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Disjoint Mapping Network for Cross-modal Matching of Voices and Faces

We propose a novel framework, called Disjoint Mapping Network (DIMNet), for cross-modal biometric matching, in particular of voices and faces. Different from the existing methods, DIMNet does not explicitly learn the joint relationship between the modalities. Instead, DIMNet learns a shared representation for different modalities by mapping them individually to their common covariates. These shared representations can then be used to find the correspondences between the modalities. We show empirically that DIMNet is able to achieve better performance than other current methods, with the additional benefits of being conceptually simpler and less data-intensive.


Introduction
A person's face is predictive of their voice. Biologically, this is to be expected: the same genetic, physical and environmental influences that affect the face also affect the voice. More directly, the vocal tract that generates voice also partially shapes the face.
Human perception apparently recognizes this association. Humans have been shown to be able to associate voices of unknown individuals to pictures of their faces [5].
They also show improved ability to memorize and recall voices when previously exposed to pictures of the speaker's face, but not imposter faces [12,19,18]. Cognitively, studies indicate that neuro-cognitive pathways for voices and faces share common structure [3], possibly following parallel pathways within a common recognition framework [2,1].
Among other things, the above studies lend credence to the hypothesis that it may be possible to find associations between voices and faces algorithmically as well. With this in perspective, this paper focuses on the task of devising computational mechanisms for cross-modal matching of voice recordings and images of the speakers' faces.
The specific problem setting we look at is one wherein we have an existing database of samples of people's voices and images of their faces, and we must automatically and accurately determine which voices match to which faces. This problem has seen significant research interest, in particular since the recent introduction of the VoxCeleb corpus [15], which comprises collections of video and audio recordings of a large number of celebrities. The existing approaches [14,13,6] have generally attempted to directly relate subjects' voice recordings and their face images, in order to find the correspondences between the two. Nagrani et al. [14] formulate the mapping as a binary se-lection task: given a voice recording, one must successfully select the speaker's face from a pair of face images (or the reverse -given a face image, one must correctly select the subject's voice from a pair of voice recordings). They model the mapper as a neural network that is trained through joint presentation of voices and faces to determine if they belong to the same person. In [6,13], the authors attempt to learn common embeddings (vector representations) for voices and faces that can be compared to one another to identify associations. The networks that compute the embeddings are also trained through joint presentation of voices and faces, to maximize the similarity of embeddings derived from them if they belong to the same speaker. In all cases, current prevailing methods attempt to directly relate voices to faces. In effect, the voice and face are implicitly assumed to directly inform about one another.
In reality, though, it is unclear how much these models capture the direct influence of the voice and face on one another, and how much is explained through implicit capture of higher-level variables such as gender, age, ethnicity etc., which individually predict the two. These higher-level variables, which we will refer to as covariates 1 can, in fact, explain much of our ability to match voices to faces (and vice versa) under the previously mentioned "select-from-apair" test (where a voice must be used to distinguish the speaker's face from a randomly-chosen imposter). For instance, simply matching the gender of the voice and the face can result in an apparent accuracy of match of up to 75% in a gender-balanced testing setting. Even in a seemingly less constrained "verification" test, where one must only verify if a given voice matches a given face, matching them based on gender alone can result in an equal error rate of 33% (Appendix A). Also, matching the voice and the face by age (e.g. matching older-looking faces to older-sounding voices) can result in match performance that's significantly better than random.
The authors of previous studies [14,6] attempt to disambiguate the effect of multiple covariates through stratified tests that separate the data by covariate value. The results show that at least some of the learned associations are explained by the covariate, indicating that their learning approaches do utilize the covariate information, albeit only implicitly. The tests generally only evaluate one or two commonly known covariates, namely gender, age, and ethnicity (or nationality), and do not necessarily imply that the residual stratified results cannot be explained by other untested covariates.
In this paper we propose an alternate learning approach to learn mappings between voices and faces that does not consider any direct dependence between the two, but instead explicitly exploits their individual dependence on the covariates. In contrast to existing methods in which supervision is provided through the correspondence of speech and facial image data to one another, our learning framework, Disjoint Mapping Network (DIMNet), obtains supervision from covariates such as gender, ethnicity, identity (ID) etc., applied separately to voices and faces, to learn a common embedding for the two. The comparison between the existing approaches and DIMNets are illustrated in Figure 1.
DIMNet comprises individual feature learning modules which learn identically-dimensioned features for data from each modality, and a unified input-modality-agnostic classifier that attempts to predict covariates from the learned feature. Data from each modality are presented separately during learning; however the unified classifier forces the feature representations learned from the individual modalities to be comparable. Once trained, the classifier can be removed and the learned feature representations are used to compare data across modalities.
The proposed approach greatly simplifies the learning process and, by considering the modalities individually rather than as coupled pairs, makes much more effective use of the data. Moreover, if multiple covariates are known, they can be simultaneously used for the training through multi-task learning in our framework.
We evaluate the voice-to-face and face-to-voice matching performance obtained with the resulting embeddings through a number of different metrics, including selection of a face or voice from a pair of choices, given input of the other modality, verification that a given pair of instances belong to the same person, one-out-of-N matching, and retrieval from a large corpus (the last being, perhaps, the most realistic test in the real world).
We find that in all of our tests for which similar results have been reported by other researchers [14,13,6], our embeddings achieve comparable or better performance than those previously reported. We find that of all the covariates, ID provides the strongest supervision. The results obtained from supervision through other covariates too match what may be expected, although they are better than those reported under similar evaluation settings in prior results, indicating, perhaps, that direct supervision through covariates is more effective in these settings. Finally we show that visualizations that although significantly-better-than-random results are obtained in terms of retrieval performance, the distributions of the actual embeddings learned do not overlap for the two modalities, showing that there remains significant scope for improvement, either through increased data or through algorithmic enhancement.
Our contributions are summarized as follows. • We propose DIMNets, a framework that formulates the problem of cross-modal matching of voices and faces as learning common embeddings for the two through individual supervisions from one or more covariates, in contrast to current approaches that attempt to map voices to faces directly. • In this framework, we can make use of multiple kinds of label information provided through covariates. • We achieve the state-or-art results on multiple tasks. We are able to isolate and analyze the effect of the individual covariate on the performance.

