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.

[EX Q]: Training methods for deep networks are primarily variants on stochastic gradient descent.   Techniques that use (approximate) second-order information are rarely used because of the computational cost and noise associated with those approaches in deep learning contexts.   However, in this paper, we show how feedforward deep networks exhibit a low-rank derivative structure.   This low-rank structure makes it possible to use second-order information without needing approximations and without incurring a significantly greater computational cost than gradient descent.   To demonstrate this capability, we implement Cubic Regularization (CR) on a feedforward deep network with stochastic gradient descent and two of its variants.   There, we use CR to calculate learning rates on a per-iteration basis while training on the MNIST and CIFAR-10 datasets.   CR proved particularly successful in escaping plateau regions of the objective function.   We also found that this approach requires less problem-specific information (e.g. an optimal initial learning rate) than other first-order methods in order to perform well.
[EX A]: We show that deep learning network derivatives have a low-rank structure, and this structure allows us to use second-order derivative information to calculate learning rates adaptively and in a computationally feasible manner.

[EX Q]: Decades of research on the neural code underlying spatial navigation have revealed a diverse set of neural response properties. The Entorhinal Cortex (EC) of the mammalian brain contains a rich set of spatial correlates, including grid cells which encode space using tessellating patterns. However, the mechanisms and functional significance of these spatial representations remain largely mysterious. As a new way to understand these neural representations, we trained recurrent neural networks (RNNs) to perform navigation tasks in 2D arenas based on velocity inputs. Surprisingly, we find that grid-like spatial response patterns emerge in trained networks, along with units that exhibit other spatial correlates, including border cells and band-like cells. All these different functional types of neurons have been observed experimentally. The order of the emergence of grid-like and border cells is also consistent with observations from developmental studies. Together, our results suggest that grid cells, border cells and others as observed in EC may be a natural solution for representing space efficiently given the predominant recurrent connections in the neural circuits.

[EX A]: To our knowledge, this is the first study to show how neural representations of space, including grid-like cells and border cells as observed in the brain, could emerge from training a recurrent neural network to perform navigation tasks.

[EX Q]: Knowledge Graph Embedding (KGE) is the task of jointly learning entity and relation embeddings for a given knowledge graph. Existing methods for learning KGEs can be seen as a two-stage process where (a) entities and relations in the knowledge graph are represented using some linear algebraic structures (embeddings), and (b) a scoring function is defined that evaluates the strength of a relation that holds between two entities using the corresponding relation and entity embeddings. Unfortunately, prior proposals for the scoring functions in the first step have been heuristically motivated, and it is unclear as to how the scoring functions in KGEs relate to the generation process of the underlying knowledge graph. To address this issue, we propose a generative account of the KGE learning task. Specifically, given a knowledge graph represented by a set of relational triples (h, R, t), where the semantic relation R holds between the two entities h (head) and t (tail), we extend the random walk model (Arora et al., 2016a) of word embeddings to KGE. We derive a theoretical relationship between the joint probability p(h, R, t) and the embeddings of h, R and t. Moreover, we show that marginal loss minimisation, a popular objective used by much prior work in KGE, follows naturally from the log-likelihood ratio maximisation under the probabilities estimated from the KGEs according to our theoretical relationship. We propose a learning objective motivated by the theoretical analysis to learn KGEs from a given knowledge graph. The KGEs learnt by our proposed method obtain state-of-the-art performance on FB15K237 and WN18RR benchmark datasets, providing empirical evidence in support of the theory.

[EX A]:
We present a theoretically proven generative model of knowledge graph embedding.