Modern neural networks are usually highly over-parameterized. Behind the wide usage of over-parameterized networks is the belief that, if the data are simple, then the trained network will be automatically equivalent to a simple predictor. Following this intuition, many existing works have studied different notions of "ranks" of neural networks and their relation to the rank of data. In this work, we study the rank of convolutional neural networks (CNNs) trained by gradient descent, with a specific focus on the robustness of the rank to image background noises. Specifically, we point out that, when adding background noises to images, the rank of the CNN trained with gradient descent is affected far less compared with the rank of the data. We support our claim with a theoretical case study, where we consider a particular data model to characterize low-rank clean images with added background noises. We prove that CNNs trained by gradient descent can learn the intrinsic dimension of clean images, despite the presence of relatively large background noises. We also conduct experiments on synthetic and real datasets to further validate our claim.

The recent successes in analyzing images with deep neural networks are almost exclusively achieved with Convolutional Neural Networks (CNNs). The training of these CNNs, and in fact of all deep neural network architectures, uses the backpropagation algorithm where the output of the network is compared with the desired result and the difference is then used to tune the weights of the network towards the desired outcome. In a 2022 preprint, Geoffrey Hinton suggested an alternative way of training which passes the desired results together with the images at the input of the network. This so called Forward Forward (FF) algorithm has up to now only been used in fully connected networks. In this paper, we show how the FF paradigm can be extended to CNNs. Our FF-trained CNN, featuring a novel spatially-extended labeling technique, achieves a classification accuracy of 99.16% on the MNIST hand-written digits dataset. We show how different hyperparameters affect the performance of the proposed algorithm and compare the results with CNN trained with the standard backpropagation approach. Furthermore, we use Class Activation Maps to investigate which type of features are learnt by the FF algorithm.

Receiver operating characteristic (ROC) curve is an informative tool in binary classification and Area Under ROC Curve (AUC) is a popular metric for reporting performance of binary classifiers. In this paper, first we present a comprehensive review of ROC curve and AUC metric. Next, we propose a modified version of AUC that takes confidence of the model into account and at the same time, incorporates AUC into Binary Cross Entropy (BCE) loss used for training a Convolutional neural Network for classification tasks. We demonstrate this on two datasets: MNIST and prostate MRI. Furthermore, we have published GenuineAI, a new python library, which provides the functions for conventional AUC and the proposed modified AUC along with metrics including sensitivity, specificity, recall, precision, and F1 for each point of the ROC curve.