The Proposed Framework
Our objective is to learn common representations of voices and faces, that permit them to be compared to one another. In the following sections we first describe how we learn them from their relationship to common covariates. Subsequently, we describe how we will use them for comparison of voices to faces.

Leveraging Covariates to Learn Embeddings
The relationship between voices and faces is largely predicted by covariates -factors that individually relate to both the voice and the face. To cite a trivial example, a person's gender relates their voice to their face: male subjects will have male voices and faces, while female subjects will have female voices and faces. More generally, many covariates may be found that relate to both voice and face [8].
Our model attempts to find common representations for both face images and voice recordings by leveraging their relationship to these covariates (rather than to each other). We will do so by attempting to predict covariates from voice and face data in a common embedding space, such that the derived embeddings from the two types of data can be compared to one another.
Let V represent a set of voice recordings, and F represent a set of face images. Let C be the set of covariates we consider. For the purpose of this paper, we assume that all covariates are discrete valued (although this is not necessary). Every voice recording in V and every face in F can be related to each of the covariates in C. For every covariate C ∈ C we represent the value of that covariate for any voice recording v as C(v), and similarly the value of the covariate for any face f as C(f ). For example, C could be ID, gender, or nationality. When C is ID, C(v) and C(f ) are the ID of voice v and face f , respectively.
Let F v (v; θ v ) : v → R d be a voice embedding function with parameters θ v that maps any voice recording v into a d-dimensional vector. Similarly, let F f (f ; θ f ) be a face embedding function that maps any face f into a d-dimensional vector. We aim to learn θ v and θ f such that the embeddings of the voice and face for any person are comparable.
For each covariate C ∈ C we define a classifier H C (x; φ C ) with parameter φ C , which assigns any input x ∈ R d to one of the values taken by C. The classifier H C (·) is agnostic to which modality its input x was derived from; thus, given an input voice v, it operates on features F v (v; θ v ) derived from the voice, whereas given a face f , it operates on F f (f ; θ f ).
For each v (or f ) and each covariate C, we define a loss L(H C (F v (v; θ v ); φ C ), C(v)) between the covariate predicted by H C (.) and the true value of the covariate for v, C(v). We can now define a total loss L over the set of all voices V and the set of all faces F, over all covariates as In order to learn the parameters of the embedding functions, θ f and θ v , we perform the following optimization. Figure 2 shows the network architecture we use to train our embeddings. It comprises three components. The first, labelled Voice Network in the figure, represents F v (v; θ v ) and is a neural network that extracts d-dimensional embeddings of the voice recordings. The second, labelled Face Network in the figure, represents F f (f ; θ f ) and is a network that extracts d-dimensional embeddings of face recordings. The third component, labelled Classification Networks in the figure, is a bank of one or more classification networks, one per covariate considered. Each of the classification networks operates on the d-dimensional features output by the embedding networks to classify one covariate, e.g. gender.

