domain: phys spi chim

text:
Soline Boyer-Richard
email: soline.richard@insa-rennes.fr

Claudine Katan

Boubacar Traore

Reinhard Scholz

Jean-Marc Jancu

Jacky Even
email: jacky.even@insa-rennes.fr

Symmetry-Based Tight Binding Modeling of Halide Perovskite Semiconductors

Based on a general symmetry analysis, this paper presents an empirical tightbinding (TB) model for the reference Pm-3m perovskite cubic phase of halide perovskites of general formula ABX 3 . The TB electronic band diagram, with and without spin orbit coupling effect of MAPbI 3 has been determined based on state of the art density functional theory results including many body correction (DFT+GW). It affords access to various properties, including distorted structures, at a significantly reduced computational cost. This is illustrated with the calculation of the band-to-band absorption spectrum, the variation of the band gap under volumetric strain, as well as the Rashba effect for a uniaxial symmetry breaking. Compared to DFT approaches, this empirical model will help to tackle larger issues, such as the electronic band structure of large nanostructures, including many-body effects, or heterostructures relevant to perovskite device modeling suited to the description of atomic-scale features.

The potential of 3D Hybrid halide Organic Perovskites (HOP) for photovoltaics has been revealed four years ago. [START_REF] Kojima | Organometal Halide Perovskites as Visible-Light Sensitizers for Photovoltaic Cells[END_REF][START_REF] Im | 6.5% Efficient Perovskite Quantum-Dot-Sensitized Solar Cell[END_REF][START_REF] Lee | Efficient Hybrid Solar Cells Based on Meso-Superstructured Organometal Halide Perovskites[END_REF][START_REF] Kim | Lead Iodide Perovskite Sensitized All-Solid-State Submicron Thin Film Mesoscopic Solar Cell with Efficiency Exceeding 9%[END_REF] Huge progresses have been made to improve the crystalline quality of thin films in solar cell devices, as well as the growth of monocrystalline samples for fundamental studies. [START_REF] Nie | High-Efficiency Solution-Processed Perovskite Solar Cells with Millimeter-Scale Grains[END_REF][START_REF] Shi | Low Trap-State Density and Long Carrier Diffusion in Organolead Trihalide Perovskite Single Crystals[END_REF] Many other possible applications of 3D HOP have recently emerged, and layered HOP have gained renewed interest for light emission and are also promising for photovoltaics. [START_REF] Tsai | High-Efficiency Two-Dimensional Ruddlesden-Popper Perovskite Solar Cells[END_REF] 3D HOP can be considered as a new class of semiconductors, with specific optoelectronic properties. To facilitate the analysis and understanding of their properties, some of us have recently proposed to consider the high temperature 3D HOP cubic perovskite phase as a platform to develop a solid-state physics approach. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Solid-State Physics Perspective on Hybrid Perovskite Semiconductors[END_REF][START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF][START_REF] Even | Theoretical Insights into Hybrid Perovskites for Photovoltaic Applications[END_REF] It can be exploited within the framework of Density functional theory (DFT), which has now been used extensively for 3D HOP, [START_REF] Even | Solid-State Physics Perspective on Hybrid Perovskite Semiconductors[END_REF][START_REF] Mosconi | First-Principles Modeling of Mixed Halide Organometal Perovskites for Photovoltaic Applications[END_REF][START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF][START_REF] Brivio | Relativistic Quasiparticle Self-Consistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers[END_REF][START_REF] Du | Efficient Carrier Transport in Halide Perovskites: Theoretical Perspectives[END_REF][START_REF] Giorgi | Cation Role in Structural and Electronic Properties of 3D Organic-Inorganic Halide Perovskites: A DFT Analysis[END_REF] but we know from the field of classical IV, III-V or II-VI semiconductors that developing symmetry analyses and empirical Hamiltonians has many advantages. For instance, the one-band or multiband (k.p) effective mass approaches close to the band edge, [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF][START_REF] Yu | Effective-Mass Model and Magneto-Optical Properties in Hybrid Perovskites[END_REF] have already been used in 3D HOP to investigate various perturbations among which the effect of lattice distortions, [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] the Rashba-Dresselhaus effect, [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF][START_REF] Kepenekian | Rashba and Dresselhaus Effects in Hybrid Organic -Inorganic Perovskites: From Basics to Devices[END_REF] the Bethe-Salpeter equation for the simulation of the exciton, [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Theoretical Insights into Hybrid Perovskites for Photovoltaic Applications[END_REF] the radiative lifetime, [START_REF] Fang | Photophysics of Organic-Inorganic Hybrid Lead Iodide Perovskite Single Crystals[END_REF] the selection rules for electron-phonon interactions, [START_REF] Nie | Light-Activated Photocurrent Degradation and Self-Healing in Perovskite Solar Cells[END_REF] and the influence of band gap renormalization, plasma screening and phase space filling on the exciton absorption. [START_REF] Even | Theoretical Insights into Hybrid Perovskites for Photovoltaic Applications[END_REF] These models are close to the free-electron model and built upon the assumption that electrons propagate almost freely with wave functions close to plane waves.

