Reda Jourani
email: jourani@irit.fr

Khalid Daoudi
email: khalid.daoudi@inria.fr

Régine André-Obrecht

Driss Aboutajdine

Discriminative speaker recognition using Large Margin GMM

Keywords: Large margin training, Gaussian mixture models, discriminative learning, speaker recognition, session variability modeling

Most state-of-the-art speaker recognition systems are based on discriminative learning approaches. On the other hand, generative Gaussian mixture models (GMM) have been widely used in speaker recognition during the last decades. In an earlier work, we proposed an algorithm for discriminative training of GMM with diagonal covariances under a large margin criterion. In this paper, we propose an improvement of this algorithm which has the major advantage of being computationally highly efficient, thus well suited to handle large scale databases. We also develop a new strategy to detect and handle the outliers that occur in the training data. To evaluate the performances of our new algorithm, we carry out full NIST speaker identification and verification tasks using NIST-SRE'2006 data, in a Symmetrical Factor Analysis compensation scheme. The results show that our system significantly outperforms the traditional discriminative Support Vector Machines (SVM) based system of SVM-GMM supervectors, in the two speaker recognition tasks.

Introduction

Generative (or informative) training of Gaussian Mixture Models (GMM) using maximum likelihood estimation and maximum a posteriori estimation (MAP) [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture models[END_REF] has been the paradigm of speaker recognition for many decades. Generative training does not however directly address the classification problem because it uses the intermediate step of modeling system variables, and because classes are modeled separately. For this reason, discriminative training approaches have been an interesting and valuable alternative since they focus on adjusting boundaries between classes [START_REF] Keshet | Automatic speech and speaker recognition: Large margin and kernel methods[END_REF][START_REF] Louradour | Feature space mahalanobis sequence kernels: Application to svm speaker verification[END_REF], and lead generally to better performances than generative methods. Hybrid learning approaches have also gained a big interest. For instance, Support Vector Machines (SVM) combined with GMM supervectors are among state-of-the-art approaches in speaker verification [START_REF] Campbell | Support vector machines using GMM supervectors for speaker verification[END_REF][START_REF] Campbell | SVM based speaker verification using a GMM supervector kernel and NAP variability compensation[END_REF].

In speaker recognition applications, mismatch between the training and testing conditions can decrease considerably the performances. The session variability remains the most challenging problem to solve. The Factor Analysis techniques [START_REF] Kenny | Eigenvoice modeling with sparse training data[END_REF][START_REF] Kenny | Speaker and session variability in GMM-based speaker verification[END_REF], e.g., Symmetrical Factor Analysis (SFA) [START_REF] Matrouf | A straightforward and efficient implementation of the factor analysis model for speaker verification[END_REF][START_REF] Fauve | State-of-the-Art Performance in Text-Independent Speaker Verification through Open-Source Software[END_REF], were proposed to address that problem in GMM based systems. While the Nuisance Attribute Projection (NAP) [START_REF] Solomonoff | Advances in Channel Compensation for SVM Speaker Recognition[END_REF] compensation technique is designed for SVM based systems.

Recently a new discriminative approach for multiway classification has been proposed, the Large Margin Gaussian mixture models (LM-GMM) [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF]. The latter have the same advantage as SVM in term of the convexity of the optimization problem to solve. However they differ from SVM because they draw nonlinear class boundaries directly in the input space, and thus no kernel trick/matrix is required. While LM-GMM have been used in speech recognition, they have not been used in speaker recognition (to the best of our knowledge). In an earlier work [START_REF] Jourani | Large Margin Gaussian mixture models for speaker identification[END_REF], we proposed a simplified version of LM-GMM which exploit the fact that traditional GMM systems use diagonal covariances and only the mean vectors are MAP adapted. We then applied this simplified version to a "small" speaker identification task. While the resulting training algorithm is more efficient than the original one, we found however that it is still not efficient enough to process large databases such as in NIST Speaker Recognition Evaluation (NIST-SRE) campaigns [13].

In order to address this problem, we propose in this paper a new approach for fast training of Large-Margin GMM which allow efficient processing in large scale applications. To do so, we exploit the fact that in general not all the components of the GMM are involved in the decision process, but only the k-best scoring components. We also exploit the property of correspondence between the MAP adapted GMM mixtures and the world model mixtures. Moreover, we develop a new strategy to detect outliers and reduce their negative effect in training. This strategy leads to a further improvement in performances.

In order to show the effectiveness of the new algorithm, we carry out full NIST speaker identification and verification tasks using NIST-SRE'2006 (core condition) data. We evaluate our fast algorithm in a Symmetrical Factor Anal-ysis compensation scheme, and we compare it with the NAP compensated GMM supervector Linear Kernel system (GSL-NAP) [START_REF] Campbell | SVM based speaker verification using a GMM supervector kernel and NAP variability compensation[END_REF]. The results show that our Large Margin compensated GMM outperform the state-of-the-art discriminative approach GSL-NAP, in the two speaker recognition tasks.

The paper is organized as follows. After an overview on Large-Margin GMM training with diagonal covariances in section 2, we describe our new fast training algorithm in section 3. To make the paper self-contained, the GSL-NAP system and SFA are described in sections 4 and 5, respectively. Experimental results are reported in section 6.

2 Overview on Large Margin GMM with diagonal covariances (LM-dGMM)

In this section we start by recalling the original Large Margin GMM training algorithm developed in [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF][START_REF] Sha | Large margin training of acoustic models for speech recognition[END_REF]. We then recall the simplified version of this algorithm that we introduced in [START_REF] Jourani | Large Margin Gaussian mixture models for speaker identification[END_REF].

In Large Margin GMM [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF][START_REF] Sha | Large margin training of acoustic models for speech recognition[END_REF], each class c is modeled by a mixture of ellipsoids in the D-dimensional input space. The m th ellipsoid of the class c is parameterized by a centroid vector µ cm (mean vector), a positive semidefinite (orientation) matrix Ψ cm and a nonnegative scalar offset θ cm ≥ 0. These parameters are then collected into a single enlarged matrix Φ cm :

Φ cm = Ψ cm -Ψ cm µ cm -µ T cm Ψ cm µ T cm Ψ cm µ cm + θ cm . ( 1
)
A GMM is first fit to each class using maximum likelihood estimation. Let {o n,t } Tn t=1 (o n,t ∈ R D ) be the T n feature vectors of the n th segment (i.e. n th speaker training data). Then, for each o n,t belonging to the class y n , y n ∈ {1, 2, ..., C} where C is the total number of classes, we determine the index m n,t of the Gaussian component of the GMM modeling the class y n which has the highest posterior probability. This index is called proxy label.

The training algorithm aims to find matrices Φ cm such that "all" examples are correctly classified by at least one margin unit, leading to the LM-GMM criterion:

∀c = y n , ∀m, z T n,t Φ cm z n,t ≥ 1 + z T n,t Φ ynmn,t z n,t , (2)
where z n,t = o n,t 1 . Eq. ( 2) states that for each competing class c = y n the match (in term of Mahalanobis distance) of any centroid in class c is worse than the target centroid by a margin of at least one unit.