Disjoint Mapping Networks
The training data comprise voice recordings and face images. Voice recordings are sent to the voice-embedding network, while face images are sent to the face-embedding network. This switching operation is illustrated by the switch at the input in Figure 2. In either case, the output of the embedding network is sent to the covariate classifiers.
As can be seen, at any time the system either operates on a voice, or on a face, i.e. the operations on voices and faces are disjoint. During the learning phase too, the updates of the two networks are disjoint -loss gradients computed when the input is voice only update the voice network, while loss gradients derived from face inputs update the face network, while both contribute to updates of the classification networks.
In our implementation, specifically, F v (·) is a convolutional neural network that operates on Mel-Spectrographic ... ... ... Figure 2: Our DIMNet framework. The input training data can be either voice or face, and there is no need for voices and faces to form pairs. Modality switch is to control which embedding network (voice or face) to process the data. While the embeddings are obtained, a multi-task classification network is applied to supervise the learning.

Multi-task Classification Network via Covariates
representations of the speech signal. The output of the final layer is pooled over time to obtain a final d-dimensional representation. F f (·) is also a convolutional network with a pooled output at the final layer that produces a ddimensional representation of input images. The classifiers H C (·) are all simple multi-class logistic-regression classifiers comprising a single softmax layer. Finally, in keeping with the standard paradigms for training neural network systems, we use the cross-entropy loss to optimize the networks. Also, instead of the optimization in Equation 2, the actual optimization performed is the one below. The difference is inconsequential.

Training the DIMNet
All parameters of the network are trained through backpropagation, using stochastic gradient descent. During training, we construct the minibatches with a mixture of speech segments and face images, as the network learns more robust cross-modal features with mixed inputs. Taking voice as an example, we compute the voice embeddings using F v (v; θ v ), and obtain the losses using classifiers H C (·) for all the covariates. We back-propagate the loss gradient to update the voice network as well as the covariate classifiers. The same procedure is also applied to face data: the backpropagated loss gradients are used to update the face network and the covariate classifiers. Thus, the embedding functions are learned using the data from their modalities individually, while the classifiers are learned using data from all modalities.

Using the Embeddings
Once trained, the embedding networks F v (v; θ v ) and F f (f ; θ f ) can be used to extract embeddings from any voice recording or face image.
Given a voice recording v and a face image f , we can now compute a similarity between the two through the co- We can employ this distance to evaluate the match of any face image to any voice recording. This enables us, for instance, to attempt to rank a collection of faces f 1 , · · · , f K in order of estimated match to a given voice recording v, according to S(v, f i ), or conversely, to rank a collection of voices v 1 , · · · , v K according to their match to a face f , on order of decreasing S(v i , f ).

Experiments
We ran experiments on matching voices to faces, to evaluate the embeddings derived by DIMNet. The details of the experiments are given below.

