Sensitivity and stability of pretrained CNN filters

Convolutional Neural Network (CNN) is a powerful and successful deep learning technique for a variety of computer vision and image analysis applications. Interpreting and explaining the decisions of CNNs is one of the most challengeable tasks despite its significant success in various image analysis tasks. Topological Data Analysis (TDA) is an approach that exploits algebraic invariants from topology to analyse high dimensional and noisy datasets as well as growing challenges of big data applications. Persistent homology (PH) is an algebraic topology method for measuring topological features of shapes and/or functions at different distance or similarity resolutions. This work is an attempt to investigate the algebraic properties of pretrained CNN convolutional layer filters based on random Gaussian/Uniform distribution. We shall investigate the stability and sensitivity of the condition number of CNN filters during and post the model training with focus on class discriminability of the PH features of the convolved images. We shall demonstrate a strong link between the condition number of the CNN filters and their discriminating power of the PH representation. In particular, we shall establish that if small perturbation added to the original images then feature maps with well-conditioned filters will produce similar topological features to the original image. Our investigation and findings are based on training CNN's with Digits, MNIST and CIFAR-10 datasets. Our ultimate interest in applying the results of these findings in designing appropriate CNN models for classifications of ultrasound tumor scan images. Preliminary results for these applications are encouraging.