In speaker recognition, most of state-of-the art systems use diagonal covariances GMM. In these GMM based speaker recognition systems, a speakerindependent world model or Universal Background Model (UBM) is first trained with the EM algorithm [START_REF] Bishop | Pattern recognition and machine learning[END_REF] from tens or hundreds of hours of speech data gathered from a large number of speakers. The background model represents speaker-independent distribution of the feature vectors. When enrolling a new speaker to the system, the parameters of the UBM are adapted to the feature distribution of the new speaker. It is possible to adapt all the parameters, or only some of them from the background model. Traditionally, in the GMM-UBM approach, the target speaker GMM is derived from the UBM model by updating only the mean parameters using a maximum a posteriori (MAP) algorithm [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture models[END_REF], while the (diagonal) covariances and the weights remain unchanged.

Making use of this assumption of diagonal covariances, we proposed in [START_REF] Jourani | Large Margin Gaussian mixture models for speaker identification[END_REF] a simplified algorithm to learn GMM with a large margin criterion. This algorithm has the advantage of being more efficient than the original LM-GMM one [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF][START_REF] Sha | Large margin training of acoustic models for speech recognition[END_REF] while it still yielded similar or better performances on a speaker identification task. In our Large Margin diagonal GMM (LM-dGMM) [START_REF] Jourani | Large Margin Gaussian mixture models for speaker identification[END_REF], each class (speaker) c is initially modeled by a GMM with M diagonal mixtures (trained by MAP adaptation of the UBM in the setting of speaker recognition). For each class c, the m th Gaussian is parameterized by a mean vector µ cm , a diagonal covariance matrix Σ m = diag(σ 2 m1 , ..., σ 2 mD ), and the scalar factor θ m which corresponds to the weight of the Gaussian.

With this relaxation on the covariance matrices, for each example o n,t , the goal of the training algorithm is now to force the log-likelihood of its proxy label Gaussian m n,t to be at least one unit greater than the log-likelihood of each Gaussian component of all competing classes. That is, given the training examples {(o n,t , y n , m n,t )} N n=1 , we seek mean vectors µ cm which satisfy the LM-dGMM criterion:

∀c = y n , ∀m, d(o n,t , µ cm ) + θ m ≥ 1 + d(o n,t , µ ynmn,t ) + θ mn,t , (3)
where

d(o n,t , µ cm ) = D i=1 (o n,ti -µ cmi ) 2 2σ 2 mi .
Afterward, these M constraints are fold into a single one using the softmax inequality min m a m ≥log m exp(-a m ). The segment-based LM-dGMM criterion becomes thus:

∀c = y n , 1 T n Tn t=1 -log M m=1 exp(-d(o n,t , µ cm ) -θ m ) ≥ 1 + 1 T n Tn t=1 d(o n,t , µ ynmn,t ) + θ mn,t . (4)
The loss function to minimize for LM-dGMM is then given by:

L = N n=1 c =yn max 0 , 1 + 1 T n Tn t=1 d(o n,t , µ ynmn,t ) + θ mn,t + log M m=1 exp(-d(o n,t , µ cm ) -θ m ) .
(

) 5
3 LM-dGMM training with k-best Gaussians

Description of the new LM-dGMM modeling

Despite the fact that our LM-dGMM is computationally much faster than the original LM-GMM of [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF][START_REF] Sha | Large margin training of acoustic models for speech recognition[END_REF], we still encountered efficiency problems when dealing with high number of Gaussian mixtures. Indeed, even for an easy 50 speakers identification task as the one presented in [START_REF] Jourani | Large Margin Gaussian mixture models for speaker identification[END_REF], we could not run the training in a relatively short time with our current implementation. This would imply that large scale applications such as NIST-SRE, where hundreds or thousands of target speakers are available, would be infeasible in reasonable time (for instance, 5460 target speakers are included in the NIST-SRE'2010 core condition, with 610748 trials to process involving 13325 test segments [START_REF]The NIST Year 2010 Speaker Recognition Evaluation Plan[END_REF] ).

In order to develop a fast training algorithm which could be used in large scale applications, we propose to drastically reduce the number of constraints to satisfy in Eq. ( 4). By doing so, we would drastically reduce the computational complexity of the loss function and its gradient, which are the quantities responsible for most of the computational time. To achieve this goal we propose to use another property of state-of-the-art GMM systems, that is, decision is not made upon all mixture components but only using the k-best scoring Gaussians.

In other words, for each o n and each class c, instead of summing over the M mixtures in the left side of equation Eq. ( 4), we would sum only over the k Gaussians with the highest posterior probabilities selected using the GMM of class c. In order to further improve efficiency and reduce memory requirement, we exploit the property reported in [START_REF] Reynolds | Speaker verification using adapted Gaussian mixture models[END_REF] about correspondence between MAP adapted GMM mixtures and UBM mixtures. We use the UBM to select one unique set S n,t of k-best Gaussian components per frame o n,t , instead of (C-1) sets. This leads to a (C -1) times faster and less memory consuming selection. Thus, the higher the number of target speakers is, the greater computation and memory saving is. More precisely, we now seek mean vectors µ cm that satisfy the large margin constraints in Eq. ( 6):

∀c = y n , 1 T n Tn t=1 -log m∈Sn,t exp(-d(o n,t , µ cm ) -θ m ) ≥ 1 + 1 T n Tn t=1 d(o n,t , µ ynmn,t ) + θ mn,t . (6)
The loss function becomes:

L = N n=1 c =yn max 0 , 1 + 1 T n Tn t=1 d(o n,t , µ ynmn,t ) + θ mn,t + log m∈Sn,t exp(-d(o n,t , µ cm ) -θ m ) . (7)
This loss function remains convex and can still be solved using dynamic programming.

Handling of outliers

We have adopted in our previous work [START_REF] Daoudi | Speaker Identification Using Discriminative Learning of Large Margin GMM[END_REF] the strategy of [START_REF] Sha | Large margin Gaussian mixture modeling for phonetic classification and recognition[END_REF] to detect outliers and reduce their negative effect on learning. Outliers are detected using the initial GMM models. The original strategy consists on computing the accumulated hinge loss incurred by violations of the large margin constraints in Eq. ( 6):

h n = c =yn max 0 , 1 + 1 T n Tn t=1 d(o n,t , µ ynmn,t ) + θ mn,t + log m∈Sn,t exp(-d(o n,t , µ cm ) -θ m ) , (8)
and then re-weighting1 the hinge loss terms in Eq. ( 7) by using segment weights

s n = min 1, 1 h n .
We propose in this paper a novel and better strategy that outperforms the previous one. We keep the global large margin constraints segmental, but we will apply now a frame (feature vectors) weighting scheme. For each feature vector o n,t , we calculate (C -1) weights s c n,t relative to each class c = y n . for each o n,t and each competing class c, we compute the loss incurred by violations of the large margin constraints:

h c n,t = 1 + d(o n,t , µ ynmn,t ) + θ mn,t + log m∈Sn,t exp -d(o n,t , µ cm ) -θ m T n . (9) h c
n,t measures the decrease in the loss function when an initially misclassified feature vector is corrected during the course of learning. We associate outliers with values of h c n,t > 1, and in this case we multiply this term by the frame weight s c n,t = 1 h c n,t