Datasets.
Our experiments were conducted on the Voxceleb [15] and VGGFace [16] datasets. The voice dataset, Voxceleb, consists of 153,516 audio segments from 1,251 speakers. Each audio segment is taken from an online video clip with an average duration of 8.2 seconds. For the face dataset, we used a manually filtered version of VGGFace. After face detection, there remain 759,643 images from 2,554 subjects. Note that we do not use video frames for training, so the number of training face images in our dataset is smaller than that in [14].
We use the intersection of the two datasets, i.e. subjects who figure in both corpora, for our final corpus, which thus includes 1,225 IDs with 667 males and 558 females from 36 nationalities. The data are split into train/validation/test sets following the settings in [14], shown in Table 1.
We use ID, gender and nationality as our covariates, all of which are provided by the datasets. Preprocessing. We employ separated data preprocessing pipelines for audio segments and face images. For audio segments, we use an energy-based voice activity detector [17] to isolate speech-bearing regions of the recordings. Subsequently, 64-dimensional log mel-spectrograms are generated, using an analysis window of 25ms, with hop  # of samples   train  validation  test  total  speech segments 112,697  14,160  21,799  148,656  face images  313,593  36,716  58,420  408,729  IDs  924  112  189  1,225  genders  2  2  2  2  nationalities  32  11  18  36  testing instances  -4,678,897 6,780,750 11,459,647   Table 1: Statistics for the data appearing in VoxCeleb and VGGFace.
of 10ms between frames. We perform mean and variance normalization of each mel-frequency bin. For training, we randomly crop out regions of varying lengths of 300 to 800 frames (so the size of the input spectrogram ranges from 300 × 64 to 800 × 64 for each minibatch, around 3 to 8 seconds). For the face data, facial landmarks in all images are detected using MTCNN [24]. The cropped RGB face images of size 128×128×3 are obtained by similarity transformation. Each pixel in the RGB images is normalized by subtracting 127.5 and then dividing by 127.5. We perform data augmentation by horizontally flipping the images with 50% probability in minibatches (effectively doubling the number of face images).
Training. The detailed network configurations are given in Table 2. For the voice network, we use 1D convolutional layers, where the convolution is performed along the axis that corresponds to time. The face network employs 2D convolutional layers. For both, the convolutional layers are followed by batch normalization (BN) [4] and rectified linear unit activations (ReLU) [7]. The final face embedding is obtained by averaging the feature maps from the final layer, i.e. through average pooling. The final voice embedding is obtained by averaging the feature maps at the final convolutional layer along the time axis alone. The classification networks are single-layer softmax units with as many outputs as the number of unique values the class can take (2 for gender, 32 for nationalities, and 924 for IDs in our case). The networks are trained to minimize the cross entropy loss. We follow the typical settings of stochastic gradient descent (SGD) for optimization. Minibatch size is 256. The momentum and weight decay values are 0.9 and 0.001 respectively. To learn the networks from scratch, the learning rate is initialized at 0.1 and divided by 10 after 16K iterations and again after 24K iterations. The training is completed at 28K iterations. Testing. We use the following protocols for evaluation: • 1:2 Matching. Here, we are give a probe input from one modality (voice or face), and a gallery of two inputs from the other modality (face or voice), including one that belongs to the same subject as the probe, and another of an "imposter" that does not match the probe. The task is to identify which entry in the gallery matches the probe. We report performance in terms of matching accuracynamely what fraction of the time we correctly identify the right instance in the gallery.  To minimize the influence of random selection, we construct as many testing instances as possible through exhaustive enumeration all positive matched pairs (of voice and face). To each pair, we include a randomly drawn imposter in the gallery. We thus have a total of 4,678,897 trials in the validation set, and 6,780,750 trials in the test set.
• 1:N Matching. This is the same as the 1:2 matching, except that the gallery now includes N − 1 imposters. Thus, we must now identify which of the N entries in the gallery matches the probe. Here too results are reported in terms of matching accuracy. We use the same validation and test sets as the 1:2 case, by augmenting each trial with N − 2 additional imposters. So the number of trials in validation and test sets is the same as earlier.
• Verification. We are given two inputs, one a face, and another a voice. The task is to determine if they are matched, i.e. both belong to the same subject. In this problem setting the similarity between the two is compared to a threshold to decide a match. The threshold can be adjusted to trade off false rejections (F R ), i.e. wrongly rejecting true matches, with false alarms (F A ), i.e. wrongly accepting mismatches. We report results in terms of equal error rate, i.e. when F R = F A . We construct our validation and test sets from those used for the 1:2 matching tests, by separating each trial into two, one comprising a matched pair, and the other a mismatched pair. Thus, our validation and test sets are exactly twice as large as those for the 1:2 test.
• Retrieval. The gallery comprises a large number of instances, one or more of which might match the probe. The task is to order the gallery such that the entries in the gallery that match the probe lie at the top of the ordering.
Here, we report performance in terms of Mean Average Precision (MAP) [11]. Here we use the entire collection of 58,420 test faces as the gallery for each of our 21,799 test voices, when retrieving faces from voices. For the reverse (retrieving voices from faces), the numbers are reversed.
Each result is obtained by averaging the performances of 5 models, which are individually trained.
Covariates in Training and Testing. We use the three covariates provided in the dataset, namely identity (I), gender (G), and nationality (N) for our experiments. The treatment of covariates differs for training and test.
• Training. For training, supervision may be provided by any set of (one two or three) covariates. We consider all combinations of covariates, I, G, N, (I,G), (I,N), (G,N) and (I,G,N). Increasing the number of covariates effectively increases the supervision provided to training. All chosen covariates were assigned a weight of 1.0. • Testing. As explained in Appendix A, simply recognizing a covariate such as gender can result in seemingly significant matching performance. For instance, just recognizing the subjects' gender from their voice and images can result in a 33% EER for verification, and 25% error in matching for the 1 : 2 tests. In order to isolate the effect of covariates on performance hence we also stratify our test data by them. Thus we construct 4 testing groups based on the covariates, including the unstratified (U) group, stratified by gender (G), stratified by nationality (N), and stratified by gender and nationality (G, N).
In each group the test set itself is separated into multiple strata, such that for all instances within any stratum the covariate values are the same. Note that the models trained against each of our covariatesupervision settings can be evaluated against each of our stratified test sets.

