The statistics of the real visual world presents a long-tailed distribution: a few classes have significantly more training instances than the remaining classes in a dataset. This is because the real visual world has a few classes that are common while others are rare. Unfortunately, the performance of a convolutional neural network is typically unsatisfactory when trained using a long-tailed dataset. To alleviate this issue, we propose a method that discriminatively learns an embedding in which a simple Bayesian classifier can balance the class-priors to generalize well for rare classes. To this end, the proposed approach uses a Gaussian mixture model to factor out class-likelihoods and class-priors in a long-tailed dataset. The proposed method is simple and easy-to-implement in existing deep learning frameworks. Experiments on publicly available datasets show that the proposed approach improves the performance on classes with few training instances, while maintaining a comparable performance to the state-of-the-art on classes with abundant training examples. Deep convolutional neural networks (CNN) have achieved impressive results in large-scale visual recognition tasks BID13 BID25 BID29 BID18 BID8 BID10 . However, despite the significant impact in visual perception, the vast majority of these advancements learn from artificially balanced largescale datasets that are not representative of the real visual world BID19 BID5 BID3 BID21 BID15 BID23 . The statistics of the real visual world follow a long-tailed distribution BID36 BID32 BID24 BID33 BID34 . This means that a few classes are predominant in the world while others are rare. Consequently, representative real-world datasets have a few classes with significantly more training instances than the remaining classes in the set; see Fig. 1 (a) for an illustration of a long-tailed dataset. We refer to classes with abundant training instances as classes in the head, and unrepresented classes as classes in the tail. As BID32 note, the main motivation for visual recognition is to understand and learn from the real visual world. Thus, while the state-of-the-art can challenge humans in visual recognition tasks, it misses a mechanism that effectively learns from long-tailed datasets. As BID32 found, training models using long-tailed datasets often leads to unsatisfying performance. This is because classifiers tend to generalize well for classes in the head, but lack generalization for classes in the tail. To alleviate this issue, learned classifiers need to generalize for classes in the tail while maintaining a good performance for all the classes. Recent efforts that aim to learn from long-tailed datasets consider penalities in the optimization-learning problem BID9 , sampling-based methods BID7 , and transfer-learning algorithms BID33 BID34 . In contrast with these solutions, the proposed method aims to learn an embedding in which the distribution of the real visual world allows a simple Bayesian classifier to predict robustly given a long-tailed dataset. Long-tailed datasets have class-prior statistics that heavily skew towards classes in the head. This skew can bias classifiers towards classes in the head, and consequently can reduce generalization for classes in the tail. To remove this skew, we appeal to Bayesian classifiers that can explicitly Figure 1: (a) The real visual world yields long-tailed datasets. Classes in the head are common (e.g., cats) while classes in the tail are rare (e.g., white reindeers). (b) The proposed approach builds a generative (Bayesian) classifier over a learned embedding to compute class-posterior probabilities. In an empirical Bayesian framework, posteriors are computed through class likelihoods and priors fit to the data (e.g., sample means, variances, and counts assuming Gaussian Mixture Models). We introduce an end-to-end pipeline for jointly learning embeddings and Bayesian models built upon them. (c) Bayesian models are particularly well-suited for long-tailed datasets because class priors and likelihoods can be fixed to be uniform and isotropic, ensuring that the learned representation is balanced across the head and tail.factor out the likelihood and prior when computing posteriors over class labels. Thus, the main goal of this work is to learn a feature embedding in which class prior statistics do not affect/skew class likelihoods. The proposed approach uses a simple Gaussian mixture model (GMM) to describe the statistics of a long-tailed dataset. This is because it enables a clean factorization of the class-likelihoods and class-priors. Moreover, it easily fits within an empirical Bayesian classification framework, because a GMM enables the computation of closed-form maximum likelihood estimation (MLE) of class-specific means, covariance matrices, and priors. We show that such closed-form estimates can be integrated into existing deep learning optimizers without much effort. By fixing the covariance matrices of all the classes to be the identity and the priors over each class to be uniform, we can explicitly enforce that both rare classes in the tail and dominant