An alternative starting point is to use localized atomic orbitals as a basis set to perform bandstructure calculations; this is the tight binding (TB) model also known as the linear combination of atomic orbitals (LCAO) approach. It is closer to chemical intuition. This method has been introduced for periodic crystals by Slater and Koster [START_REF] Slater | Simplified LCAO Method for the Periodic Potential Problems[END_REF] and was extensively used in a wide range of materials from transition metals to semiconductor crystals. [START_REF] Jancu | Empirical Spds * Tight-Binding Calculation for Cubic Semiconductors : General Method and Material Parameters[END_REF] The first description of perovskite crystals using LCAO approach concerns metal oxides [START_REF] Kahn | Electronic Energy Bands in Strontium Tintanate[END_REF][START_REF] Honig | ReO3 Band Structure in the Tight-Binding Approximation[END_REF] in the 60's. A detailed description of the LCAO parameterization of perovskite structures was given by Granovskii [START_REF] Granovskii | Formulas of the Tight-Binding Interpolation Method for Crystals of Perovskite Type[END_REF] and takes into account s, p and d type orbitals based on the symmetry analysis by Egorov. [START_REF] Egorov | Consistent Treatment of Symmetry in the Tight Binding Approximation[END_REF] Noteworthy, the TB approach is a semi-empirical method that preserves all the symmetry properties of the crystal. However, it is not designed to yield the crystal structure by minimizing the total energy, and an additional (semi-)empirical or DFT-based approach must be used to yield the atomic positions. [START_REF] Robert | Electronic, Optical, and Structural Properties of (In,Ga)As/GaP Quantum Dots[END_REF] Recently, some of us have demonstrated that TB provides an efficient tool for fundamental studies of buried III-V nanostructures with millions atoms, [START_REF] Robert | Electronic, Optical, and Structural Properties of (In,Ga)As/GaP Quantum Dots[END_REF] colloidal nanostructures including image charge effects, [START_REF] Benchamekh | Tight-Binding Calculations of Image-Charge Effects in Colloidal Nanoscale Platelets of CdSe[END_REF] many-body interactions in semiconductors, [START_REF] Benchamekh | Microscopic Electronic Wave Function and Interactions between Quasiparticles in Empirical Tight-Binding Theory[END_REF] and optoelectronic properties of nanostructures including either with direct or indirect band gaps. [START_REF] Robert | Electronic Wave Functions and Optical Transitions in (In,Ga)As/GaP Quantum Dots[END_REF] Besides, stateof-the-art simulations of semiconductor devices, including atomistic effects at the TB level, have also been developed for both inorganic [START_REF] Klimeck | Atomistic Simulation of Realistically Sized Nanodevices Using NEMO 3-D -Part I: Models and Benchmarks[END_REF][START_REF] Klimeck | Atomistic Simulation of Realistically Sized Nanodevices Using NEMO 3-D -Part II: Applications[END_REF][START_REF] Auf Der Maur | Trap-Assisted Tunneling in InGaN/GaN Single-Quantum-Well Light-Emitting Diodes[END_REF] and organic semiconductors. [START_REF] Fallahpour | Modeling and Simulation of Energetically Disordered Organic Solar Cells[END_REF] For HOP, Kim and co-workers already proposed a local TB Hamiltonian, developed on an eight functions basis with spin-orbit coupling (SOC). [START_REF] Kim | Switchable S = 1/2 and J = 1/2 Rashba Bands in Ferroelectric Halide Perovskites[END_REF] It was designed to model the Rashba effect close to the R-point of the perovskite reciprocal lattice and provides clear evidence of sizeable Rashba effect, thus confirming the initial DFT results addressing the importance of spatial and time reversal symmetries in this new family of semiconductors. [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] Unfortunately, the chosen reduced TB scheme was not designed to afford the entire band structure across the whole Brillouin zone (BZ). Knudson and coworkers also used an extended Huckel TB model to analyze the effects of band folding in tin-based 2D HOP. [START_REF] Knutson | Tuning the Band Gap in Hybrid Tin Iodide Perovskite Semiconductors Using Structural Templating[END_REF] Given this background and our preliminary results, [START_REF] Even | Theoretical Insights into Hybrid Perovskites for Photovoltaic Applications[END_REF] we are convinced that the present work will open up new avenues for understanding HOP devices.

The aim of this article is to build a simple but general TB model for 3D HOP, starting from the symmetry properties of the cubic reference structure of the halide perovskite lattice (Fig. 1a,b), and incorporating their unusual physical properties, like the giant SOC of lead-based HOP. The design of this novel TB model for 3D HOP allows us to illustrate how it can accurately model their properties, using the prototype MAPbI 3 compound. Despite the simplicity of the chosen sp 3 model (Fig. 1c), experimental data like effective masses and band gaps are accurately reproduced within this scheme. We further show that absorption coefficient and oscillator strengths can also be deduced from this TB model. Finally, in order to demonstrate the versatility of this approach, we extend the basic cubic perovskite TB model to study the impact of strain, and the Rashba effect related to the loss of inversion symmetry. We start by considering the reference cubic structures of the 3D ABX 3 family (Fig. 1b), where A is an organic or inorganic cation located at the center of the unit cell, B is a metal atom (Pb, Sn, …) and X are halogen atoms (X=I, Br, Cl). Obviously, the interpretation of experiments dedicated to electronic and optical properties of HOP as well as their efficient simulation may greatly benefit from symmetry analyses. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] To extensively report on symmetry properties, we herein describe in details the high-temperature 3D perovskite cubic reference phase. But it has been shown that, for example, symmetry breaking is relevant to understand the influence of crystal structure change at a phase transition corresponding to a group-subgroup relationship. [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF][START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] From a crystallographic point of view, at sufficiently high temperature, 3D HOP have a simple cubic lattice with an octahedron pattern and a central cation. Organic cations automatically break the cubic symmetry, but the cubic space group Pm-3m is restored by dynamical averaging of the cation stochastic reorientations at high temperature. Thus, to perform our symmetry analysis within this TB scheme, only the inorganic atoms of each octahedron are considered (Fig. 1a,b).

For each metal atom B and halogen atom X, we consider one s and three p orbitals (Fig. 1c). No basis functions are considered for any of the A cations, consistently with DFT studies of the electronic states close to the band edges that do not reveal significant weight stemming from A cations. [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF] An isolated octahedron cluster BX 6 belongs to O h point group symmetry (Fig. 1a), as the cubic crystal ABX 3 (Fig. 1b). However, translational symmetry of crystal modifies orbitals hybridization. For an isolated cluster without SOC, s orbitals and p orbitals of B atom correspond to A 1g and T 1u irreducible representation (IR), respectively. Meanwhile, s and p orbitals of the six X atoms correspond to + + and

+ + + + 2 + IR,
respectively. s(B) and p(X) orbitals can therefore be hybridized. However, if we consider the same orbitals of the cubic crystal ABX 3 , no hybridization between s(B) and p(X) is possible at the Γ-point, while it is possible at the R-point as illustrated Fig. 2. [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] The aim of the present TB study is to build an empirical model taking into account these specific symmetry considerations, and thus provide accurate simulation of experimental data close to the band gap.