Cross-modal Matching
In this section we report results on the 1:2 and 1:N matching tests. In order to ensure that the embedding networks do indeed leverage on accurate modelling of covariates, we first evaluate the classification accuracy of the classification networks for the covariates themselves. Table 3 shows the results.
The rows of the table show the covariates used to supervise the learning. Thus, for instance, the row labelled "DIMNet-I" shows results obtained when the networks have been trained using ID alone as covariate, the row labelled "DIMNet-G" shows results when supervision is provided by gender, "DIMNet-IG" has been trained using ID and gender, etc.
The columns of the  The results in Table 3 show that gender is learned very well, and in all cases gender recognition accuracy is quite high. Nationality, on the other hand, is not a well-learned classifier, presumably because the distribution of nationalities in the data set is highly skewed [14], with nearly 65% of all subjects belonging to the USA. It is to be expected therefore that nationality as a covariate will not provide sufficient supervision to learn good embeddings. 1 : 2 matching. Table 4 shows the results for the 1:2 matching tests. In the table, the row labelled "SVHF-Net" gives results obtained with the model of Nagrani et al. [14].
The columns are segregated into two groups, one labelled "voice → face" and the other labelled "face → voice". In the former, the probe is a voice recording, while the gallery comprises faces. In the later the modalities are reversed. Within each group the columns represent the stratification of the test set. "U" represents test sets that are not stratified, and include the various covariates in the same proportion that they occur in the overall test set. The columns labelled "G" and "N" have been stratified by gender and nationality, respectively, while the column "G,N" represents data that have been stratified by both gender and nationality. In the stratified tests, we have ensured that all data within a test instance have the same value for the chosen covariate. Thus, for instance, in a test instance for voice → face in the "G" column, the voice and both faces belong to the same gender. This does not reduce the overall number of test instances, since it only requires ensuring that the gender of the imposter matches that of the probe instance.
We make several observations. First, DIMNet-I per- forms better than SVHF-Net, improving the accuracies by 2.45%-4.02% for the U group, and 7.01%-8.38% for the G group. It shows that mapping voices and faces to their common covariates is an effective strategy to learn representations for cross-modal matching.
Second, DIMNet-I produces significantly better embeddings that DIMNet-G and DIMNet-N, highlighting the rather unsurprising fact that ID provides the more useful information than the other two covariates. In particular, DIMNet-G respectively achieves 72.90% and 72.47% for voice to face and face to voice matching using only gender as a covariate. This verifies our hypothesis that we can achieve almost 75% matching accuracy by only using the gender. These numbers also agree with the performance expected from the numbers in Table 3 and the analysis in Appendix A. As expected, nationality as a covariate does not provide as good supervision as gender. DIMNet-IG is marginally better than DIMNet-I, indicating that gender supervision provides additional support over ID alone.
Third, we note that while DIMNet-I is able to achieve good performance on the dataset stratified by gender, DIMNet-G only achieves random performance. The performance achieved by DIMNet-G on the U dataset is hence completely explained by gender matching. Once again, the numbers match our expectations (Appendix A). 1:N matching. We also experiments for N > 2. Unlike Nagrani et al. [14], who train a different model for each N in this setting, we use the same model for different N . The results are shown in the Figure 3, which shows accuracy as a function of N for various models.
All the results are consistent with that in Table 4. As expected, performance degrades with increasing N . In general, DIMNets using ID for supervision outperform SVHF-Net. We obtain the best results when both ID and gender are used as covariates. The results obtained using only gender as covariate is however much worse, and, once again, consistent with an analysis such as that in Appendix A.

Cross-modal Verification
For verification, we need to determine whether an audio segment and a face image are from the same ID or not. We  Table 5.
In general, DIMNets that use ID as a covariate achieve an EER of about 25%, which is considerably larger than the 33% expected if the verification were based on gender matching alone. Note that previously reported results [14] on this task have generally been consistent with matching based on gender; ID-based DIMNet, however, clearly learns more than gender. Once again, nationality is seen to be an ineffective covariate, while gender alone as a covariate produces results that are entirely predictable.