classes in the head have equal weight for Bayesian classification. In simple terms: we learn a discriminative embedding of training data such that Bayesian classifiers with balanced priors produce accurate class posteriors. As a point of clarity, the proposed approach does not learn an embedding in the traditional Bayesian sense, which might define a prior distribution over embeddings that is then combined with training data to produce a posterior embedding. Rather, it learns a single embedding that is discriminatively trained to produce accurate features for Bayesian classifiers. See Fig. 1 for an illustration about the proposed approach. A GMM not only is useful for learning an embedding using a long-tailed dataset, but also provides flexibility at the evaluation stage. This is because it enables the measurement of generalization for classes in the tail by simply setting equal class-prior probabilities. In addition, it enables the possibility of giving more importance to the most frequent classes by adjusting their respective class-prior probabilities. In sum, the proposed approach aims to learn an embedding in which a GMM enables a Bayesian classifier to generalize well for classes in the tail by balancing out class-priors. The proposed method is simple, easy-to-train using deep learning frameworks, and increases classification performance for classes in the tail. The experiments on publicly available datasets show that this approach tends to perform better on classes in the tail than the competing methods, while performing comparable to the state-of-the-art on classes with abundant training instances. The main challenges for learning models using long-tailed datasets comprise learning parameters that generalize from a few-shots and avoiding classifier bias. While the proposed approach aims to tackle these two problems simultaneously, methods that tackle each of these problems independently are still relevant. As such, this section not only covers prior work on learning using imbalanced datasets, but also covers relevant solutions for few-shot learning. Given that the proposed approach is based on a GMM model, this section also covers recent approaches that use class-centroid representations for incremental learning and for improving discriminative properties. Simple techniques that deal with imbalanced datasets use random sampling to artificially create a more balanced training set BID7 ). For instance, random oversampling effectively "repeats" training instances from the classes in the tail, while random undersampling "removes" instances from the classes with abundant training instances. Thus, these techniques address imbalanced datasets by means of artificially balancing the training set. An alternative approach to deal with long-tailed datasets use transfer learning techniques. BID34 proposed MetaModelNet, a meta-learning algorithm that learns the evolution of parameters when gradually including more training samples. MetaModelNet improves the performance of CNN models since it transfers the parameter-evolution knowledge from data-rich classes to categories in the tail. Rather than artificially modifying the training set or use transfer learning, the proposed approach aims to learn an embedding that allows classifiers to generalize when learning from a long-tailed dataset. Consequently, the proposed method can complement sampling or transfer-learning-based methods. Recent approaches in this category aim to learn good parameters from a few training instances BID26 BID6 . A recent approach that considers an imbalanced dataset to tackle few-shot learning is the work by BID6 . Their proposed approach learns a feature embedding from the classes with the most samples in the dataset. Then, the approach "hallucinates" samples for classes with a few training instances in the learned embedding. While this work learns from an imabalanced dataset, it considers a different setting that that of the proposed approach. The work by BID6 assumes that classes with few instances are added incrementally. The proposed approach differs in this regard, since the introduced method aims to learn the embedding using the entire long-tailed dataset, generalize, and avoid any bias towards the classes with abundant training instances. To achieve generalization given a few shots in an incremental learning context, BID22 proposed iCaRL, a deep-learning-based incremental classifier. Similarly to the proposed approach, iCaRL represents each class using a single centroid in an embedding learned using a regular CNN model. However, instead of using the learned softmax classifier, it uses a nearestclass-mean BID16 classifier. Unlike iCaRL that uses features from a learned CNNsoftmax model, the proposed approach learns an embedding using a generative model. It is worth noting that the proposed approach uses a GMM, which by default includes a nearest-class-mean classifier as part of the learning problem. The use of class-centroids in learning representations is also useful to improve discriminative properties. BID35 proposed a loss that aims to minimize intra-class variation in