. The new loss function becomes thus:

L = N n=1 c =yn max 0 , Tn t=1 s c n,t h c n,t . (10)
We solve this unconstrained non-linear optimization problem using the second order optimizer LBFGS [START_REF] Nocedal | Numerical optimization[END_REF]. In summary, our new and fast training algorithm of LM-dGMM is the following:

-For each class (speaker), initialize with the GMM trained by MAP adaptation of the UBM, -select Proxy labels using these GMM, -select the set of k-best UBM Gaussian components for each training frame, -compute the point weights s c n,t , -using the LBFGS algorithm, solve the unconstrained non-linear minimization problem:

min L. ( 11
)

Evaluation phase

During test, we use the same principle as in the training to achieve fast scoring. Given a test segment of T frames, for each test frame o t we use the UBM to select the set E t of k-best scoring proxy labels.

In an identification task, we compute the LM-dGMM likelihoods using only these k labels. The decision rule is thus given as:

y = argmin c T t=1 -log m∈Et exp(-d(o t , µ cm ) -θ m ) . (12)
In a verification task, we compute a match score depending on both the target model {µ cm , Σ m , θ m } and the UBM {µ U m , Σ m , θ m } for the test hypothesis (trial). The average log likelihood ratio is calculated using only the k labels:

LLR avg = 1 T T t=1 log m∈Et exp(-d(o t , µ cm ) -θ m ) -log m∈Et exp(-d(o t , µ U m ) -θ m ) . (13)
This quantity provides a score for the test segment to be uttered by the target model/speaker c.

The GSL-NAP system

In this section we briefly describe the GMM supervector linear kernel SVM system (GSL) [START_REF] Campbell | Support vector machines using GMM supervectors for speaker verification[END_REF] and its associated channel compensation technique, the Nuisance attribute projection (NAP) [START_REF] Solomonoff | Advances in Channel Compensation for SVM Speaker Recognition[END_REF].

SVM-GMM supervector

Given an M -components GMM adapted by MAP from the UBM, one forms a GMM supervector by stacking the D-dimensional mean vectors, leading to an M D supervector. This GMM supervector can be seen as a mapping of variablelength utterances into a fixed-length high-dimensional vector, through GMM modeling:

φ(x) =    µ x1 . . . µ xM    , (14)
where the GMM {µ xm , Σ m , w m } is trained on the utterance x.

For two utterances x and y, the Kullback-Leibler divergence kernel is defined as:

K(x, y) = M m=1 √ w m Σ -(1/2) m µ xm T √ w m Σ -(1/2) m µ ym . (15)
The UBM weight and variance parameters are used to normalize the Gaussian means before feeding them into a linear kernel SVM training. This system is referred to as GSL in the rest of the paper.

Nuisance attribute projection (NAP)

NAP is a pre-processing method that aims to compensate the supervectors by removing the directions of undesired sessions variability, before the SVM training [START_REF] Solomonoff | Advances in Channel Compensation for SVM Speaker Recognition[END_REF]. NAP transforms a supervector φ to a compensated supervector φ: φ = φ -S(S T φ),

using the eigenchannel matrix S, which is trained using several recordings (sessions) of various speakers.

In the following, (h, s) will indicate the session h of the speaker s. Given a set of expanded recordings:

{φ(1, s 1 ) • • • φ(h 1 , s 1 ) • • • φ(1, s N ) • • • φ(h N , s N )}, (17)
of N different speakers, with h i different sessions for each speaker s i , one first removes the speakers variability by subtracting the mean of the supervectors within each speaker {φ(s i )}:

∀s i , ∀h, φ(h, s i ) = φ(h, s i ) -φ(s i ). ( 18
)
The resulting supervectors are then pooled into a single matrix:

C = φ(1, s 1 ) • • • φ(h 1 , s 1 ) • • • φ(1, s N ) • • • φ(h N , s N ) , (19)
representing the intersession variations. One identifies finally the subspace of dimension R where the variations are the largest by solving the eigenvalue problem on the covariance matrix CC T , getting thus the projection matrix S of a size M D × R. This system is referred to as GSL-NAP in the rest of the paper.

5 Symmetrical Factor Analysis (SFA)

In this section we describe the symmetrical variant of the Factor Analysis model (SFA) [START_REF] Matrouf | A straightforward and efficient implementation of the factor analysis model for speaker verification[END_REF][START_REF] Fauve | State-of-the-Art Performance in Text-Independent Speaker Verification through Open-Source Software[END_REF] (Factor Analysis was originally proposed in [START_REF] Kenny | Eigenvoice modeling with sparse training data[END_REF][START_REF] Kenny | Speaker and session variability in GMM-based speaker verification[END_REF]). In the mean supervector space, a speaker model can be decomposed into three different components:

a session-speaker independent component (the UBM model), -a speaker dependent component, -a session dependent component.

The session-speaker model, can be written as [START_REF] Matrouf | A straightforward and efficient implementation of the factor analysis model for speaker verification[END_REF]:

M (h,s) = M + Dy s + Ux (h,s) , (20)
where -M (h,s) is the session-speaker dependent supervector mean (an M D vector), -M is the UBM supervector mean (an M D vector), -D is a M D × M D diagonal matrix, where DD T represents the a priori covariance matrix of y s , -y s is the speaker vector (speaker offset), an M D vector assumed to follow a standard normal distribution N (0, I), -U is the session variability matrix of low rank R (an M D × R matrix), -x (h,s) are the channel factors (session offset), an R vector (theoretically, not dependent on s) assumed to follow a standard normal distribution N (0, I).

Dy s and Ux (h,s) represent respectively the speaker dependent component and the session dependent component [START_REF] Fauve | State-of-the-Art Performance in Text-Independent Speaker Verification through Open-Source Software[END_REF].

The factor analysis modeling starts by estimating the U matrix, using different recordings per speaker. The matrix U is theoretically similar to the channel matrix S of NAP, and it also requires many recordings to identify accurately the subspace where intersession variability is high. However, the matrix U estimation is computationally less efficient than the matrix S one. Given the fixed parameters (M, D, U), the target models are then compensated by eliminating the session mismatch directly in the model domain. Whereas, the compensation in the test is performed at the frame level (feature domain).

We perform experiments on the NIST-SRE '2006 [19] speaker identification and verification tasks and compare the performances of the baseline GMM, the LM-dGMM and the SVM systems, with and without using channel compensation techniques. The comparisons are made on the male part of the NIST-SRE'2006 core condition (1conv4w-1conv4w). In the identification task, performances are measured in term of the speaker identification rate. In the verification task, they are assessed using Detection Error Tradeoff (DET) plots and measured in terms of equal error rate (EER) and minimum of detection cost function (minDCF) which is calculated following NIST criteria [START_REF] Przybocki | NIST Speaker Recognition Evaluation Chronicles[END_REF].