Cross-modal Retrieval
We also perform retrieval experiments using voice or face as query. Table 6 lists the mean average precision of the retrieval for various models.
The columns in the table represent the covariate being retrieved. Thus, for example, in the "ID" column, the objective is to retrieve gallery items with the same ID as the query, whereas in the "gender" column the objective is to retrieve the same gender. We note once again that ID-based DIMNets produce the best features for retrieval, with the best performance obtained with DIMNet-IG. Also, as may be expected, the covariates used in training result in the best retrieval of that covariate. Thus, DIMNet-G achieves an mAP of nearly 98% on gender, though on retrieval of ID it is very poor. On the other hand, DIMNet-I is not nearly as successful at retrieving gender. As in other experiments, nationality remains a poor covariate in general.

Discussions and Concluding Remarks
We have proposed that it is possible to learn common embeddings for multi-modal inputs, particularly voices and faces, by mapping them individually to common covariates. In particular, the proposed DIMNet architecture is able to extract embeddings for both modalities that achieve or surpass state of art performance that has been achieved by directly mapping faces to voices, on several tasks.
The approach also provides us the ability to tease out the influence of each of the covariates of voice and face data, in determining their relation. The results show that the strongest covariate, not unexpectedly, is ID. The results also indicate that prior results by other researchers who have attempted to directly match voices to faces may perhaps not be learning any direct relation between the two, but implicitly learning about the common covariates, such as ID. The advantage, of course, being that this is done without explicit presentation of this information.
Our experiments also show that although we have achieved possibly the best reported performance on this task, thus far, the performance is not anywhere close to prime-time. In the 1 : N matching task, performance degrades rapidly with increasing N , indicating a rather poor degree of true match.
To better understand the problem, we have visualized the learned embeddings from DIMNet-I in Figure 4 to provide more insights. The visualization method we used is multi-dimensional scaling (MDS) [23], rather than the cur-rently more popular t-SNE [20]. This is because MDS tends to preserve distances and global structure, while t-SNE attempts to retain statistical properties and highlights clusters, but does not preserve distances.
From the figure, we immediately notice that the voice and face data for a subject are only weakly proximate. While voice and face embeddings for a speaker are generally relatively close to each other, they are often closer to other subjects. Interestingly, the genders separate (even though gender has not been used as a covariate for this particular network), showing that at least some of the natural structure of the data is learned. Figure 4 shows embeddings obtained from both training and test data. We can observe similar behaviors in both, showing that the the general characteristics observed are not just the outcome of overfitting to training data.
Thus, while DIMNets do learn to place the two modalities close together, there is still significant room for improvement. Ideally, the voice and face embeddings of the same ID should overlap. However, as also apparent from the results and the figures shown, it does not appear that we actually learn compact overlapping representations for both modalities. Rather, retrieval appears to be based on proximity, and the actual ceiling for the performance achieved with such embeddings may be low. While we cannot directly compare our features to those obtained in other reported research, we assume that they too have similar characteristics.
Part of the issue may lie with the data itself -they are noisy, and in many ways insufficient. Better results may perhaps be achieved with more data. Another may be with the actual feature-extraction architectures employed. We do not believe that our architectures have been optimized to the extent possible, and further improvements may be obtained by exploring more expressive architectures. Also, it may be possible to force compactness of the distributions of voice and face embeddings through modified loss functions such as the center loss [22] or angular softmax loss [10,9], or through an appropriately designed loss function that is specific to this task. We continue to explore all these avenues.

Appendix A. Expected performance based on gender matching
In this appendix we discuss the performance to be expected in the matching and verification tests, when the matching is done based purely on gender.
We assume below that in any distribution of humansubject data, the division of subjects between male and female genders to be half and half.
It is to be noted that gender is merely an illustrative example here; the analysis can be extended to other covariates. For a more detailed analysis of covariates with more values and unbalanced distributions, please refer to [21].

A.1. Accuracy of 1:2 matching based on gender
We show that the equal-error-rate for 1:2 matching can be as high as 25%, through gender matching alone.
The problem is as follows: a probe input (voice or face), and a gallery consisting of two inputs (face or voice), one of which is from the same subject as the probe. We must identify which of the two is the true match.