Our TB model is based on a sixteen basis functions without SOC and thirty-two basis functions when taking SOC into account. The symmetries are accounted for under simple and double space groups, respectively. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF][START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF] The crystal pattern contains one B metal atom at = 0,0,0 , and three X halide atoms at = , 0,0 , = 0, , 0 and = 0,0, . Without SOC, nine parameters are considered. This encompasses four different diagonal matrix elements related to the atomic energies of the chemical constituents B and X:

• = * |$| • = * |$| = * |$| = *! |$|! • = * |$| = * |$| = * |$| • = + , -$-, . = + , -$-, . = +! , -$-! , .
where / = 1, 2, 3 for each X atom.

Five transfer matrix elements refer to the overlap integral of first neighbor atomic functions:

• = + -$ '() -, .
where / = 1, 2, 3 for each X atom

• = * |$ '() | = * |$ '() | = * |$ '() |! • = * |$ '() | = * |$ '() | = *! |$ '() | • = * |$ '() | = * |$ '() | = *! |$ '() |! • = * |$ '() | = * |$ '() | = *! |$ '() |! = *! |$ '() |! = * |$ '() | = * |$ '() | .
These are schematically represented Fig. 1c. Noteworthy, the overlap integrals between perpendicular p orbitals are vanishing. The transfer matrix elements between s and p orbitals are nonzero if and only if the axis of the p orbital contains a s orbital center. The complete expression of the 16-band Hamiltonian $ without SOC is given in the supporting information (SI).

In presence of SOC, the triply degenerate p states whose orbital angular momentum is 3 = 1 split into lower # = and upper # = bands, respectively doublet and quartet, whereas SOC is ineffective on s states (3 = 0). [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF][START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF] The 32-band TB Hamiltonian reads:

$ 4 && = $ + $ 567 .
$ 567 is a block-diagonal matrix. For each atom, " = 0 for B and " = 1, 2, 3 for X,

                    - - - - - - = 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 1 0 0 0 0 2 i i i i i i i i H SOC α λ .
Two different SOC values are taken into account: < = = > ? for p(B) orbitals and < = = > @ = > A = > B for p(X) orbitals. The 32-band Hamiltonian is larger than the one obtained by Kim et al. [START_REF] Kim | Switchable S = 1/2 and J = 1/2 Rashba Bands in Ferroelectric Halide Perovskites[END_REF] (vide supra) but remains very empty.

To determine the parameters of the TB Hamiltonian relevant to the cubic phase of MAPbI 3 , we fit the TB band diagram on the following experimental and theoretical data: (i) the experimental band gap, C = 1.61 E , as obtained by photoluminescence measurement at room temperature, [START_REF] Yamada | Near-Band-Edge Optical Responses of Solution-Processed Organic-inorganic Hybrid Perovskite CH 3 NH 3 PbI 3 on Mesoporous TiO 2 Electrodes[END_REF] (ii) the reduced effective mass, F = 0.104 H , as determined by magneto optical measurement, [START_REF] Miyata | Direct Measurement of the Exciton Binding Energy and Effective Masses for Charge Carriers in Organic-Inorganic Tri-Halide Perovskites[END_REF] and (iii) the band diagram obtained within the quasiparticle self-consistent GW theory (QSGW), with SOC included in the self-consistent cycle. [START_REF] Brivio | Relativistic Quasiparticle Self-Consistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers[END_REF] Valence (VB) and/or conduction (CB) band densities of states determined by photoelectron spectroscopy with hard X-rays, [START_REF] Lindblad | Electronic Structure of TiO2 /CH3NH3PbI3 Perovskite Solar Cell Interfaces[END_REF] ultraviolet and inverse photoemission spectroscopies, [START_REF] Endres | Valence and Conduction Band Densities of States of Metal Halide Perovskites: A Combined Experimental-Theoretical Study[END_REF] as well as pressure study of the band gap, [START_REF] Kong | Simultaneous Band-Gap Narrowing and Carrier-Lifetime Prolongation of Organic-inorganic Trihalide Perovskites[END_REF] provide further information to assess model parameters. Noteworthy, the complete calculation of the electronic band diagram including SOC presented in this work, takes less than 1 sec on any basic desktop computer without any computational pitfall.

A first set of TB parameters (Table S1, with corresponding computed properties shown Figure S1) allowed to obtain a fit of quite good quality with C = 1.620 E at the R-point (see Brillouin zone sketched Fig. 2a) and a reduced effective mass of 0.116 H , consistently with the above mentioned experimental results. The SOC value between the predominantly lead-based spinorbit split-off bands (CB) amounts to Δ = = 1.3 E , in line with the findings of earlier reports. [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF][START_REF] Jin | Topological Insulator Phase in Halide Perovskite Structures[END_REF][START_REF] Even | Electronic Model for Self-Assembled Hybrid Organic/perovskite Semiconductors: Reverse Band Edge Electronic States Ordering and Spin-Orbit Coupling[END_REF][START_REF] Pedesseau | Electronic Properties of 2D and 3D Hybrid Organic / Inorganic Perovskites for Optoelectronic and Photovoltaic Applications[END_REF] Noteworthy, this value is consistent with the SOC splitting between p-like orbitals reported in atomic energy level tables which amounts to 1.75 eV for lead in its 6s 2 6p electronic configuration. [START_REF] Wood | Interferometric Measurement of Pb Ii Spectrum[END_REF] Meanwhile, this first set of TB parameters is obtained with Δ = = 1.2 E . This value seems rather high as compared to the 0.94 eV reported for the 5s 2 5p 5 electronic configuration of iodine. 49 This prompts us to examine the way this parameter influences the computed band diagram.

< = < = -1
Unfortunately, unlike Δ = that can be rather easily extracted from the splitting of the doublet and quartet CB (Fig. 2e andS1), Δ = related to iodine p-like states cannot be easily extracted from the highest lying VB, as a consequence of significant hybridization with the lead 6s orbitals (Fig. 2b). Thus, we chose Δ = = 0.9 E , close to the value reported in atomic tables, and refine all other TB parameters except Δ = = 1.3 E . Corresponding values and band diagram are given in Table 1 and Fig. 2c, respectively. The band gap remains at the R-point with C = 1.603 E .