CNN-softmax models. Unlike the center-loss approach, the proposed method minimizes the intra-class variation automatically by finding the GMM parameters in the learned embedding. Different from the center-loss that requires a mechanism to estimate the class-centroids, the proposed approach uses back-propagation to learn the GMM parameters. A recent approach that aims to generalize by using class-centroid representations are the Prototypical Networks (proto-nets) by BID26 . Proto-nets estimate the class centroids from a slice of a mini-batch-like subset of the training set. Then, they evaluate the loss from the complementary slice of the mini-batch-like subset, and update the feature encoder weights. The proposed approach has two main differences with proto-nets. First, the proposed approach is based on generative models describing the statistics of an imbalanced dataset, rather than learning an embedding tailored for a nearest-class-mean classifier that requires specific parameter-update rules. Second, the proposed approach uses regular batching mechanisms and updates parameters using back-propagation. Thus, in constrast with proto-nets, the proposed approach avoids modifying components in the deep learning frameworks. The goal of this work is to learn an embedding that allows a simple Bayesian classifier to robustly operate given a long-tailed training dataset. Specifically, this work aims to learn an encoder f w (·), parameterized by its set of weights w, that produces a good representation for Bayesian classification given a long-tailed dataset. In order to learn the aforementioned encoder, the proposed approach requires a model that describes the distribution of the data. Let x = f w (I) be the encoded feature for image I and y be its corresponding class label. Thus, the distribution of the training set can be described with the following joint probability: DISPLAYFORM0 where p(f w (I) | y; θ y ) represents the likelihood of observing the feature vector x as part of class y, and π y is the prior probabilities for class y. The likelihood is a function with parameters θ y (e.g., parameters of a multivariate Gaussian) that describes the distribution of the feature vectors in the embedding. Thus, the joint probability of the data p(x, y; w, θ) is a function with parameters composed by the the encoder w parameters, and the Bayesian parameters θ which include the likelihood parameters θ y , and priors π y . In practice, the likelihood parameters proves most crucial as it is not sensitive to class priors, which can be misleading in the long-tailed setting (as discussed in Section 3.1).Given the above joint probability model, the posterior probability for class y given a feature vector x can be computed using Bayes rule as follows: DISPLAYFORM1 where θ is a concatenation of the likelihood parameters and priors of all the classes. Thus, the class posterior probability is a function that depends on the encoder parameters w and the Bayesian parameters θ. The overall objective of this work is to jointly learn the weights w of the feature encoder and the Bayesian parameters θ to guarantee a good classification performance. Given a training dataset of images and label pairs D = {(x i , y i )}, we propose to learn parameters by maximizing the Bayesian class-posterior probability of the true class labels: DISPLAYFORM2 where MLE is a function that computes the closed form maximum likelihood estimates of the parameters of our Bayesian model, a procedure commonly known as Empirical Bayes BID1 .To use existing solvers for learning deep networks, we reformulate the problem shown in Eq. FORMULA2 as an unconstrained optimization by using a Lagrangian penalty BID2 ) that penalizes solutions which violate the constraint: DISPLAYFORM3 where λ ≥ 0. In this formulation, the optimization explicitly searches over the feature encoder parameters w and the Bayesian parameters θ so as to maximize class posterior probabilities. The last term penalizes deviations of the θ parameters from their MLE estimates, effectively acting as a regularizer. GMMs: The likelihood models are crucial to determine the parameters that allows the proposed approach to learn the feature encoder given a long-tailed dataset. We propose to use a multivariate Gaussian probability density function as the likelihood model. Given this likelihood model, the proposed approach implicitly uses a Gaussian mixture model to represent the distribution of the training set. Using a multivariate Gaussian brings benefits to the proposed formulation. This is because its parameters (the centroid µ, covariance matrix Σ, and prior π) have an intuitive meaning and closed-form-maximum-likelihood estimators. Interestingly, as discussed by van den BID31 and BID20 , a mixture of multivariate Gaussians can be used to theoretically motivate the success of deep learning. Balancing: The use of a GMM not only brings simplicity into the formulation, but also allows the feature encoder to generalize better for classes in the tail. The generalization aspect of a GMM We compare the effect of gradient-based updates for a traditional softmax