For front-end processing, we follow the same procedure as in [START_REF] Fauve | State-of-the-Art Performance in Text-Independent Speaker Verification through Open-Source Software[END_REF]. The feature extraction is carried out by the filter-bank based cepstral analysis tool Spro [21]. Bandwidth is limited to the 300-3400Hz range. 24 filter bank coefficients are first computed over 20ms Hamming windowed frames at a 10ms frame rate and transformed into Linear Frequency Cepstral Coefficients (LFCC) [START_REF] Davis | Comparison of parametric representations for monosyllabic word recognition in continuously spoken sentences[END_REF]. Consequently, the feature vector is composed of 50 coefficients including 19 LFCC, their first derivatives, their 11 first second derivatives and the delta-energy. The LFCCs are preprocessed by Cepstral Mean Subtraction and variance normalization [START_REF] Viikki | Cepstral domain segmental feature vector normalization for noise robust speech recognition[END_REF]. We applied an energy-based voice activity detection to remove silence frames, hence keeping only the most informative frames. Finally, the remaining parameter vectors are normalized to fit a zero mean and unit variance distribution.

We use the state-of-the-art open source software ALIZE/Spkdet [START_REF] Fauve | State-of-the-Art Performance in Text-Independent Speaker Verification through Open-Source Software[END_REF][START_REF] Bonastre | ALIZE/SpkDet: a state-of-the-art open source software for speaker recognition[END_REF] for GMM, SFA, GSL and GSL-NAP modeling. A male-dependent UBM is trained using all the telephone data from the NIST-SRE'2004. Then we train a MAP adapted GMM for the 349 target speakers belonging to the primary task. The identification is made on a list of 539554 trials (involving 1546 test segments), whereas the verification task uses a shorter list of 22123 trials (involving 1601 test segments) for test. Score normalization techniques are not used in our experiments. The so MAP adapted GMM define the baseline GMM system, and are used as initialization for the LM-dGMM one. The GSL system uses a list of 200 impostor speakers from the NIST-SRE'2004, on the SVM training. The LM-dGMM-SFA system is initialized by model domain compensated GMM, which are then discriminated using feature domain compensated data. The session variability matrix U of SFA and the channel matrix S of NAP, both of rank R = 40, are estimated on NIST-SRE'2004 data using 2934 utterances of 124 different male speakers. Table 1 presents the speaker identification accuracy scores of the various systems. Table 2 presents the speaker verification scores (EER and minDCF). We show performances using GMMs with 256 and 512 Gaussian components (M = 256, 512). All the scores are obtained with the 10 best proxy labels selected using the UBM, k = 10. The actual large margin systems adopt a segmental weighting approach.

The results of Table 1 and Table 2 show that, without SFA channel compensation, the LM-dGMM system outperforms the classical generative GMM one, however it does yield worse performances than the discriminative approach GSL. Nonetheless, when applying channel compensation techniques, compensated models outperform the non-compensated ones as expected, but the LM-dGMM-SFA system significantly outperforms the GSL-NAP and GMM-SFA ones in the two tasks. Our best system achieves 91.27% speaker identification rate, while the best GSL-NAP achieves 87.77%. This leads to a 3.5% improvement. In verification, the LM-dGMM-SFA and GSL-NAP achieve respectively 5.02% and 5.90% equal error rates, and 2.18 * 10 -2 and 2.73 * 10 -2 minDCF values. This shows that LM-dGMM-SFA yields relative reductions of EER and minDCF of about 14.92% and 20.15% over the GSL-NAP system. Moreover, The performances of the GMM-SFA system show that LM-dGMM-SFA yields relative reductions of speaker identification rate and EER of about 0.57% and 9.22% over this system. It is known that SVM performances are sensitive to the C parameter. We have thus used different values of C and reported the best scores of the SVM systems in Table 1 andTable 2. This can be seen in Table 3 and Table 4 which show the scores obtained using different values of C for GSL and GSL-NAP with M = 256. We also report in these tables the average number of support vectors.

Figure 1 displays the EER and minDCF performances of all systems, with and without channel compensation, for models with 512 Gaussian components (M = 512). Figure 2 shows DET plots for LM-dGMM and GSL systems with and without channel compensation, for models with 512 Gaussian components. One can see that LM-dGMM-SFA outperforms GSL and GSL-NAP at all operating points.

Table 5 gives the EER scores of LM-dGMM and LM-dGMM-SFA systems using the two weighting strategies, for models with 512 Gaussian components. One can see that the frame weighting approach further improves the LM-dGMM (+SFA) performance. All these results show that our fast Large Margin GMM discriminative learning algorithm not only allows efficient training but also achieves better speaker recognition (identification and verification) performances than a state-of-the-art discriminative technique.

Conclusion

We proposed a new algorithm for discriminative learning of diagonal GMM under a Large-Margin criterion. Our algorithm is highly efficient which makes it well suited to process large scale databases such as in NIST-SRE campaigns. We also developed a frame weighting strategy to detect and handle outliers in training data. This strategy yields further improvement in performances. We carried out experiments on full speaker identification and verification tasks under the NIST-SRE'2006 core condition. Combined with the SFA channel compensation technique, the resulting algorithm significantly outperforms the state-of-the-art speaker recognition discriminative approach GSL-NAP. An-other major advantage of our method is that it outputs diagonal GMM models. Thus, broadly used GMM techniques/softwares such as SFA or ALIZE/Spkdet can be readily applied in our framework. Our future work will consist in improving margin selection. Like in SVM, this should indeed significantly improve the performances. We emphasize also that, while we have applied our algorithm to speaker recognition, it can be actually applied in any other classification task which involves supervised learning of diagonal GMM.