Close to the band gap, effective masses are: H K * = 0.215 H for the VB and H N * = 0.218 H for the CB, which leads to a reduced effective mass of 0.108 H , i.e. remaining consistent with the experimental value taken as reference. [START_REF] Miyata | Direct Measurement of the Exciton Binding Energy and Effective Masses for Charge Carriers in Organic-Inorganic Tri-Halide Perovskites[END_REF] As a matter of fact, all parameters have to be refined, because the SOC splitting of iodine has a direct impact on the band gap and effective masses, as has been evidenced earlier. [START_REF] Katan | Interplay of Spin-orbit Coupling and Lattice Distortion in Metal Substituted 3D Tri-Chloride Hybrid Perovskites[END_REF] This is further illustrated by keeping all parameters fixed expect Δ = , which is decreased to 0.45 and 0 eV. Corresponding band gaps (reduced effective masses) increase to C = 1.65 eV (µ = 0.111 m 0 ) and C = 1.68 eV (µ = 0.112 m 0 ), respectively (Figure S2). The comparison of Figure 2c with Figures S1 andS2 clearly evidences that the smaller the splitting Δ = the smaller the dispersion of the p(X) bands lying about 2 to 4 eV below the top of the valence band. For Δ = = 0, ten of them are totally flat and the quality of the fit with the QSGW significantly deteriorates. We thus proceed with the TB parameters given Table 1 to further investigate the potential of this sp 3 TB model.

Prior to that, we should underline two limitations of the present TB model. They are a direct consequence of the desired simplicity, namely a model simple enough to allow symmetry consideration to be taken into account close to the band gap. Firstly, the present TB approach is limited to first nearest neighbor interactions and this directly influences the flexibility of the TB model, namely it further contributes to the flatness of some of the band dispersions (vide supra).

Secondly, the sharp curvature of VB in the vicinity of 5 eV below the top of the VB (Fig. 2c), much less pronounced in both DFT and QSGW calculations, [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Brivio | Relativistic Quasiparticle Self-Consistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers[END_REF] is an indirect consequence of the low hole effective mass that has to be considered to fit both the band edge dispersion and experimental data. However DFT based computations also fail to reproduce the experimental VB density of states in the same energy range. [START_REF] Lindblad | Electronic Structure of TiO2 /CH3NH3PbI3 Perovskite Solar Cell Interfaces[END_REF][START_REF] Endres | Valence and Conduction Band Densities of States of Metal Halide Perovskites: A Combined Experimental-Theoretical Study[END_REF] Among others, the optical absorption of a given material is a key parameter for its potential relevance in the field of photovoltaics. The present model allows calculating optical absorption by deriving the TB Hamiltonian. An electromagnetic wave The absorption coefficient shows a sharp optical absorption edge, corresponding to the VB1(R)-CB1(R) optical transition at the R-point. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF] It is consistent with experimental data recorded by De Wolf [START_REF] De Wolf | Organometallic Halide Perovskites: Sharp Optical Absorption Edge and Its Relation to Photovoltaic Performance[END_REF] and comparable to the absorption spectrum of GaAs. The calculated absorption slope is modified above 2.75 eV, which corresponds to the VB1(M)-CB1(M) optical transition at M-point and a saddle point in the R M direction of the BZ. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] The additional absorption feature near 2.75 eV also contains contributions from the VB2(R)-CB1(R) and VB1(R)-CB2(R) secondary transitions. [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF] This again points the multi-valley and multi-bandgap nature [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF] of the optical absorption of HOP leading to high photovoltaic efficiency in very thin layers.

P = Q E ' R SPT PUV) + * E U' R SPT PUV) WE P =&

R M

Noteworthy, the shape of the spectrum is similar to that computed by Brivio and co-workers. [START_REF] Brivio | Relativistic Quasiparticle Self-Consistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers[END_REF] It might not be surprising, as part of the TB parameterization relies on their calculated band structure. Nonetheless, there is a marked difference for the higher optical feature, which is found above 3 eV in the QSGW spectrum, [START_REF] Brivio | Relativistic Quasiparticle Self-Consistent Electronic Structure of Hybrid Halide Perovskite Photovoltaic Absorbers[END_REF][START_REF] Leguy | Experimental and Theoretical Optical Properties of Methylammonium Lead Halide Perovskites[END_REF] while it is close to the experimental value in the TB spectrum computed with parameters of Table 1, without need of any empirical rescaling. In fact, an empirical rescaling was used for QSGW data, to correctly match the optical spectrum over a large energy range. [START_REF] Leguy | Experimental and Theoretical Optical Properties of Methylammonium Lead Halide Perovskites[END_REF] In addition, when using our first set of TB parameters (Table S1) in which the SOC splitting for iodine atoms is unrealistically high, similar discrepancies with experimental observation are found (Figure S1). Thus, using the TB parameters of Table 1 also improves quantitatively the higher optical features by comparison to state of the art DFT approaches.

General symmetry properties of the halide perovskite lattice, [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] , , 0 , 0, , and , 0, which all contribute equally to the observed absorption. The electronic energy eigenfunctions are described as Bloch waves to obtain the effects of point and translational symmetries on the band diagram:

q (,r = E 'r i (,r (1)
where s is the band index, the position vector and r the wave vector inside the BZ. The E 'r phase factor is important to understand the bonding or antibonding character of the wavefunctions. [START_REF] Katan | Interplay of Spin-orbit Coupling and Lattice Distortion in Metal Substituted 3D Tri-Chloride Hybrid Perovskites[END_REF] At the R-point corresponding to the optical band edge (Fig. 2a, Fig. 3 and Table 2), the symmetry adapted basis functions associated to each IR are (without SOC):

• two u y functions: | ,

√ | + | + |!
• two u y functions:

√€ 2|! -| -| , √ | -|
• three u { y functions:

√ | + |! , √ | + |! , √ | + |
• three u } y functions:

√ |! -| , √ | -|! , √ | -| • six u { U functions: | , | , |! , | , | , |
This confirms that the totally symmetric u y correspond to s(B) -p(X) hybridized atomic orbitals at the top of the VB at point R, whereas the bottom of the CB u { U correspond to p(B) states (Fig. 2b). [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] In order to account for the influence of the lattice strain, we use a simple variation of the five transfer matrix elements as a function of the interatomic distance •: ' = ' ' ?