classifier versus our Bayesian embedding model. Recall that our approach learns an embedding for which Bayesian classifiers produce accurate class posteriors. During softmax training, an "easy" example of a class will tend to not generate a strong gradient update, and so is not useful for learning (left). This might be considered paradoxical: when children learn a new concept (for say, a never-before-seen animal), an easy or "protypical" example might be most informative for learning. On the other hand, in our framework, an easy example of a class will change its centroid, generating a strong signal for updating our learned representation (right).model comes from the fact that a class is described with a single centroid. The benefit of this class representation is that estimating the centroid with a handful of examples is simple and produces a good estimate. Perhaps more importantly, a GMM allows us to access specific parameters that control the probabilistic "footprint" of each class in the embedded space. We can set these parameters to ensure balanced footprints by fixing the covariance matrices to be the identity and the class priors to be uniform -see Fig. 1-(c) . The remaining parameters to be estimated are then the class means µ = (µ 1 , . . . , µ nc ). Given this setting and considering that deep-learning frameworks use mini batches, the unconstrained problem shown in Eq. (4) becomes: DISPLAYFORM0 where y i is the true class label/index for the i-th data point, m is the batch size, M is the set of class indices in the batch, n j is the number of samples of the j-th class in the batch, and i is the index running over instances in the batch. Other probabilistic models: Our analysis and experiments focus on Gaussian Mixture Models, but the general learning problem from Eq. (4) holds for other probabilistic models. For example, deep embeddings can be learned for rectified (nonnegative) or binary features BID0 BID4 . For such embeddings, likelihood models based on rectified Gaussians or multivariate Bernoulli distributions may be more appropriate BID27 BID30 . Such models do not appear to have closed form maximum likelihood estimates, and so may be challenging to formulate precisely as a constrained optimization problem. Relationship to softmax: The GMM-based formulation has a direct relationship with softmax classifiers. This relationship can be obtained by expanding the squared distance terms in the classposterior probability, yielding the following: DISPLAYFORM0 where v j = µ j and DISPLAYFORM1 2 is a common term between the numerator and denominator. This relationship thus indicates that the proposed approach fits linear classifiers with restricted biases. This relationship is useful for an easy implementation in many deep learning frameworks. This is because this approach can be implemented using a dense layer without the bias terms. In addition, this relationship shows that the proposed approach requires fewer parametersto-learn in comparison with classical CNN-softmax models. An more intuitive comparison between GMMs and softmax classifiers can be made with respect to to their parameter updates. Intuitively, during softmax training, an "easy" example of a class will not generate a model update. In some sense, this might be considered paradoxical. When children learn a new concept (for say, a neverbefore-seen animal), they tend to be presented with an easy or "protypical" example. On the other hand, an easy example of a class will change its centroid, generating a signal for learning -see FIG1 . This section presents a series of experiments evaluating the learned embedding computed using the proposed method and long-tailed datasets. Since the goal of the experiments is to evaluate the feature encoder, all the experiments trained all the baselines or competing methods and the proposed one from scratch. An additional goal of the experiments is to show that the proposed approach can be adapted to any CNN architecture. For this reason, the experiments also used legacy and recent CNN architectures. Datasets: One evaluation aspect of the experiments is to measure the performance on smalland medium-scale datasets. The experiments included MNIST (LeCun et al., 1998) and CIFAR 10 ( BID12 ) as the small-scale datasets (each with ten classes); and CIFAR 100 BID12 ) and Tiny ImageNet 1 as the medium-scale datasets (with hundred and two hundred classes, respectively). The balanced MNIST dataset contains 60,000 and 10,000 28x28 training and testing images depicting hand-written digits, respectively. The CIFAR 10 dataset contains 50,000 and 10,000 32x32 training and testing images, respectively. The CIFAR 100 dataset contains 500 and 100 32x32 training and testing images per class, respectively. Lastly, Tiny ImageNet has 500 and 50 64x64 training and testing images for every class, respectively. However, the experiments used a 224x224 image instead. See Sec. 4.1 for details on how the experiments processed these datasets to evaluate classifiers using long-tailed datasets. The experiments included recent approaches that deal with imbalanced datasets. These