Applications and recent advances of polynomial neural networks in medical imaging

Polynomial Neural Networks (PoNNs) are a novel variant of the recently introduced Kolmogorov-Arnold Networks (KANs), designed to harness the approximation properties of orthogonal polynomials for modeling complex nonlinear functions. The KAN framework represents a new path in neural architecture design by introducing learnable 1-D functions as its fundamental block instead of traditional weights, biases, and fixed activation functions. In their elemental form, both PoNNs and KANs are analogous with the classic Multilayer Perceptron (MLP) but extend its expressiveness to model more complex data distributions. This paper presents applications of two simple polynomial neural architectures built upon Hermite and Legendre polynomials, which serve as classification heads on top of a classical convolutional neural network (ConvNet). The ConvNet acts as an automatic feature extractor, while the PoNN head performs the final classification of the input medical image. We directly compare the performance of these polynomial heads against a standard KAN head, with all models sharing the same convolutional backbone. All architectures are trained from scratch and evaluated on curated medical datasets from the MedMNIST collection. Medical images present a unique computer vision challenge for a new neural architecture due to their diverse acquisition modalities, from optics and ultrasound to computed tomography, each introducing inherent device specific noise. Moreover, in the feature space, medical classes may exhibit significant overlap. The experimental setup involves both binary and multiclass classification tasks, including pneumonia recognition and blood cell recognition. Finally, we present recent advances in the development of a Convolutional Polynomial Neural Network (Convolutional-PoNN). This architecture replaces each parameter in a classical convolutional kernel with a learnable 1-D Hermite or Legendre polynomial. The Convolutional-PoNN is also trained and evaluated on medical datasets from the MedMNIST collection.