Inspired by the diversity of biological neurons, quadratic artificial neurons can play an important role in deep learning models. The type of quadratic neurons of our interest replaces the inner-product operation in the conventional neuron with a quadratic function. Despite promising results so far achieved by networks of quadratic neurons, there are important issues not well addressed. Theoretically, the superior expressivity of a quadratic network over either a conventional network or a conventional network via quadratic activation is not fully elucidated, which makes the use of quadratic networks not well grounded. Practically, although a quadratic network can be trained via generic backpropagation, it can be subject to a higher risk of collapse than the conventional counterpart. To address these issues, we first apply the spline theory and a measure from algebraic geometry to give two theorems that demonstrate better model expressivity of a quadratic network than the conventional counterpart with or without quadratic activation. Then, we propose an effective training strategy referred to as ReLinear to stabilize the training process of a quadratic network, thereby unleashing the full potential in its associated machine learning tasks. Comprehensive experiments on popular datasets are performed to support our findings and confirm the performance of quadratic deep learning. We have shared our code in \url{https://github.com/FengleiFan/ReLinear}.

Deep learning is a main focus of artificial intelligence and has greatly impacted other fields. However, deep learning is often criticized for its lack of interpretation. As a successful unsupervised model in deep learning, various autoencoders, especially convolutional autoencoders, are very popular and important. Since these autoencoders need improvements and insights, in this paper we shed light on the nonlinearity of a deep convolutional autoencoder in perspective of perfect signal recovery. In particular, we propose a new type of convolutional autoencoders, termed as Soft-Autoencoder (Soft-AE), in which the activations of encoding layers are implemented with adaptable soft-thresholding units while decoding layers are realized with linear units. Consequently, Soft-AE can be naturally interpreted as a learned cascaded wavelet shrinkage system. Our denoising numerical experiments on CIFAR-10, BSD-300 and Mayo Clinical Challenge Dataset demonstrate that Soft-AE gives a competitive performance relative to its counterparts.