'

where ' and

• are respectively the transfer matrix element and the lattice parameter, without strain. [START_REF] Yu | Fundamentals of Semiconductors[END_REF] The band gap variation as a function of the lattice strain, shown Fig. 4, is reversed as compared to classical semiconductors. This is a consequence of the reverse ordering of the CB and VB in halide perovskites. [START_REF] Even | Importance of Spin-Orbit Coupling in Hybrid Organic/Inorganic Perovskites for Photovoltaic Applications[END_REF] In order to precisely track the origin of the band gap variation, it is possible to calculate analytically the eigenvalues after block diagonalization of the Hamiltonian. The For a small variation ∆ of the lattice parameter , the variations of 7ƒ o and †ƒ o are given by:

a †ƒ o ‹ U{OE † •?Ž@ A ‰ • Ž@ U• •? A y{OE † •?Ž@ A ∆^ (2)
and

a 7ƒ o ‹ U € † Ž?•@ A ‰ • Ž? U• •@ A y € † Ž?•@ A ∆^ (3)
This clearly shows that the band gap variation as a function of the strain is related to the transfer integrals between the s and p orbitals of B and X atoms, and . The most important variation is related to the VB maximum. The same conclusion was drawn qualitatively in a recent experimental paper reporting on pressure dependence of the band gap of MAPbI 3 . [START_REF] Kong | Simultaneous Band-Gap Narrowing and Carrier-Lifetime Prolongation of Organic-inorganic Trihalide Perovskites[END_REF] Indeed, the authors assumed that the CB has mainly a nonbonding Pb p character. It is possible to quantify this aspect with the TB modelling, and state that the p orbitals of X atoms and the s orbital of the B atom are energetically closer, thus enhancing the coupling (see expressions (2) and ( 3)). More, is significantly larger than (Table 1

= + + ∆ = /3 2 + ‰ - + ∆ = /3 + 16 2 u OE U F / …
The variation of 7ƒ as a function of the strain now reads as follows:

a 7ƒ o ‹ -16 ‰ - -2∆ = /3 + 16 ∆
The additional ∆ •"? term at the denominator is much smaller than -. This explains why the band gap variation as a function of a volumetric strain, shown Fig. 4, follows the same trend with and without SOC. Moreover, using the experimental value of the bulk modulus reported by Rakita et al., [START_REF] Rakita | Mechanical Properties of APbX 3 (A = Cs or CH 3 NH 3 ; X = I or Br) Perovskite Single Crystals[END_REF] " = 14•-, the TB band gap decrease computed for a pressure of 32•is -0.04eV, which compares well with the experimental value of -0.03eV. A refined TB modelling of strain effect is possible by slightly tuning the dependence of the transfer matrix elements as a function of the interatomic distances, [START_REF] Jancu | Tetragonal and Trigonal Deformations in Zinc-Blende Semiconductors: A Tight-Binding Point of View[END_REF] but this is beyond the scope of the present paper.

From symmetry consideration, it appears clearly that the s-p bonding does not stabilize the VB maximum t y at the Γ-point like at the R-point; this is a direct consequence of the phase factor in the definition of the Bloch functions (expression (1)). [START_REF] Katan | Interplay of Spin-orbit Coupling and Lattice Distortion in Metal Substituted 3D Tri-Chloride Hybrid Perovskites[END_REF] Finally, in order to illustrate a simple approach of the Rashba effect within a TB model, we shift the position ˜ of the third iodide atom X along the z axis, by a quantity i (˜ = + i) (Fig. 5a). This shift corresponds to a space group change from Pm-3m to P4mm and the creation of a permanent electric dipole along the ˜ axis. [START_REF] Kepenekian | Rashba and Dresselhaus Effects in Hybrid Organic -Inorganic Perovskites: From Basics to Devices[END_REF] Using the variation of the above mentioned transfer matrix elements as a function of interatomic distances, it is straightforward to show that, for small values of i, two phase factors are modified (see supporting information for the definition of ℎ ™ , š ™ ):

š ™ ›oeoeoeoe• š ™ -4iℎ ™ E U'k ž ^ and ℎ ™ ›oeoeoeoe• ℎ ™ -4iš ™ E U'k ž ^
The resulting electronic band diagrams along the g Ÿ 0, , -u -g , 0, path of the cubic BZ are given in Fig. 5b,c, for a displacement of the axial iodine atom along Oz u=0.05.

Without SOC, the phase transition lifts partially the degeneracy in the CB. [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] Adding SOC lifts totally the degeneracy (Fig. 5c). [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF] The Rashba effect is characteristic of a further lifting of the spinor degeneracy in the vicinity of the R-point, both in CB and VB, first revealed in the context of HOP in ref. [START_REF] Even | Modelling of the Phase Transitions of Lead and Tin Halide Perovskites for Photovoltaic Cells[END_REF]

Figure 1 .

 1 Figure 1. Schematic representation of (a) a single BX 6 octahedron, (b) the BX 3 cubic lattice and (c) s and p orbitals with the main transfer matrix elements for the sp 3 TB model.

  The sixteen corresponding atomic orbitals centered at are denoted | for s orbitals and | , | , |! for p orbitals, " = 0 for the B atom and " = 1, 2 or 3 for each of the three halide atoms (see graphical abstract).

Figure 2 .

 2 Figure 2. (a) Reciprocal space 3D view showing the first Brillouin Zone (BZ) of the Pm3m

|

  ↑ , | ↑ , |! ↑ up spins and | ↓ , | ↓ , |! ↓ down spin states are coupled by the Pauli matrix $ 56 written as follows in the basis :| ↑ , | ↑ , |! ↑ , | ↓ , | ↓ , |! ↓ ;:

  creates a perturbation of the Hamiltonian H due to optical excitation $ The photon absorption rate between an initial VB state |[ , of energy ' , and a final CB state |\ , of energy 4 , is given by Fermi's golden rule. ] ^_ = ℏ |*\|$ X |[ | ab 4 -' -ℏde with *\|$ X |[ = Nf ? Y ? +\-E 'R SPT P E P =& . Z P -[ The optical matrix element g '→4 = +\-E 'R SPT P E P =& . Z P -[ = +i jk -E P =& . Z P -i lk Z N m is calculated as Z N m = Y ? ℏ ∇ k $ m , deriving the crystal Hamiltonian for each value of k in the first BZ in the polarization direction E P =& . Left and right side of this bra-ket are eigenstates obtained solving the TB Hamiltonian for each value of m in the BZ.To perform the BZ integration, the resulting discrete transitions are dressed with a 10 meV Gaussian broadening in order to get smooth spectral functions. Fig.3ashows the absorption spectrum resulting from this calculation for cubic MAPbI 3 . The validity of this absorption spectrum extends not further than 3.3eV.