approaches include iCaRL (Rebuffi et al., 2017), center loss BID35 , and a plain softmax classifier. The experiments also consider a variation of iCaRL. This variation does not use normalized feature vectors as originally proposed by BID22 . While prototypical networks BID26 are similar to the proposed approach, they require a balanced dataset with a few training instances for every class. Since prototypical networks do not assume a long-tailed dataset, these experiments did not include it as a competing method. The experiment also considered a method that uses a full GMM model (i.e., full covariance, means, and priors) of a softmax representation of the training set. As discussed in Sec. 2, MetaModelNet BID34 ) deals with long-tailed datasets by operating at the classifier-parameter level, since it is a meta-learning algorithm. Thus, MetaModelNet does not learn an embedding, and consequently complements the proposed method. 2 . This open-source project implements various legacy architectures, and several preprocessing imaging techniques (e.g., random translations and shifts). The experiments used the following CNN architectures: LeNet (LeCun et al., 1998) for MNIST; CifarNet BID12 ) for CIFAR 10; AllCNN (Springenberg et al., 2014) for CIFAR 100; and VGG 16 (Simonyan & Zisserman, 2015) for Tiny ImageNet. We implemented center loss BID35 and verified correctness using a balanced setting. We implemented the proposed approach in TFM using a fully connected layer with a restrictive bias. This is possible thanks to the relationship with linear classifiers discussed in Sec. 3.2. The regularizer was implemented using plain Tensorflow operations and was added as a regularizer function for the fully connected layer with restrictive bias. We will release the code upon publication. The hyperparameters for center-loss and the proposed approach were estimated using a validation set for every dataset. See Sec. A in the Appendix for the specific parameters. The main motivation of this work is to learn from a realistic dataset representing the statistics of the real visual world. Recall that realistic datasets are long-tailed since the visual world has a few predominant classes while others are rare. As such, the performance evaluation of the visual recognition system in this setting needs to be discussed, since common evaluation methods may not be adequate given this context. Intuitively, since the visual world yields long-tailed datasets, then the test set should ideally be long-tailed as well. While this rationale is logical given the statistics of the world, it has a main drawback: a simple classifier that is biased towards classes in the head is likely to perform well using a long-tailed testing set. While this setting may reflect a good performance for the common classes in practice, achieving a good performance for classes in the tail is still desirable in real practical applications. For instance, consider a self-driving car: the vehicle may easily detect common objects or events, e.g., pedestrians walking on the sidewalk. However, children playing soccer on the street is a rare event that can occur in the real world, and it is important to evaluate autonomous systems on such rare but crucial events. Thus, although rare events are infrequent, classifiers still need to account for them. Consequently, average accuracy on a long-tailed dataset is not an adequate measure of performance across rare classes. An alternative approach using long-tailed testing sets is to evaluate per-class accuracy. This explicitly weights all classes -both in the head and tail -equally. However, this has the drawback that performance estimates of rare classes in the tail have high variability and can be unreliable. In the autonomous vehicle scenario above, we might encounter very few (or even no) examples of children playing street soccer in any finite testset. This means that performance estimates fort tail classes can be unreliable. We propose an evaluation approach that addresses the bias towards the head and intra-class variation of classes in the tail. The proposed evaluation protocol requires a training and an evaluation procedure. The training setting includes several training trials that used different versions of long-tailed training sets. The evaluation procedure uses a balanced dataset. The use of a balanced testing set addresses the issue of classifiers that are biased towards the head since the class-priors are uniform and both classes in the head and tail contribute to the performance measure. Training a classifier using different long-tailed sets accounts for intra-class variation for classes in the tail. Consequently, aggregates of performance from these different trials account for the intra-class variation noise from classes in the tail. The experiments report a per-class accuracy average, the average class-accuracy, and their standard deviations over three different trails. Because the considered datasets are balanced, the experiments "long-tailed" these datasets following the procedure proposed by BID34 . For every class, the procedure computed the number of