The Loss Landscape of Shallow ReLU-like Neural Networks: Stationary Points, Saddle Escaping, and Network Embedding

In this paper, we investigate the loss landscape of one-hidden-layer neural networks with ReLU-like activation functions trained with the empirical squared loss. As the activation function is non-differentiable, it is so far unclear how to completely characterize the stationary points. We propose the conditions for stationarity that apply to both non-differentiable and differentiable cases. Additionally, we show that, if a stationary point does not contain "escape neurons", which are defined with first-order conditions, then it must be a local minimum. Moreover, for the scalar-output case, the presence of an escape neuron guarantees that the stationary point is not a local minimum. Our results refine the description of the saddle-to-saddle training process starting from infinitesimally small (vanishing) initialization for shallow ReLU-like networks, linking saddle escaping directly with the parameter changes of escape neurons. Moreover, we are also able to fully discuss how network embedding, which is to instantiate a narrower network within a wider network, reshapes the stationary points.

Latent Noise Segmentation: How Neural Noise Leads to the Emergence of Segmentation and Grouping

Deep Neural Networks (DNNs) that achieve human-level performance in general tasks like object segmentation typically require supervised labels. In contrast, humans are able to perform these tasks effortlessly without supervision. To accomplish this, the human visual system makes use of perceptual grouping. Understanding how perceptual grouping arises in an unsupervised manner is critical for improving both models of the visual system, and computer vision models. In this work, we propose a counterintuitive approach to unsupervised perceptual grouping and segmentation: that they arise because of neural noise, rather than in spite of it. We (1) mathematically demonstrate that under realistic assumptions, neural noise can be used to separate objects from each other, and (2) show that adding noise in a DNN enables the network to segment images even though it was never trained on any segmentation labels. Interestingly, we find that (3) segmenting objects using noise results in segmentation performance that aligns with the perceptual grouping phenomena observed in humans. We introduce the Good Gestalt (GG) datasets -- six datasets designed to specifically test perceptual grouping, and show that our DNN models reproduce many important phenomena in human perception, such as illusory contours, closure, continuity, proximity, and occlusion. Finally, we (4) demonstrate the ecological plausibility of the method by analyzing the sensitivity of the DNN to different magnitudes of noise. We find that some model variants consistently succeed with remarkably low levels of neural noise ($\sigma<0.001$), and surprisingly, that segmenting this way requires as few as a handful of samples. Together, our results suggest a novel unsupervised segmentation method requiring few assumptions, a new explanation for the formation of perceptual grouping, and a potential benefit of neural noise in the visual system.