Figure 3 .

 3 Figure 3. Absorption spectrum (black straight line) of MAPbI 3 computed with the sp 3 TB model

  are translated in the TB model to block diagonal Hamiltonians at high symmetry points Γ, R and M. Indeed point groups of vectors m o , , and m p , , 0 are O h and D 4h , respectively. Isotropic optical transitions are predicted because the star of m o contains only itself whereas the star of m p contains three arms: 8

Figure 4 .

 4 Figure 4. Variation of the band gap as a function of the volumetric strain (sketch in the center), computed with the sp 3 TB model and parameters given inTable 1: with (blue straight line) or

  ∆a/a (%)

  based on DFT calculations. Noteworthy, the band structure along the u -g ¡ path does not exhibit such a degeneracy lifting, since the Oz polar axis corresponds to the C 4 axis of the structure.[START_REF] Kepenekian | Rashba and Dresselhaus Effects in Hybrid Organic -Inorganic Perovskites: From Basics to Devices[END_REF]

Figure 5 .

 5 Figure 5. (a) Displacement of the axial iodine atom along the z axis: ˜ = + i. This shift corresponds to a space group change from Pm-3m to P4mm. (b) Corresponding electronic band diagram of MAPbI 3 , along the g Ÿ -u -g path of the cubic BZ, obtained within the TB scheme and parameters of Table 1 and i = 0.05: (b) without SOC and (c) with SOC.

Table 1 .

 1 Tight binding parameters in eV for MAPbI 3 band diagram with SOC.

	.10	1.19	0.70 -3.65	0.55	-9.01 -13.01	2.34	-1.96	1.30	0.90

  ).

	With SOC, the CB minimum splits in a double degenerate CB minimum ( 7ƒ o , u € U E / … )
	and a quadruple degenerate band lying at higher energy ( 7ƒ o , u OE U F / … ): 8
	7ƒ o =	+	-2∆ = /3 2	+	‰	-	-2∆ = /3 + 16 2	u € U E / …
	and
	7ƒ o

ACKNOWLEDGMENT This project has received funding from the European Union's Horizon 2020 research and innovation program under the grant agreement No 687008. The information and views set out in this publication are those of the authors and do not necessarily reflect the official opinion of the European Union. Neither the European Union institutions and bodies nor any person acting on their behalf may be held responsible for the use which may be made of the information contained therein. Work at ISCR and FOTON was performed using grant from Cellule Energie du CNRS (SOLHYBTRANS Project) and the University of Rennes 1 (Action Incitative, Défis Scientifiques Emergents 2015).

Here, we propose to associate a basis of symmetrized linear combination of atomic orbitals (SLCAO) used in the TB model to each IR obtained by symmetry. We start without SOC. The center of the BZ t 0,0,0 and u , , are both points where the group of wave vector is O h , whereas the wave vector at g , , 0 belongs to the D 4h group, with only sixteen symmetries instead of forty-eight for O h . In the O h group, s orbitals have v Kw = Γ y or R y IR while p [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF][START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] In the D 4h group, s orbitals have v Kw = M y IR while p [START_REF] Even | Molecular Disorder and Translation/rotation Coupling in the Plastic Crystal Phase of Hybrid Perovskites[END_REF] To account for the site symmetry, one determines the site reducible representation v ')N . At the Γ-point, the site symmetry correspond to t y for the B atom, and t y + t y for X atoms. The product v ')N ⨂v Kw provides the IR of eigenfunctions. At Γ-point, the SLCAO leads to t y , t { U , t y + t y , 2t { U + t } U IR for s(B), p(B), s(X) and p(X) orbitals, respectively, and there is no common IR between s(B) and p(X). [START_REF] Even | Pedestrian Guide to Symmetry Properties of the Reference Cubic Structure of 3D All-Inorganic and Hybrid Perovskites[END_REF] At the R-point, a similar analysis leads to u y , u { U , u { U , u y + u y + u { y + u } y IR for the same orbitals. At the Mpoint, this analysis leads to g y , g U + g } U , g y + g } U , g U + g } U + g y + g y + g y + g { y + g } y . This decomposition allows determining which orbitals are hybridized at each point of the BZ (Table 2).

Irreducible representations (IR) of eigenfunctions within the TB model at various points of the BZ without SOC.

At point M , , 0 , the basis functions associated to each IR read:

• one g y function:

• one g y function:

The electronic states at the M-point correspond to a saddle point of the Fermi surface, [START_REF] Even | Analysis of Multivalley and Multibandgap Absorption and Enhancement of Free Carriers Related to Exciton Screening in Hybrid Perovskites[END_REF] and contribute to an additional feature in the optical absorption spectrum (Fig. 3). The symmetry decomposition of Bloch states at the M-point is fully translated in the TB Hamiltonian (See supporting information for the full expression of this Hamiltonian). For example, at point M, the TB Hamiltonian is block-diagonal with a 3 × 3 block for g y , a 6 × 6 block for g } U , a 2 × 2 block for g U and five diagonal terms for g y + g y + g { y + g } y .

At point Γ, the basis functions associated to each IR read:

streamlining the findings, e.g. providing a natural explanation for the effect of SOC or in terms of orbital hybridization such as s-p transfer integrals.

The powerfulness of the present TB Hamiltonian is further exemplified with the Rashba effect by considering an uniaxial symmetry breaking. Compared to first-principles approaches, this semi-empirical model challenges us to tackle more difficult questions in terms of size with complex heterostructures, nanostructures or composite materials as well as diversity of physical phenomenon under investigation. We strongly believe that such a TB model, suited for an atomic-scale description, is relevant to perovskite device modeling, as it is for conventional semiconductors.

ASSOCIATED CONTENT

Supporting information available:

• Full expression of tight binding Hamiltonians without SOC

• Simulations with alternative TB parameters.

• Effect of p(X) SOC

AUTHOR INFORMATION

Corresponding Authors *soline.richard@insa-rennes.fr, *jacky.even@insa-rennes.fr

Notes

The authors declare no competing financial interest.