A.1.1 Perfect gender identification
Consider the situation where we are able to identify the gender of the subject of the data (face or voice) perfectly.
There are two possibilities: (a) both probe instances are the same gender, and (b) they are different genders. Each of the two possibilities occurs with a probability of 0. 5 We employ the following simple strategy: If the two gallery instances are different genders, then we select the instance whose gender matches the probe. In this case, clearly, the probability of error is 0. If the two instances are the same gender, we select one of them randomly with a probability of 0.5. The probability of error here is 0.5.

A.1.2 Imperfect gender identification
Now let us consider the situation where gender identification itself is imperfect, and we have error rates e f and e v in identifying the gender of faces and voices, respectively. Assume the error rates are known. We will assume below that gallery entries are faces, and probe entries are voices.
(The equations are trivially flipped to handle the converse case).
Since we are aware that we sometimes make mistakes in identifying gender, we modify our strategy as follows: when the two gallery items are found to have different genders, we select the entry with the same gender as the probe P of the time (so that if the gender classification was correct, we would have a match error rate of (1-P )). When both gallery items are found to be the same gender, we choose randomly.
The actual error can now be computed as follows. The gallery items are both of the same gender in 0.5 of the trials, and of mismatched gender in the remaining 0.5 of the trials.
When both gallery items have the same gender, regardless of the strategy chosen, the probability of error is 0.5 (by symmetry).
When both gallery items are of mismatched gender, we have 8 combinations of correctness of gender-classification. Table 7 lists all eight, along with the probability of matching error (in the final column). Taking type 1 as an example, we type probe gallery1 gallery2 P rob(error, type) have probability (1 − e v )(1 − e f ) 2 that the gender of both probe and galleries are correctly classified. In this case, our strategy gives us an error of (1 − P ). For type 2, the gender of probe and one of the gallery items is correctly classified, while the other gallery item is misclassified, we have an error of 0.5. If we go through all the cases, the total error P rob(error) can be computed as P rob(error) = 0.25 + 0.5 8 type=1 P rob(error, type) Our objective is to minimize P rob(error), so we must choose P to minimize the above term. I.e. we must solve Its easy to see that the solution for P is 1.0 if its multiplicative factor is negative in the above equation, and 0 otherwise, i.e. Although complicated looking, the solution is, in fact, quite intuitive. When gender classification is either better than random for both modalities (i.e. e f , e v > 0.5) or worse than random for both (e f , e v < 0.5), the best strategy is to select the gallery item that matches the gender of the probe. If either of these is random (i.e. either e f or e v is 0.5) , the choice of P does not matter, and the error is 0.5. If one of the two is correct more than half the time, and the other is wrong more than half the time (e.g. e f < 0.5, e v > 0.5), the optimal choice is to select the gallery item that is classified as mismatched in gender with the probe.

A.2. Accuracy of 1:N matching based on gender
We now consider the best achievable performance on 1:N matching, when the only information known is the gender of the voices and faces.

A.2.1 Perfect gender identification
Consider the situation where the gender of the faces and voices in each test trial is perfectly known.
We employ the following strategy: we randomly select one of the gallery instances that have the same gender as the probe instance. If there are K imposter gallery instances of the same gender as the probe instance, the expected accuracy is 1 K+1 . The probability of randomly having K of N − 1 imposters of the same gender as the probe is given by The overall accuracy is given by: giving us the error

A.2.2 Imperfect gender identification
Consider now that the gender recognition is erroneous for voices with probability e v and for faces with probability e f .
Note that regardless of the error in gender recognition, the probability of any noisy gallery entry having any gender remains 0.5.
To account for the possible error in gender classification, we consider the following stochastic policy: with probability P we select one of the gallery entries with the same gender assigned to probe (by the gender classifier), and with probability 1 − P we choose one of the entries with the opposite gender assigned to the probe.
Let α represent the probability that the genders assigned to probe and the corresponding gallery entry by their respective classifiers are identical.
The equation above considers both possibilities: that both the probe and its matching gallery entry are correctly classified, and that both of them are misclassified. It follows that the probability and its matching gallery entries are assigned different genders is 1 − α.
Given that we have selected the correct gender for retrieval from the the gallery (i.e. that the gender we have selected is the same as that assigned to the gallery entry matching the probe by the face classifier), using the same analysis as in Section A.2.1, we obtain the following probability of being correct: The probability of selecting the correct gender is given by Since the probability of being correct when we choose the wrong gender is 0, the overall probability of being correct is Maximizing the probability requires us to solve which gives us the optimal P as P = 1 if α > 0.5 0 otherwise and the optimal error as