Fig. 1

 1 Fig.1EER and minDCF performances for GMM, LM-dGMM and GSL systems with and without channel compensation

Fig. 2

 2 Fig.2DET plots for LM-dGMM and GSL systems with and without channel compensation

Table 1

 1 Speaker identification rates with GMM, Large Margin diagonal GMM and GSL models, with and without channel compensation

		System		Speaker identification rate 256 Gaussians 512 Gaussians
		GMM		75.87%	77.88%
		LM-dGMM	77.62%	78.40%
		GSL		81.50%	82.21%
		GSL-NAP		87.26%	87.77%
		GMM-SFA	89.26%	90.75%
		LM-dGMM-SFA	89.65%	91.27%
	Table 2 EERs(%) and minDCFs(x100) of GMM, Large Margin diagonal GMM and GSL
	systems with and without channel compensation
	System	256 Gaussians EER minDCF(x100)	512 Gaussians EER minDCF(x100)
	GMM	9.43%	4.26	9.74%	4.18
	LM-dGMM	8.97%	3.97	9.66%	4.12
	GSL	7.39%	3.41	7.23%	3.44
	GSL-NAP	6.40%	2.72	5.90%	2.73
	GMM-SFA	6.15%	2.41	5.53%	2.18
	LM-dGMM-SFA 5.58%	2.29	5.02%	2.18
	Table 3 GSL performance using different values of C and average number of support vectors
	(M = 256)
	C	Number of support vectors	Identification rate	EER	minDCF(x100)
	2 -4	46		78.33%	7.81%	3.71
	2 -3	49		78.20%	7.85%	3.72
	2 -2	51		78.20%	7.85%	3.72
	2 -1	52		78.20%	7.83%	3.72
	2 0	52		81.50%	7.40%	3.41
	2 1	52		81.50%	7.39%	3.41
	2 2	52		81.50%	7.39%	3.41
	2 3	52		81.50%	7.39%	3.41
	2 4	52		81.50%	7.40%	3.41
	Table 4 GSL-NAP performance using different values of C and average number of support
	vectors (M = 256)
	C	Number of support vectors	Identification rate	EER	minDCF(x100)
	2 -4	63		84.22%	6.77%	2.99
	2 -3	70		84.15%	6.77%	3.00
	2 -2	75		84.09%	6.78%	2.98
	2 -1	77		84.15%	6.80%	2.98
	2 0	78		87.26%	6.40%	2.72
	2 1	78		87.19%	6.44%	2.71
	2 2	78		87.19%	6.44%	2.71
	2 3	78		87.19%	6.44%	2.71
	2 4	78		87.19%	6.44%	2.71

Table 5

 5 Segmental weighting strategy vs frame weighting strategy

	System		EER
	Segmental weighting	LM-dGMM LM-dGMM-SFA	9.66% 5.02%
	Frame weighting	LM-dGMM LM-dGMM-SFA 4.89% 9.47%

Note that by setting the segment weights to one, i.e., no handling of outliers is done, the experiments show that the performances degrade.

Acknowledgements The authors would like to thank the anonymous reviewers for their helpful comments.


Claudia Toma
email: claudia.toma@uclouvain.be

Vincent Yzerbyt

Olivier Corneille

Cooperation, Competition and Interpersonal Projection 1 Anticipated Cooperation vs. Competition Moderates Interpersonal Projection

Keywords: interpersonal projection, interdependence, self-representation, multi-level analysis Cooperation, Competition and Interpersonal Projection 2 Cooperation, Competition and Interpersonal Projection 3 Anticipated Cooperation vs. Competition Moderates Interpersonal Projection

Two experiments investigated the impact of anticipated interdependence on people's projection of their characteristics onto an unknown target. After participants had rated themselves on a list of personality traits, they were led to expect a situation of cooperation or competition with another participant and rated this participant on the same list of traits. In both experiments, projection of self-attributed traits was stronger under cooperation than competition. This effect was independent of trait valence, whether defined a priori (Expt 1) or as an idiosyncratic measure (Expt 2). Experiment 2 also revealed that the moderation of interpersonal projection by interdependence was not driven by changes in participants' self-representation. These findings suggest that the anticipated interdependence context influences the way we perceive similarity with unknown others. We discuss possible cognitive and motivational mechanisms underlying this effect.

that can heavily bias social judgments about others.

Social projection has received considerable attention over the last decade. Yet, its boundary conditions have been relatively understudied. Most of the research examined the role of the target characteristics as a moderator of social projection. For instance, it has been shown that projection is stronger for ingroup than for outgroup members [START_REF] Cadinu | Self-anchoring and differentiation processes in the minimal group setting[END_REF][START_REF] Clement | Social categorization moderates social projection[END_REF][START_REF] Krueger | Social categorization and the truly false consensus[END_REF] and for targets already sharing a certain degree of similarity (see the similarity contingency model of social inference: Ames, 2004a[START_REF] Ames | Strategies for social inference: A similarity contingency model of projection and stereotyping in attitude prevalence estimates[END_REF]. The current research moves beyond the traditional focus on target characteristics to develop our understanding about the role of self-other interdependence in social projection.

More specifically, we predicted social projection to be stronger under conditions of interpersonal cooperation than competition. The latter hypothesis received indirect support in a recent intergroup study conducted by Riketta and Sacrameto (2008). These authors showed that people are more likely to project their personality traits toward an outgroup that is cooperating rather than competing with the ingroup. However, we do not know yet if the same effect would be observed when anticipated interdependence is manipulated at the interpersonal rather than intergroup level. First, research on the discontinuity effect (e.g., Schopler & Insko, 1992) invites caution about equating intergroup and interpersonal effects, especially so when competitive relationships are involved. Secondly, and perhaps even more importantly, people are known to anticipate more variability among individuals than among groups (e.g., [START_REF] Gidron | Implicit quantification of personality traits[END_REF] and this may constrain social projective effects when they are observed at the interpersonal level. Third, intergroup projection is likely to involve judgments about the ingroup as an intermediate step between self and outgroup judgments. There is little reason to expect such an intermediate judgment to play a role in the context of interpersonal projection. This being said, there are also reasons to believe that interdependence may play a role in interpersonal projection. Research has documented that whether one cooperates or competes with a target person is an important determinant of the direction of comparison and social judgments. Comparison is a key mechanism in social judgment and classically involves one of two processes: assimilation or contrast (see [START_REF] Biernat | Simultaneous assimilation and contrast effects in judgments of self and other[END_REF]. [START_REF] Stapel | Cooperation, competition and the effects of others on me[END_REF] have shown that, when confronted with a relevant comparison target, participants assimilate their self-perception toward the target when expecting cooperation but contrast their self-perception away from the target when expecting competition. In line with this line of work, we expected more projection under conditions of cooperation than competition. Of note, however, past research took Cooperation, Competition and Interpersonal Projection 5 the other person as the point of reference. The focus was thus the distance of the self from the target person. In the present work, and building upon the tradition of work on social projection, our prediction concerned the extent to which participants would see the other person as being similar to them.

The egocentric comparison model of social prediction [START_REF] Mussweiler | When egocentrism breeds distinctness-comparison processes in social prediction: Comment on Karniol (2003)[END_REF] is also relevant to our current hypothesis. This model proposes that, in order to predict the behavior and the personality traits of others, people relate self-knowledge to the target through a comparison process. Two processes should be distinguished here as a function of the nature of the interdependence. In a context of cooperation, people may engage a process of similarity testing, yielding target judgments that are consistent with the self.

In a context of competition, people may rely on a process of dissimilarity testing, leading to target judgments that are inconsistent with the self. Similar to other models of social judgment [START_REF] Dunning | Evidence of egocentric comparison in social judgment[END_REF][START_REF] Smith | Exemplar-based model of social judgment[END_REF], this model conceptualizes social projection as a flexible comparison process in which the self is the primary representational basis for projection, even under conditions of distinctiveness (e.g., competition).