Pauline Wauquier
email: pauline.wauquier@inria.fr

Mikaela Keller
email: mikaela.keller@univ-lille3.fr

A METRIC LEARNING APPROACH FOR GRAPH-BASED LABEL PROPAGATION

The efficiency of graph-based semi-supervised algorithms depends on the graph of instances on which they are applied. The instances are often in a vectorial form before a graph linking them is built. The construction of the graph relies on a metric over the vectorial space that help define the weight of the connection between entities. The classic choice for this metric is usually a distance measure or a similarity measure based on the euclidean norm. We claim that in some cases the euclidean norm on the initial vectorial space might not be the more appropriate to solve the task efficiently. We propose an algorithm that aims at learning the most appropriate vectorial representation for building a graph on which the task at hand is solved efficiently.

INTRODUCTION

Transductive or semi-supervised graph-based learning algorithms, such as [START_REF] Zhu | Semi-supervised learning using gaussian fields and harmonic functions[END_REF]; [START_REF] Zhou | Learning with local and global consistency[END_REF]; [START_REF] Zhu | Semi-supervised learning literature survey[END_REF]; [START_REF] Liu | Robust multi-class transductive learning with graphs[END_REF], take advantage of both annotated and unannotate data to solve automated labeling tasks. It relies on the homophily property: entities that are close in the graph are supposed to have similar behaviors.

However, those graph-based learning algorithms are dependent on the graph on which they are applied. Indeed the effectiveness of graph-based learning algorithms relies on the relevance of the data representation for the targeted task [START_REF] Maier | Influence of graph construction on graphbased clustering measures[END_REF]; [START_REF] Jebara | Graph construction and b-matching for semi-supervised learning[END_REF]; de [START_REF] Andr | Influence of graph construction on semi-supervised learning[END_REF]).

In general the graph representation of the data is built in two steps [START_REF] Liu | Robust multi-class transductive learning with graphs[END_REF]; [START_REF] Maier | How the result of graph clustering methods depends on the construction of the graph[END_REF]; de [START_REF] Andr | Influence of graph construction on semi-supervised learning[END_REF]. In this representation, every data point is seen as a vertex of the graph and the first building step is to compute the edge weight between every pair of vertices. This weight, refleting the similarity degree between two data points is usually computed as a function of the euclidean scalar product between the data points. In the second step of the graph construction edge with low similarity degree are discarded. This is done by applying a non-linear transformation on the edges, like selecting a threshold on the weight value (a method referred as -graph), a fixed number k of neighbors for each node (k-nn graph) or by applying a kernel on the weights (making high weights higher and low weights lower).

We claim that there might be cases where the euclidean space in which the data points lie will not produce an optimal graph for solving the targeted task.

To answer this concern, we propose an approach to build a graph from our data but also adapted to a specific labeling task. We learn a new representation of our data based on constraints related to the task: Data points should be close in the graph if they share similar labels. As the data may be too complex to be projected with a linear approach, we will use a deep neural network in order to learn the most appropriate representation of our data for the task.

Previous work related to ours fall into two main categories: representation learning algorithms and metric learning approaches. Among representation learning algorithms, some attempt at learning the best representation for a supervised task in a semi-supervised setting. In particular, the work presented in [START_REF] Chopra | Learning a similarity metric discriminatively, with application to face verification[END_REF] 2005), a deformed kernel is learn depending on the data geometry which will be used to the classification task resolve.

Our approach stands in between metric learning and representation learning. It focuses on a specific existing metric w but learns to project the data in a space in which w is meaningful for the targeted task.

ALGORITHM'S DESCRIPTION

Let us define D = {(x i , y i )} i=1..n a dataset of examples such that ∀i ∈ {1, ..., n}, x i ∈ R d and y i ∈ {1, ..., c}. Let X be the projection of D on R d and C 1 , ..., C c be a partition of X, such that ∀k ∈ {1, ..., c},

C k = {x i ∈ X|(x i , y i ) ∈ D ∧ y i = k} and ∀h = k, C k ∩ C h = ∅.
Let us assume that D is partioned into a training set D train and a test set D test ; we can define X l to be the projection of D train on R d and X u the projection of D test on R d , X = X l ∪ X u . Labels of elements x ∈ X u are hidden. If T is the set of triplet constraints defined as:

T = {(x, x + , x -)|(x, y), (x + , y + ), (x -, y -) ∈ D ∧ y = y + = y -}, then let us define T = T train ∪ T test such that T train = {(x, x + , x -) ∈ T |x, x + , x -∈ X l } and T test = T \T train .
Let φ : R d ← R q be a non-linear function. Let us consider a distance measure d1 on R q . We want the projection φ to respect the following constraint with respect to distance d:

∀(x, x + , x -) ∈ T, d(φ(x), φ(x + )) < d(φ(x), φ(x -)).
At training time, and using a hinge loss we can reformulate our set of constraints as the following cost function to minimize:

C(φ|T train ) = (x,x+,x-)∈Ttrain max(0, 1 -[d(φ(x), φ(x -) -d(φ(x), φ(x + ))])
The objective function of our problem is:

φ = arg min φ C( φ|T train )
In order to learn φ, we train a Siamese neural network containing three replicated non-linear neural network φ (Fig. 1), by stochastic gradient descent optimization.

Let us define W ∈ R n×n as the matrix whose components W ij = d(φ(x i ), φ(x j )). W can be seen as the adjacency matrix of a complete graph. We can use the computed graph in order to predict the hidden label, through the well known label propagation algorithm [START_REF] Zhu | Learning from labeled and unlabeled data with label propagation[END_REF]; [START_REF] Zhou | Learning with local and global consistency[END_REF]); In order to remove the non relevant edges of the graph, a pruning phase is usually performed on the graph. Different pruning methods can be applied. The creation of a k -nn graph, where only the edges for the k nearest neighbors of each instances are kept, is a popular one. Another pruning technique is the extraction of the -graph, where edges are kept depending on a threshold on their weights values. The obtained matrix W , pruned or not, is then row-normalized.

At each step of the label propagation algorithm, the label of an instance is decided according to the labels of its neighbors. Let us define F m ∈ R n×c such that F m ik is the probability at iteration m for x i to belong to the class C k ; let's define

F 0 ik = 1 if x i ∈ X l ∩ C k 0 otherwise . Let'
F m+1 ic = (W F m ) ic if x i ∈ X u F 0 im if x i ∈ X l .
Let LP (x i ) be the label predicted by the label propagation algorithm for x i ∈ X u . The label is computed as

LP (x i ) = arg max k F ∞ ik .