samples to draw from the balanced set using an exponential distribution. Thus, as the class index grows, the number of training instances decreases according to the exponential distribution. Given the computed number of samples to draw, the procedure randomly selects these instances from the balanced set to generate a training long-tailed dataset version. FIG2 shows a visualization of the training-instance distribution of the resultant long-tailed datasets. The experiments used the balanced testing sets because the goal is to measure generalization and overall performance. The goal of this experiment is to evaluate the learned embedding using a long-tailed dataset. To do so, the experiments used the long-tailed datasets described above. In particular, the target is to measure any classification improvement for classes in the tail with respect to the regular softmax classifier. Since most of the baselines rely on class-centroids to classify, the experiments use a FIG3 : Left column: Relative classification accuracy gain of the competing and proposed methods with respect to a softmax classifier using long-tailed datasets. Overall, the proposed method tends to achieve a comparable accuracy to that of a softmax classifier while delivering an increase for tail classes. Right column: The performance of a softmax classifier. The performance for classes in the head is higher than that of the classes in the tail.nearest-class-mean BID16 classifier. Thus, the experiments computed the deepfeatures for the training and testing sets after learning the feature encoder f w (·); a deep feature is the output of f w (·) which is the input tensor to the classifier or softmax layer. Then, the experiments computed a class centroid using the long-tailed training set for every method. To measure the classification improvements, the experiments trained all the methods with three different long-tailed datasets and used the balanced testing sets. Then, the experiment computed an average class-accuracy from the three trials for every baseline. To measure the relative performance with respect to a softmax classifier, the experiment computed the ratio between the average classaccuracy of a competing method (i.e., iCaRL BID22 , center loss BID35 , and the proposed method) and the average class-accuracy of a softmax classifier; the softmax classifier is the reference because it tends to bias towards classes in the head BID34 . The results on the MNIST dataset (first row) show that most of the competing methods perform comparable to that of a softmax classifier. However, the GMM method underperforms for classes in the tail. For this dataset, a softmax classifier does not present a significant bias towards classes in the head. Thus, these results indicate that the competing and proposed methods, with the exception of the GMM, operate well given a dataset with minor visual variations (e.g., illumination variations, pose, occlusion, among others). Consequently, these results effectively work as a sanity check of the proposed and competing methods (i.e., iCaRl and Centerloss). The results on CIFAR 10 (second row) show that the proposed approach and competing methods tend to perform comparable to a softmax classifier for classes in the head (i.e., the first three classes). In addition, the results show that the GMM also underperforms for classes in the tail. However, the proposed approach and competing methods tend to increase relative performance for classes in the tail. In this dataset, the proposed approach achieved an average class-accuracy of 68%, which is the highest compared to all the methods. The plots in the third row show the results on CIFAR 100. The plot in the left shows that the proposed method achieves a comparable performance with respect to a softmax classifier for classes in the head (i.e., the first twenty classes). On the other hand, the competing methods have a larger decrease in accuracy for classes in the head. The GMM in this dataset again underperforms for classes in the tail. The plot in the left shows that the proposed approach tends to increase the relative performance for classes in the tail. Overall, they tend to be larger than those of the competing methods and a softmax classifier. Lastly, the plot at the bottom shows the results on Tiny ImageNet. The plot in the left shows similar observations. The proposed approach maintains a comparable performance with respect to a softmax classifier for classes in the head. However, it delivers an increase in relative performance for classes in the tail. The GMM approach suffers for classes in the tail because the covariance estimates are poor due to the lack of data. To highlight the previous observations, TAB0 shows a break down of the average relative performance for classes in the head (H column) and in the tail (T column); this experiment excludes the GMM approach. To measure an average relative performance for classes in the head, the experiments used a weighted average of the relative performance considering all the classes. The average used the fraction of instances for a given class in the training set as its