A.3. EER of verification based on gender
Here we show that the equal-error-rate for verification (determining if the the subjects in two recordings are the same) can be as high as 33%, through gender matching alone.
The problem is as follows: we are given a pair of inputs, one (features extracted from) a face, and the other a voice. We must determine whether they are both from the same speaker.
The test set include some number of "positives", where both do belong to the same subject, and some "negatives", where both do not. If a positive is falsely detected as a negative, we have an instance of false rejection. If a negative is wrongly detected as a positive, we have an instance of false acceptance.
Let F R represent the 'false rejection rate", i.e. the fraction of all positives that are wrongly rejected. Let F A represent the "false acceptance rate", i.e. the fraction of negatives that are wrongly accepted. Any classifier can generally be optimized to trade off F R against F A . The "Equal Error Rate" (EER) is achieved when F R = F A .
Among the "positive" test pairs, both voice and face in each pair have the same gender. We assume the "negative" test instances are drawn randomly, i.e., 0.5 of all negative pairs have the same gender, while the remaining 0.5 do not.

A.3.1 Perfect gender identification
Consider the situation where we know the subject's gender for both the voices and faces (or, alternately, are able to identify the gender from the voice or face perfectly).
We employ the following strategy: if the gender of the voice and face are different, we declare it as a negative 100% of the time. If the two are from the same gender, we randomly call it a positive P of the time, where 0 ≤ P ≤ 1.0.
Using this strategy, the false acceptance rate is: Here we're considering that using our strategy we never make a mistake on the 50% of negative pairs that have mismatched genders, but are wrong P of the time on the negative pairs with matched genders. Among the positives, where all pairs are gender matched, our strategy of accepting only a fraction P of them as positives will give us a false rejection rate F R = 1 − P .
The equal error rate is achieved when F R = F A , i.e.
0.5P = 1 − P, giving us P = 2 3 , i.e. the best EER is achieved when we accept gender-matched pairs two-thirds of the time.
The EER itself is 0.5P = 1 3 . Thus, merely by being able to identify the gender of the subject accurately, we are able to verification EER of 0.33.

A.3.2 Imperfect gender identification
Now let us consider the situation where gender identification itself is imperfect, and we have error rates e f and e v in identifying the gender of the face and the voice, respectively. Assume these error rates are known.
To account for this, we modify our strategy: when we find the genders of the voice and face to match, we accept the pair as positive P of the time, but when they are mismatched we still accept them as positive Q of the time.
Let α represent the probability that we will correctly call the polarity of the gender match between the voice and the face. I.e. α is the probability that if the two have the same gender, we will correctly state that they have the same gender, or if they are of opposite gender, we will correctly state they are of opposite gender.
This combines two terms: that we call the genders of both the voice and face correctly, and that we call them both wrongly (which also results in finding the right polarity of the relationship). Its easy to see that 0 ≤ α ≤ 1, and to verify that when gender identification is perfect, α = 1.0. The probability of calling the polarity of the gender relationship wrongly is 1 − α.
Among the positive test pairs, all pairs are gender matched. We will correctly call α of these as gender matched. Using our strategy, our error on these instances is (1 − P ). We will incorrectly call 1 − α of these as gender mismatched, and the error on these instances is (1 − Q). So the overall false rejection rate is given by Among the negative pairs, half are gender matched, and half are gender mismatched. Using the same logic as above, the error on the gender-matched negative pairs is αP + (1 − α)Q. Among the gender mismatched pairs the error is αQ+ (1 − α)P . The overall false acceptance rate is given by F A = 0.5(αP +(1−α)Q)+0.5(αQ+(1−α)P ) = 0.5(P +Q).
Equating F A and F R as the condition for EER, we obtain Since at EER, the EER equals F A , and we would like to minimize it, we obtain the following solution to determine the optimal P and Q: For α > 0.5 it is easy to see that the solution to this is obtained at For α < 0.5 the optimal solution is at That is, when the probe and gallery classifiers are likely to make the same error more than half the time, the optimal solution is to always reject pairs detected as having mismatched genders, and to accept matched-gender pairs 2 1+2α of the time. The optimal EER is 1 1+2α . When they are more likely to make different errors, the optimal solution is to always reject pairs detected as having matched genders, and to accept mismatched-gender pairs 2 3−2α of the time. The optimal EER now is 1 3−2α . Note that if an operating point other than EER were chosen to quantify performance (e.g. F A = βF R for β = 1, or for some fixed F A or F R ), the above analysis can be modified to accommodate it, provided a feasible solution exists.