THEORETICAL GUARANTEES

We claim that under some initial assumptions, the algorithm described in section 2 provides a graph representation of the data that is optimal for classification through the label propagation algorithm.

In the following we substantiate our claim by showing that we can find an value for pruning the complete graph obtained from our data in the projected space in such a way that the resultinggraph is made of c connected components homogeneous in class. We show that we can ensure the existence of such an if we suppose that each testing triplet is in a close neighbourhood of at least one similarly labeled training triplet.

We prove our claim through three main steps. We first state that we can bound the distance between the projection of points depending on their distance in the initial space. In a second step we show that if a triplet is properly projected in the new space, then triplets that are projected nearby are also properly projected. By bringing together the first two steps we are able to show that triplets that are close in the initial space to a triplet that is properly projected is also projected properly.

Knowing that, we can finally exhibit an that allows us to compute an optimal -graph for the label propagation algorithm.

Let the distance be the euclidean distance and our transformation be a multi-layered perceptrons φ, φ : R d → R q , with one hidden layer, as

φ(x) = B tanh( Ãx + α) + β and φ(x) = B tanh(Ax + α) + β, where B, B ∈ R q×p , Ã, A ∈ R p×d , α, α ∈ R p and β, β ∈ R q .
Let's suppose that B, A, α and β are learned, with no error, such that

φ = arg min φ (x,x+,x-)∈Ttrain max [0, 1 -d( φ(x), φ(x -)) + d( φ(x), φ(x + ))]
Based on φ, our first lemma claims that we can easily bound the euclidean distance between the projection of two points by a factor of their initial euclidean distance, where the factor is dependent on A and B. This can easily be done by bounding the application of each transformation of φ.

Our second lemma defines the maximal distance, in the new representation space, between similarly labeled training and testing instances of triplets allowing us to ensure the respect of the relative constraints on the testing triplets. By considering that the distance of the projected testing instances to the projected training instances is lower than a fixed value, we can define the minimal margin needed between the training triplet elements such that the testing triplet respects the relative constraint.

Just by bringing those two elements together, we can then show that there exists a maximum distance in the initial space between similarly labeled training and testing instances such that the projection of testing triplet will respect the relative constraints; this maximum distance is related to the margin of the projection of training triplets.

Let's now consider all our testing instances to be close enough in the initial space from similarly labeled instances. From previous lemmas, we know that all training and testing triplets respect the relative constraints, we can thus define a threshold distinguishing similarly labeled pair of instances from dissimilarly labeled ones. Thus, the -graph obtained based on this threshold now contains only edges between similarly labeled instances. As our graph is only composed of connected components of same classes instances, the label propagation will be optimal.

We also prove that those theoretical element are generalizable for deeper multi-layered perceptrons.

EXPERIMENTS

After proving theoretically the interest of our approach for an ideal learned metric, we experimentally evaluate the performance of our algorithm.

We evaluate our framework and compare it to different algorithms on several artificial data sets, composed of 500 instances, chosen for their increasing degree of complexity. The first artificial data set we will use is the classical circle data set. It is obtained by sampling points on two concentric circles in 2-D, the circle defines the class membership. The three other artificial data sets are developed variants of the circle data set; thus their two first features are generated through the circle data set generative model. perturbedCircle1and perturbedCircle2 data sets both get two additional features, respectively random and from a 2-D gaussian. perturbedCircle3 is complemented twice by 2-D gaussian coordinates, depending on the sign of each of the two first features. Labels of perturbedCircle2 data sets are defined depending on the circle on which instances lie and on the gaussian they are associated to. For the other artificial data sets, the label is only dependent on the first two features, as for the initial circle data set. We also evaluate our framework on the real data sets cancer and vehicule, which are real data sets of same order of complexity than the previous data sets; yet vehicule is composed of 4 classes.

For each data set, we can compute a graph where the weights are computed by the the euclidean distance in, respectively, the initial space and the learned space for LMNN algorithm [START_REF] Kilian | Distance metric learning for large margin nearest neighbor classification[END_REF]), the ExtraTrees feature selection algorithm (Geurts et al. ( 2006)) and our approach2 .

We apply an -simplification on each graph. Our network was trained for the euclidean distance with up to 30% of the possible triplets from the train set. In figure 2, we compare, for each dataset, the classification accuracy of the label propagation algorithm performed on the -graph computed on the different representation spaces, with 20% of training instances. As expected, the euclidean distance and LMNN perform ideally on the circle data set, as the initial space is locally representative of the labelling similarity; however their performance decrease on artifical data sets as the initial description space is getting blurred and noisy, like for perturbedCircle1, perturbedCircle2 and perturbedCircle3 data sets. Feature selection algorithm performs well enough on perturbedCircle1 and perturbedCircle3 data set, where the labels are non-perturbed by the added features; however it does not surpass euclidean distance and LMNN algorithm for other data sets. On those artificial data sets, our algorithm performs either ideally or better than the other algorithms, and was able to learn a more representative projection space.

Concerning the real dataset cancer, the different algorithms perform quit well, broadly similar. Finally, on the vehicule dataset, we can see the interest of learning a projection space to obtain a better label propagation; yet the non-linear projection seems more adapted to the dataset complexity.

CONCLUSION

In this paper, we introduced an algorithm to learn a representation space for dataset that is adapted to a specific task. We defined and proved a first theoretical requirement for our algorithm to be optimal; what have been proved can easily be generalized to multi-layers neural network. Experiments on artificial and real data sets confirmed the relevance of our approach for solving classification task.

  s consider the clamped label propagation algorithm, i.e. the label of training instances are not modified during the different epochs:

FigureFigure 2 :

 2 Figure 1: Trained siamese network

Concepts introduced in the following can be made true for a similarity measure instead, with a simple inversion in the constraints inequality and in the cost function.

Implemented with torch7[START_REF] Collobert | Torch7: A Matlab-like Environment for Machine Learning[END_REF]

ACKNOWLEDGEMENT

This work was partially supported by a grant from CPER Nord-Pas de Calais/FEDER DATA Advanced data science and technologies 2015-2020 and ANRT under the grant number CIFRE N o 2013/0961