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.

Q: Inspired by neurophysiological discoveries of navigation cells in the mammalian
 brain, we introduce the first deep neural network architecture for modeling Egocentric
 Spatial Memory (ESM). It learns to estimate the pose of the agent and
 progressively construct top-down 2D global maps from egocentric views in a spatially
 extended environment. During the exploration, our proposed ESM network
 model updates belief of the global map based on local observations using a recurrent
 neural network. It also augments the local mapping with a novel external
 memory to encode and store latent representations of the visited places based on
 their corresponding locations in the egocentric coordinate. This enables the agents
 to perform loop closure and mapping correction. This work contributes in the
 following aspects: first, our proposed ESM network provides an accurate mapping
 ability which is vitally important for embodied agents to navigate to goal locations.
 In the experiments, we demonstrate the functionalities of the ESM network in
 random walks in complicated 3D mazes by comparing with several competitive
 baselines and state-of-the-art Simultaneous Localization and Mapping (SLAM)
 algorithms. Secondly, we faithfully hypothesize the functionality and the working
 mechanism of navigation cells in the brain. Comprehensive analysis of our model
 suggests the essential role of individual modules in our proposed architecture and
 demonstrates efficiency of communications among these modules. We hope this
 work would advance research in the collaboration and communications over both
 fields of computer science and computational neuroscience.

A: first deep neural network for modeling Egocentric Spatial Memory inspired by neurophysiological discoveries of navigation cells in mammalian brain
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Q: Neural network training depends on the structure of the underlying loss landscape, i.e. local minima, saddle points, flat plateaus, and loss barriers. In relation to the structure of the landscape, we study the permutation symmetry of neurons in each layer of a deep neural network, which gives rise not only to multiple equivalent global minima of the loss function but also to critical points in between partner minima. In a network of $d-1$ hidden layers with $n_k$ neurons in layers $k = 1, \ldots, d$, we construct continuous paths between equivalent global minima that lead through a `permutation point' where the input and output weight vectors of two neurons in the same hidden layer $k$ collide and interchange. We show that such permutation points are critical points which lie inside high-dimensional subspaces of equal loss, contributing to the global flatness of the landscape. We also find that a permutation point for the exchange of neurons $i$ and $j$ transits into a flat high-dimensional plateau that enables all $n_k!$ permutations of neurons in a given layer $k$ at the same loss value. Moreover , we introduce higher-order permutation points by exploiting the hierarchical structure in the loss landscapes of neural networks, and find that the number of $K$-th order permutation points is much larger than the (already huge) number of equivalent global minima -- at least by a polynomial factor of order $K$. In two tasks, we demonstrate numerically with our path finding method that continuous paths between partner minima exist: first, in a toy network with a single hidden layer on a function approximation task and, second, in a multilayer network on the MNIST task. Our geometric approach yields a lower bound on the number of critical points generated by weight-space symmetries and provides a simple intuitive link between previous theoretical results and numerical observations.

A: Weight-space symmetry in neural network landscapes gives rise to numerous number of saddles and flat high-dimensional subspaces.
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Q: Motivated by applications to unsupervised learning, we consider the problem of measuring mutual information. Recent analysis has shown that naive kNN estimators of mutual information have serious statistical limitations motivating more refined methods. In this paper we prove that serious statistical limitations are inherent to any measurement method. More specifically, we show that any distribution-free high-confidence lower bound on mutual information cannot be larger than $O(\ln N)$ where $N$ is the size of the data sample. We also analyze the Donsker-Varadhan lower bound on KL divergence in particular and show that, when simple statistical considerations are taken into account, this bound can never produce a high-confidence value larger than $\ln N$. While large high-confidence lower bounds are impossible, in practice one can use estimators without formal guarantees. We suggest expressing mutual information as a difference of entropies and using cross entropy as an entropy estimator.  We observe that, although cross entropy is only an upper bound on entropy, cross-entropy estimates converge to the true cross entropy at the rate of $1/\sqrt{N}$.

A:
We give a theoretical analysis of the measurement and optimization of mutual information.
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