corresponding weight. Specifically, the weight for the i-th class is w i = ni n , where n i is the number of training instances for the i-th class and n is the total number of training instances in the long-tailed training set. Thus, this average emphasizes the relative performance of classes with abundant training instances while decreasing the contribution of the classes with scarce training data. To compute a weighted average of the relative performance for classes in the tail, the experiment calculated the weight w i for the i-th class as follows: w i = 1−wi i 1−wi . These weights emphasize the relative performance of the classes in the tail while diminishing the relative performance of classes in the head. The results in TAB0 : Classification performance improvement by using the regularizer in the proposed approach on CIFAR 10. The proposed approach with regularizer achieves a higher classification accuracy than the approach without the regularizer. Figure 5: Accuracy increase achieved by using the proposed regularizer on CIFAR 100. Overall, the proposed approach with regularizer tends to increase the accuracy across all classes compared to the proposed approach without one.show that the proposed method maintains a comparable peformance for classes in the head with respect to a softmax classifier. At the same time, the proposed method consistently improves the performance for classes in the tail. The goal of this experiment is to measure the benefits of the regularizer in the proposed method. To do so, this experiment compared the proposed method with a hyperparemeter λ = 0, leaving only the Bayesian classifier, and the configuration tested in the previous Section. Note that this setting is equivalent to only using a linear classifier with restricted bias, according to the discussion in Sec. 3.2. This experiment only considered CIFAR 10 and 100, and tested performance also considering three different long-tailed training sets for each dataset. The results of this experiment on CIFAR 10 are shown in Table 2 . The table shows the average accuracies per class for the proposed method using a regularizer with λ = 0.001 (top row), and without a regularizer (bottom row). The last column of the table shows the average classification performance. This table shows that the regularizer overall improves classification performance. This is expected since the regularizer aims to retain the centroid-parameters that are as close as possible to the batch-sample-mean centroids. Fig. 5 presents the results of this experiment on CIFAR 100. The plot shows the accuracy gains obtained by comparing the accuracies of the proposed method using a regularizer with λ = 0.0001 across all classes. Also, the plot shows the average accuracies for both methods. The plot indicates that the regularizer consistently provides an accuracy increase across classes. Thus, the regularizer is an important component that overall improves the classification performance. This work introduced a method that improves the classification performance for classes in the tail. The proposed approach is based on a Gaussian mixture model that allows a Bayesian classifier to represent the distribution of a long-tailed dataset and to compute the class-prediction probabilities. The experiments on publicly available dataset show that the proposed approach tends to increase the classification accuracy for classes in the tail while maintaining a comparable accuracy to that of a softmax classifier for classes in the head. In addition, this work introduced an evaluation method for methods that tackle the learning of concepts from a long-tailed dataset. Finally, this work demon-strated that class-centroid approaches overall tend to generalize well for classes in the tail while maintaining a comparable performance to that of a softmax classifiers for classes in the head. In order to guarantee similar training conditions for all the methods, the experiments used the same framework parameters (e.g., number of steps, learning rate, decay factors, among others) for all the considered methods. All the tested methods used an Adam optimizer BID11 ) and a batch size of 32. The experiments used a learning rate of 0.01 and 0.1 for the small-and medium-scale datasets, respectively. The experiments used the default exponential-learning-rate decay, weight decay, and drop-out parameters provided in TensorFlow Models. The hyperparameters used for center-loss are 0.5 for the centroids learning rate and a scale value of 0.001 for MNIST and CIFAR 10, 0.0001 for CIFAR 100 and Tiny ImageNet the proposed method. The hyperparameter for the proposed approach was set to 0.001 for MNIST and CIFAR 10, and 0.0001 for CIFAR 100 and Tiny ImageNet.
Approach to improve classification accuracy on classes in the tail. The main goal of this paper is to learn a ConvNet classifier which performs better for classes in the tail of the class occurrence distribution. Proposal for a Bayesian framework with a Gaussian mixture model to address an issue in classification applications, that the number of training data from different classes is unbalanced.