Overview of the studies

In the present research, we conducted two studies that manipulated anticipated interdependence (cooperation vs. competition) between the self and a totally unknown target person and we examined people's projection of their personality traits onto the target. As already noted, we predicted more social projection under conditions of interpersonal cooperation than competition. We also extended prior work on the role of interdependence in social judgments in four significant ways: First, we asked participants to judge a target person on whom they received no information. Although social projection is generally conceptualized as a comparison process, we believe that projection may also occur when no comparison information is provided about a target. As a matter of fact, it may actually be even easier for people to project their own traits onto an unknown than a known target. Of particular relevance here is [START_REF] Krueger | From social projection to social behavior[END_REF] suggestion that social projection corresponds to a heuristic use of the self in conditions of judgment uncertainty. Clearly enough, judgments should be more uncertain when no information at all is conveyed about a target. Hence, it remains to be explained why do we expect more social projection when cooperating than when competing with an unknown target? One possibility here is that people are more inclined to rely on a heuristic processing when facing a non-problematic (i.e., cooperative) than a problematic (i.e., competitive) situation. If so, social projection may be facilitated in cooperation, and this should occur even when no comparative information is communicated about the target. Another possibility is that people expect to benefit more from increased self-other similarity in situations of cooperation than competition. We will come back to these two accounts in the general discussion.

Besides the absence of comparative information about the target, a second distinctive feature of the current experiments is that we asked participants to ascribe not only positive but also negative personality traits onto the target person [START_REF] Stathi | Imagining intergroup contact promotes projection to outgroups[END_REF]. Because general evaluative tendencies can produce positive or negative correlation between trait ratings independent of social projection [START_REF] Otten | Self-anchoring and ingroup favoritism: An individualprofiles analysis[END_REF], we controlled for valence in these studies. If social projection is found to be stronger in cooperation than competition, then this effect would not be confounded with Cooperation, Competition and Interpersonal Projection 7 evaluative tendencies. The valence of traits was defined a priori in Experiment 1, and we secured idiosyncratic measures of valence in Experiment 2.

A third originality of the current experiments is that the personality traits that participants were asked to ascribe to their self and to the targets were selected so as to be unrelated to competition and cooperation. This ensured that our findings would not be due to naïve theories linking cooperative and competitive settings to specific behavioral tendencies (e.g., being aggressive in a competitive context).

Finally, the second experiment reported here addresses the possibility that increased social projection under conditions of cooperation results from a change in selfrepresentation after participants learned about the nature of the interdependence between them and the target.

Experiment 1

Method

Participants and Design

A total of 47 participants (22 males) were approached on campus. They were invited to take part in a study on spontaneous impression formation in exchange for 3 EUR, and randomly assigned to a cooperation or competition condition.

Procedure

On the first page of the questionnaire, participants rated themselves on a list of 16 personality traits (8 positive and 8 negative). These traits were borrowed from [START_REF] Riketta | They cooperate with us, so they are like me': Perceived intergroup relationship moderates projection from self to outgroups[END_REF] who pretested them to be unrelated to cooperation and competition. Participants had to indicate the extent to which each of the traits (e.g., progressive, silent, creative) characterized them on a 9-point rating scale ranging from 1 (= not at all) to 9 (= very much).

On the second page, participants read a scenario in which they had to imagine being employed in a new software company where they would have to team up with an unknown Person A (the target). They learned that the top manager of this company would offer a bonus trip to the Caribbean if they managed to sell more than 10,000 copies of computer software during the first 6 months. The relation with the target was presented as either cooperative or competitive. In the cooperation condition, participants were told that it was possible that both they and the person A win the trip, so that they should help each other in order to sell, together, the 10,000 copies of the computer software. In the competition condition, participants were told that only one of them could win the trip, so that they had to outperform person A in being the first to sell the 10,000 copies of the computer software.

On the third page of their questionnaire, participants were asked to rate the target on the same list of traits that they had used to rate the self. Finally, as a manipulation check, participants judged if the relationship between themselves and the target was cooperative or competitive. Seven participants who were wrong were excluded from analyses. When all tasks had been completed, participants were debriefed, thanked, and dismissed.

Results

Because the ratings of personality traits were nested within participants the data were analyzed by means of multilevel modeling. The traits were our level-1 variable

ACCEPTED MANUSCRIPT

Cooperation, Competition and Interpersonal Projection 9 whereas condition was our level-2 variable. The analysis used the ratings of the target as our criterion and self-ratings and valence of traits as predictors at level 1. In Experiment 1, the valence of traits was coded positive or negative on the basis of [START_REF] Riketta | They cooperate with us, so they are like me': Perceived intergroup relationship moderates projection from self to outgroups[END_REF] study. The anticipated interdependence (cooperation vs. competition) was included as a moderator variable at level 2.

Our hypothesis is that the variation of the level-1 slopes (i.e. the relation between self-and target ratings) is influenced by our level-2 variable (anticipated interdependence). In other words, we predicted the presence of a significant cross-level interaction between self-ratings and the anticipated interdependence. To test our prediction, we implemented the following models:

Level-1 Model:

Target = P0 + P1xSelf + P2xValence + E Level-2 Model: P0 = B00 + B01xInterdependence + R0 P1 = B10 + B11xInterdependence + R1 P2 = B20 + R2
with P0, B00, B10, B20 as intercepts; P1, P2, B01, B11 as regression weights; and E, R0, R1, R2 as residuals. Condition was coded -1 for competition and +1 for cooperation. We coded valence -1 for negative traits and +1 for positive traits. B01 refers to the extent to which the target is judged differently in cooperation than in competition. B10 denotes the extent to which the self is used to judge the target. The critical parameter for our hypothesis is B11 because it denotes the extent to which the tendency of the self-ratings to predict the target ratings (i.e. P1) varies as a function of anticipated interdependence.

ACCEPTED MANUSCRIPT

Cooperation, Competition and Interpersonal Projection 10 B20 refers to the extent to which the valence affects target ratings. Self-ratings were centered at the mean of each participant's ratings. It is important to note that parameters in this model (especially B10 and B11) reflect self-target covariance, not correspondence.

The method of estimation is restricted maximum likelihood and the covariance matrix is unstructured. This also applies to the subsequent models.

As predicted, the relation between self-and target-ratings (self-target projection) depended positively on the anticipated interdependence manipulation, B =.17, SE = .05, t = 3.26, p < .001. The effect of self-ratings on target-ratings was positive and significant under cooperation, B =.30, SE = .07, t =3.86, p < .001, but non-significant under competition, B = -.04, SE = .07, t < 1 1 . In other words, participants projected their personality traits onto the target more when they anticipated cooperation than when they anticipated competition (see Figure 1).

We conducted additional analyses to test for the interaction between the selfratings and the valence of traits at level 1. We estimated the following models:

Level-1 Model:

Target = P0 + P1xSelf + P2xValence + P3XSelfxValence + E Level-2 Model: P0 = B00 + B01xInterdependence + R0 P1 = B10 + B11xInterdependence + R1 P2 = B20 + R2 P3 = B30 + B31xInterdependence + R3
with P0, B00 B10, B20, B30 as intercepts; P1, P2, P3, B01, B11, B31 as regression weights; and E, R0, R1, R2, R3 as residuals.

Again, the anticipated interdependence moderated the impact of self-ratings on target-ratings, B =.17, SE = .05, t = 3.37, p < .001. Introducing valence as a moderator did not significantly affect the impact of self-ratings (B = .01, SE = .03, t < 1) nor the interaction between self-ratings and anticipated interdependence (B = .03, SE = .03, t < 1).

