We present a latent variable model for classification that provides a novel probabilistic interpretation of neural network softmax classifiers. We derive a variational objective to train the model, analogous to the evidence lower bound (ELBO) used to train variational auto-encoders, that generalises the softmax cross-entropy loss. Treating inputs to the softmax layer as samples of a latent variable, our abstracted perspective reveals a potential inconsistency between their anticipated distribution, required for accurate label predictions, and their empirical distribution found in practice. We augment the variational objective to mitigate such inconsistency and induce a chosen latent distribution, instead of the implicit assumption found in a standard softmax layer. Overall, we provide new theoretical insight into the inner workings of widely-used softmax classifiers. Empirical evaluation on image and text classification datasets demonstrates that our proposed approach, variational classification, maintains classification accuracy while the reshaped latent space improves other desirable properties of a classifier, such as calibration, adversarial robustness, robustness to distribution shift and sample efficiency useful in low data settings.

Brain tumors pose a significant global health challenge due to their high prevalence and mortality rates across all age groups. Detecting brain tumors at an early stage is crucial for effective treatment and patient outcomes. This study presents a comprehensive investigation into the use of Convolutional Neural Networks (CNNs) for brain tumor detection using Magnetic Resonance Imaging (MRI) images. The dataset, consisting of MRI scans from both healthy individuals and patients with brain tumors, was processed and fed into the CNN architecture. The SoftMax Fully Connected layer was employed to classify the images, achieving an accuracy of 98%. To evaluate the CNN's performance, two other classifiers, Radial Basis Function (RBF) and Decision Tree (DT), were utilized, yielding accuracy rates of 98.24% and 95.64%, respectively. The study also introduced a clustering method for feature extraction, improving CNN's accuracy. Sensitivity, Specificity, and Precision were employed alongside accuracy to comprehensively evaluate the network's performance. Notably, the SoftMax classifier demonstrated the highest accuracy among the categorizers, achieving 99.52% accuracy on test data. The presented research contributes to the growing field of deep learning in medical image analysis. The combination of CNNs and MRI data offers a promising tool for accurately detecting brain tumors, with potential implications for early diagnosis and improved patient care.

Training a classifier over a large number of classes, known as 'extreme classification', has become a topic of major interest with applications in technology, science, and e-commerce. Traditional softmax regression induces a gradient cost proportional to the number of classes $C$, which often is prohibitively expensive. A popular scalable softmax approximation relies on uniform negative sampling, which suffers from slow convergence due a poor signal-to-noise ratio. In this paper, we propose a simple training method for drastically enhancing the gradient signal by drawing negative samples from an adversarial model that mimics the data distribution. Our contributions are three-fold: (i) an adversarial sampling mechanism that produces negative samples at a cost only logarithmic in $C$, thus still resulting in cheap gradient updates; (ii) a mathematical proof that this adversarial sampling minimizes the gradient variance while any bias due to non-uniform sampling can be removed; (iii) experimental results on large scale data sets that show a reduction of the training time by an order of magnitude relative to several competitive baselines.