (V 3 + V v=1 m n v )). Note that it is easy to parallelize our algorithm: by using V different machines, we can learn the view-specific classifiers and weights over them (Steps 4 to 9).

Conclusion

In this paper, we tackle the issue of classifier combination when observations have different representations (or have multiple views). Our approach jointly learns weighted majority vote view-specific classifiers (i.e. at the view level) over a set of base classifiers, and a second weighted majority vote classifier over the previous set of view specific weighted majority vote classifiers. We show that the minimization of the multiview classification error is equivalent to the minimization of Bregman divergences. This embedding allowed to derive a parallel-update optimization boosting-like algorithm to learn the weights of the double weighted multiview majority vote classifier. Our results show clearly that our method allows to reach high performance in terms of accuracy and F 1 -score on three datasets in the situation where few initial labeled training documents are available. It also comes out that compared to the uniform combination of view-specific classifiers, the learning of weights allows to better capture the strengths of different views.

As future work, we would like to extend our algorithm to the semi-supervised case, where one has access to an additionally unlabeled set during the training. One possible way is to learn a view-specific classifier using pseudo-labels (for unlabeled data) generated from the classifiers trained from other views, e.g. [START_REF] Xu | Co-labeling for multi-view weakly labeled learning[END_REF]. Moreover, the question of extending our work to the case where all the views are not necessarily available or not complete (missing views or incomplete views, e.g. [START_REF] Amini | Learning from Multiple Partially Observed Views -an Application to Multilingual Text Categorization[END_REF], [START_REF] Xu | Multi-view learning with incomplete views[END_REF]), is very exciting. One solution could be to adapt the definition of the matrix M v to allow to deal with incomplete data; this may be done by considering a notion of diversity to complete M v .

Figure 2 :

 2 Figure 2: Evolution of accuracy and F 1 -score w.r.t to the number of labeled examples in the initial labeled training sets on MNIST 1 , MNIST 2 and Reuters datasets.

Table 1 :

 1 Test classification accuracy and F 1 -score of different approaches averaged over all the classes and over 20 random sets of m = 100 labeled examples per training set. Along each column, the best result is in bold, and second one in italic. ↓ indicates that a result is statistically significantly worse than the best result, according to a Wilcoxon rank sum test with p < 0.02. .8470 ± .015 ↓ .5704 ± .012 ↓ .8331 ± .016 ↓ .5320 ± .011 ↓ .7484 ± .017 ↓ .5034 ± .016 ↓ rBoost.SH .7580 ± .011 ↓ .4067 ± .009 ↓ .8247 ± .009 ↓ .5148 ± .015 ↓ .7641 ± .014 .5093 ± .010 ↓

	MωMvC 2	.8659 ± .011	.5914 ± .015	.8474 ± .012	.5523 ± .018	.7662 ± .010	.5244 ± .012

1 https://archive.ics.uci.edu/ml/datasets/Reuters+RCV1+RCV2+Multilingual,+Multiview+Text+Categorization+ Test+collection

  Table 1 reports the results obtained for m=100 training examples by different methods averaged over all classes and the 20 test results obtained over 20 random experiments 3 . From these results it becomes clear that late fusion and other multiview approaches (except MVMLsp) provide consistent improvements over training independent mono-view classifiers and with early fusion, when the size of the training set is small. Furthermore, MωMvC 2 outperforms the

We used the Python code available from https://lives.lif.univ-mrs.fr/?page_id=12

We also did experiments for Mono, Concat, Fusion, MV-MV using Adaboost. The performance of Adaboost for these baselines is similar to that of decision trees.

Acknowledgments.

This work is partially funded by the French ANR project LIVES ANR-15-CE23-0026-03 and the "Région Rhône-Alpes".