In this task, you are given the abstract of a research paper. Your task is to generate a summary of this abstract. Your summary should not be very short, but it's better if it's not more than 30 words.

Input: Consider Input: The linear transformations in converged deep networks show fast eigenvalue decay. The distribution of eigenvalues looks like a Heavy-tail distribution, where the vast majority of eigenvalues is small, but not actually zero, and only a few spikes of large eigenvalues exist.
 We use a stochastic approximator to generate histograms of eigenvalues. This allows us to investigate layers with hundreds of thousands of dimensions. We show how the distributions change over the course of image net training, converging to a similar heavy-tail spectrum across all intermediate layers.

Output: We investigate the eigenvalues of the linear layers in deep networks and show that the distributions develop heavy-tail behavior during training.


Input: Consider Input: Modeling informal inference in natural language is very challenging. With the recent availability of large annotated data, it has become feasible to train complex models such as neural networks to perform natural language inference (NLI), which have achieved state-of-the-art performance. Although there exist relatively large annotated data, can machines learn all knowledge needed to perform NLI from the data? If not, how can NLI models benefit from external knowledge and how to build NLI models to leverage it? In this paper, we aim to answer these questions by enriching the state-of-the-art neural natural language inference models with external knowledge. We demonstrate that the proposed models with external knowledge further improve the state of the art on the Stanford Natural Language Inference (SNLI) dataset.

Output: the proposed models with external knowledge further improve the state of the art on the SNLI dataset.


Input: Consider Input: While self-organizing principles have motivated much of early learning models, such principles have rarely been included in deep learning architectures. Indeed, from a supervised learning perspective it seems that topographic constraints are rather decremental to optimal performance. Here we study a network model that incorporates self-organizing maps into a supervised network and show how gradient learning results in a form of a self-organizing learning rule. Moreover, we show that such a model is robust in the sense of its application to a variety of  areas, which is believed to be a hallmark of biological learning systems.


Output: integration of self-organization and supervised learning in a hierarchical neural network