Discussion

Together, the results of Experiment 1 suggest that participants projected their view of themselves onto the target as a function of the anticipated interdependence. They judged a target on which they receive no information as more similar to themselves when they anticipated cooperation rather than competition. In competition, participants did not infer any similarity between themselves and the unknown target. Rikettta and Sacaramento (2008) found that the perceived intergroup cooperation leads to more outgroup projection than the perceived intergroup competition. Our study is consistent with this research and extends the role of interdependence in projection from the intergroup to the interpersonal level.

What is also important about these results is that projection of self-ratings onto the target occurred regardless of the valence of the traits. Contrary to [START_REF] Stathi | Imagining intergroup contact promotes projection to outgroups[END_REF] who found projection mainly on positive traits, we found that if projection occurs more in cooperation than in competition, this is the case for both positive and negative traits.

Experiment 2

We conducted a second experiment in order to further investigate the role of interdependence in social projection. The first aim of Experiment 2 was to replicate Experiment 1 using a different manipulation for the anticipated interdependence and using an idiosyncratic measure of traits valence. Since the self-concept is predominantly positive [START_REF] Alicke | Global self-evaluations as determined by the desirability and controllability of trait adjectives[END_REF][START_REF] Baumeister | The self[END_REF] and given that the self serves as an evaluative basis [START_REF] Gramzow | Self-esteem and favoritism toward novel ingroups: The self as an evaluative base[END_REF], the valence of the traits could vary as a function of their attribution to the self (Krueger, 1998;[START_REF] Sinha | Idiographic self-evaluation and bias[END_REF]. If a person X rates her/himself as very sociable and a person Y rates her/himself as not very sociable, then sociability may likely be seen as more positive by person X than by person Y. Of note, if self-ratings and the perceived valence of the traits co-vary, the interaction between selfratings and anticipated interdependence is not adjusted for valence simply by controlling for trait valence (see [START_REF] Yzerbyt | Adjusting researchers' approach to adjustment: On the use of covariates when testing interactions[END_REF]. Therefore, in Experiment 2, we took care to control for both trait valence and the interaction between trait valence and anticipated interdependence.

The second aim of this Experiment 2 was to examine whether a change in participants' self-representation accounts for the stronger projection observed in the cooperation condition compared to the competition condition. In Experiment 1, selfratings were measured before the introduction of the manipulation of interdependence. It is possible that the interdependence manipulation changed participants' selfrepresentation. It could be that participants projected their changed self (after the manipulation) to the same extent in cooperation and competition. This means that the latter change, rather than the nature of the interdependence context, may be directly responsible for our findings. Several studies have shown that the self is a flexible and context-dependent knowledge structure (i.e., cognitive schema) that serves adaptive and self-regulatory functions [START_REF] Baumeister | The self[END_REF][START_REF] Higgins | The ''self digest'': Self-knowledge serving self regulatory functions[END_REF][START_REF] Markus | The dynamic self-concept: A social psychological perspective[END_REF].

Individuals can reorganize the structure of their self-representations in response to situational demands [START_REF] Markus | Stability and malleability of the self-concept[END_REF] or age-related challenges [START_REF] Greve | Immunizing the self: Self-concept stabilization through reality-adaptive self-definitions[END_REF]. For example, research reveals that college students respond to different conditions of feedback by dynamically adjusting the content and structure of their selfrepresentations [START_REF] Markus | Stability and malleability of the self-concept[END_REF]. These studies stress the fact that people can hold both general and contextual self-representations [START_REF] Caspi | Personality development across the life course: The argument for change and continuity[END_REF][START_REF] Markus | Stability and malleability of the self-concept[END_REF].

In order to examine this issue, we used self-ratings collected before and after the manipulation of interdependence to predict target-ratings in Experiment 2. If both self 1 and self 2 predict target-ratings more under cooperation than competition, then this finding would rule out the idea that the pattern observed in Experiment 1 was solely due to a change in participants' self view in response to the interdependence context.

Method

Participants and Design

Forty-three participants (22 males) were approached on campus and invited to take part in a study on spontaneous impression formation in exchange for 3 EUR.

Procedure

The procedure and materials were very similar to those of Experiment 1. We used the same 16 personality traits. However, in order to examine the generality of our findings, we manipulated anticipated interdependence with a different scenario. This time, participants were asked to imagine being contestants in the first phase of a TV quiz in which they could win 25,000 Euros. An unknown Person A (the target) was presented as being another contestant. The relation with the target was presented as either cooperative or competitive. In the cooperation condition, participants were told that it was possible that they and the Person A could both win the money if they managed to collectively respond correctly to a maximum of questions. Their interest was thus to help each other to move up to the second phase of the show. In the competition condition, participants were told that only one of them could win the money. They had to outperform Person A in responding to the questions so as to being the one who could move up to the second phase of the game. Next, participants were again asked to rate their self and the target on the same list of traits. Finally, participants were invited to judge the valence of each personality trait, using a 9-point response scale anchored with rather negative to rather positive. As in Experiment 1, participants also indicated if the relationship between them and the target was cooperative or competitive. They were then debriefed, thanked, and dismissed.

Results and discussion

As in Experiment 1, our hypothesis predicts that the relation between the self-and the target-ratings is influenced by a level-2 variable (anticipated interdependence) and that this effect emerges for both positive and negative traits. We therefore examined whether projection occurred regardless of traits valence. We also relied on trait valence as an idiosyncratic measure. We estimated the following model:

Level-1 Model:

Target = P0 + P1xSelf + P2xValence + P3xSelfxValence + E Level-2 Model: P0 = B00 + B01xInterdependence + R0 P1 = B10 + B11xInterdependence + R1 P2 = B20 + B22xInterdependence + R2 P3 = B30 + B33xInterdependence + R3
with P0, B00, B10, B20, B30 as intercepts; P1, P2, P3, B01, B11, B22, B33 as regression weights; and E, R0, R1, R2, R3 as residuals. We coded condition -1 for competition and 1 for cooperation. The critical parameter is again B11, which denotes the extent to which projection of the self to the target (i.e., P1) varies as a function of interdependence.

Parameters P3 and B33 denote the extent to which projection depends on the valence of the traits. The self-ratings and valence were centered at the mean of each participant's ratings. We used self-ratings before (self1) and after (self2) the manipulation of interdependence in our multilevel model. Regarding self1, we expected to replicate the findings of Experiment 1. Specifically, we expected that self-ratings would be more strongly related to target-ratings in cooperation than in competition. We examined the exact same question with self2.

As predicted, the relation between self1 and target ratings (self-target projection) depended on the anticipated interdependence manipulation, B =.10, SE = .05, t = 2.34, p < .05, after controlling for the main effect of valence as well as for the self1 by valence interaction [START_REF] Yzerbyt | Adjusting researchers' approach to adjustment: On the use of covariates when testing interactions[END_REF]. The relation between self1 ratings and target ratings was positive and significant under cooperation, B =.21, SE = .06, t = 3.41, p <. 01, and non-significant under competition, B = .001, SE = .06, t < 1. In other words, participants projected their personality traits into the target when they anticipated cooperation but not when they anticipated competition (see Figure 2). Valence did not affect self1 ratings projection (B = -.01, SE = .01, t < 1), nor the interaction between self1 ratings and interdependence (B = .01, SE = .01, t < 1).

