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]:  Auto-encoding and generative models have made tremendous successes in image and signal representation learning and generation. These models, however, generally employ the full Euclidean space or a bounded subset (such as $[0,1]^l$) as the latent space, whose trivial geometry is often too simplistic to meaningfully reflect the structure of the data. This paper aims at exploring a nontrivial geometric structure of the latent space for better data representation. Inspired by differential geometry, we propose \textbf{Chart Auto-Encoder (CAE)}, which captures the manifold structure of the data with multiple charts and transition functions among them. CAE translates the mathematical definition of manifold through parameterizing the entire data set as a collection of overlapping charts, creating local latent representations. These representations are an enhancement of the single-charted latent space commonly employed in auto-encoding models, as they reflect the intrinsic structure of the manifold.   Therefore, CAE achieves a more accurate approximation of data and generates realistic new ones. We conduct experiments with synthetic and real-life data to demonstrate the effectiveness of the proposed CAE.
[EX A]: Manifold-structured latent space for generative models

[EX Q]: We introduce geomstats, a Python package for Riemannian modelization and optimization over manifolds such as hyperspheres, hyperbolic spaces, SPD matrices or Lie groups of transformations. Our contribution is threefold. First, geomstats allows the flexible modeling of many a machine learning problem through an efficient and extensively unit-tested implementations of these manifolds, as well as the set of useful Riemannian metrics, exponential and logarithm maps that we provide. Moreover, the wide choice of loss functions and our implementation of the corresponding gradients allow fast and easy optimization over manifolds. Finally, geomstats is the only package to provide a unified framework for Riemannian geometry, as the operations implemented in geomstats are available with different computing backends (numpy,tensorflow and keras), as well as with a GPU-enabled mode–-thus considerably facilitating the application of Riemannian geometry in machine learning. In this paper, we present geomstats through a review of the utility and advantages of manifolds in machine learning, using the concrete examples that they span to show the efficiency and practicality of their implementation using our package
[EX A]: We introduce geomstats, an efficient Python package for Riemannian modelization and optimization over manifolds compatible with both numpy and tensorflow .

[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.