We then tested for the role of self2 (after the manipulation of anticipated interdependence). Here too, the relation between self2 and target ratings depended on the anticipated interdependence, B =.14, SE = .05, t = 2.54, p < .01, even after controlling for valence and the self2 by valence interaction. This time, however, the relation between self2 ratings and target ratings was positive and significant under cooperation, B =.42, SE = .08, t = 5.32, p < .001, and positive and marginally significant under competition, B = .13, SE = .07, t = 1.70, p = .09. Valence did not affect self2 ratings projection (B = -.001, SE = .01, t < 1), nor the interaction between self2 ratings and anticipated interdependence (B = .01, SE = .01, t = 1.25, p =.21).

Discussion

Both self1 and self2 ratings predicted target ratings more in cooperation than in competition. Self2 ratings were more strongly associated with target ratings than self1 ratings, indicating that participants changed their self-representation somewhat after the manipulation. However, despite this change the predicted interaction self2 ratings and interdependence was significant showing that although participants saw themselves differently after the interdependence manipulation, this does not explain why people projected more in cooperation than in competition. Of importance too, the valence of the traits did not affect projection even when we relied on an idiosyncratic measure.

General Discussion

In two experiments, projection of self-attributed traits into a totally unknown target person was found to be stronger under conditions of anticipated cooperation than anticipated competition. Our data thus confirm our prediction that perceived self-other interdependence moderates interpersonal projection. Research conducted in the context of outgroup projection had already shown that perceived intergroup interdependence moderates projection [START_REF] Riketta | They cooperate with us, so they are like me': Perceived intergroup relationship moderates projection from self to outgroups[END_REF] and that positive ingroup-outgroup contacts increase outgroup projection [START_REF] Stathi | Imagining intergroup contact promotes projection to outgroups[END_REF]. The current experiments extend this work to the interpersonal level. Additionally, by having participants rate themselves and the target on both positive and negative traits, we were able to show that the observed projection was independent of the valence of the traits and the positivity of the self-view [START_REF] Alicke | Global self-evaluations as determined by the desirability and controllability of trait adjectives[END_REF][START_REF] Baumeister | The self[END_REF]. This was particularly clear for the second experiment, which involved an idiosyncratic measure of trait valence. As a third important message, Experiment 2 also revealed that this moderation of projective tendencies by interdependence was not driven by changes in self-representation. That is, the predicted moderation was observed whether we used self-ratings that were collected before or after the interdependence manipulation took place. Hence, our data make clear that changes in self-representation cannot account for the higher level of projection in a context of anticipated cooperation than competition.

Interestingly, competition did not lead to contrastive judgments. As a matter of fact, some level of social projection was observed under condition of competition in Experiment 2, at least when considering the self2 measure. This is consistent with intergroup studies showing contrastive judgments only for familiar outgroups of strong personal relevance [START_REF] Mullen | Ingroup-outgroup differences in social projection[END_REF]Riketta & Sacramento, 2005). For example, in a paradigm similar to the one used here, Riketta and Sacrameto (2005) found negative correlation between self and outgroup ratings in competition when using real groups but positive correlations when using abstract (i.e., team A) groups.

The latter findings may suggest that when available information exists about a target person or a target outgroup, people relate their self-view to this information, resulting in either assimilation or contrast effects [START_REF] Mussweiler | When egocentrism breeds distinctness-comparison processes in social prediction: Comment on Karniol (2003)[END_REF]. However, when little or no information is provided about the target, people may well be forced to rely on their self-view and project it onto the target. The latter interpretation is consistent with the present findings and with what Riketta and Sacramento (2005) found in the context of intergroup studies.

A tricky issue concerns the type of information and processing that participants relied on when reporting a judgment about a competing target. Although we can only speculate on this, we surmise that participants in both cooperation and competition used information about the self to judge the target. The lower level of projection observed in competition may imply that participants in this condition underestimated self-target similarity, as suggested by research on naïve realism and perception of exaggerated stances in conflict [START_REF] Robinson | Actual versus assumed differences in construal: "Naive realism" in intergroup perception and conflict[END_REF].

Finally, it remains to be explained why people expect an unknown target to be more similar to the self when anticipating cooperation than competition. One possibility is that projection corresponds to a heuristic process [START_REF] Krueger | From social projection to social behavior[END_REF], which is more likely to be used in cooperative contexts than when negative interdependence is elicited vis-àvis a target. In other words, the negative perspective elicited by competition may motivate participants to think more thoroughly about the accuracy of their appraisal of the target. If this were the case, then participants may withhold their judgment more under conditions of competition than cooperation. This means that they may be more prone to select the mid-point of the scales (leading to less variability and reducing correlations between self and target judgments) in competition condition. As it turns out, complementary analyses do not support this interpretation.

Another possibility is that participants hold naïve theories implying that partner similarity comes along with benefits in cooperation. In the present research, participants may have estimated that self-other similarity was more conducive to success in cooperative than competitive contexts. This may have increased projective tendencies in the cooperative condition. Preliminary evidence collected in our laboratory would seem consistent with this account. In a pilot study that examined participants' naïve theories, we found that self-other similarity was believed to increase personal success more in conditions of positive than negative interdependence. Further research should examine if and to what extent such naïve theories may contribute to the effects observed here.

Conclusion

Past interpersonal research has examined the consequences of interdependence on the way we judge others. In particular, people have been found to make more dispositional inferences about others in competitive contexts [START_REF] Ruscher | Interpersonal competition can cause individuating processes[END_REF] and to like cooperative partners more than competitive ones (e.g., Fiske & Ruscher, 1993). Surprisingly, however, no research to date has examined whether interdependence influences projective tendencies about unknown targets. The present research contributes to a more complete picture of interpersonal effects of cooperation and competition by showing that cooperation leads to more self-other perceived similarity than competition. Specifically, cooperation not only leads to perceive the self as being closer to a known target [START_REF] Stapel | Cooperation, competition and the effects of others on me[END_REF] but, as the present research demonstrates, also leads an unknown target to be perceived as more similar to the self. This implies that people expecting cooperation with unknown persons make similarity judgments that further encourage cooperative interactions. The levels of cooperation and competition that characterize our everyday relationships are often externally and sometimes arbitrarily imposed. Yet, the interdependence context may dramatically influence the way we will perceive others.

between the self and the target-ratings for all traits. We analyzed the impact of anticipated interdependence on the correspondence index and we found the mean square difference to be smaller in cooperation than in competition. This also applies to Experiment 2.

Figure 1 :

 1 Figure 1: Relation between self-ratings and target ratings by anticipated interdependence,

Figure 2 :

 2 Figure 2: Relation between self-ratings before the manipulation of anticipated

We also computed an index of correspondence that captures the mean square difference