We consider the fundamental task of optimizing a real-valued function defined
in a potentially high-dimensional Euclidean space, such as the loss function in
many machine-learning tasks or the logarithm of the probability distribution in
statistical inference. We use the warped Riemannian geometry notions to
redefine the optimisation problem of a function on Euclidean space to a
Riemannian manifold with a warped metric, and then find the function's optimum
along this manifold. The warped metric chosen for the search domain induces a
computational friendly metric-tensor for which optimal search directions
associate with geodesic curves on the manifold becomes easier to compute.
Performing optimization along geodesics is known to be generally infeasible,
yet we show that in this specific manifold we can analytically derive Taylor
approximations up to third-order. In general these approximations to the
geodesic curve will not lie on the manifold, however we construct suitable
retraction maps to pull them back onto the manifold. Therefore, we can
efficiently optimize along the approximate geodesic curves. We cover the
related theory, describe a practical optimization algorithm and empirically
evaluate it on a collection of challenging optimisation benchmarks. Our
proposed algorithm, using third-order approximation of geodesics, outperforms
standard Euclidean gradient-based counterparts in term of number of iterations
until convergence and an alternative method for Hessian-based optimisation
routines.

通过在具有截断度量的流形上进行优化，我们得到了函数的最优解，并通过构造适当的回缩映射将途经的递近测地线拉回到流形上，从而有效地沿着近似的测地线进行优化。

欧几里得函数优化中的扭曲几何信息

Warped geometric information on the optimisation of Euclidean functions

Statistical modeling of spatiotemporal phenomena often requires selecting a
covariance matrix from a covariance class. Yet standard parametric covariance
families can be insufficiently flexible for practical applications, while
non-parametric approaches may not easily allow certain kinds of prior knowledge
to be incorporated. We propose instead to build covariance families out of
geodesic curves. These covariances offer more flexibility for problem-specific
tailoring than classical parametric families, and are preferable to simple
convex combinations. Once the covariance family has been chosen, one typically
needs to select a representative member by solving an optimization problem,
e.g., by maximizing the likelihood of a data set. We consider instead a
differential geometric interpretation of this problem: minimizing the geodesic
distance to a sample covariance matrix ("natural projection"). Our approach is
consistent with the notion of distance employed to build the covariance family
and does not require assuming a particular probability distribution for the
data. We show that natural projection and maximum likelihood estimation within
the covariance family are locally equivalent up to second order. We also
demonstrate that natural projection may yield more accurate estimates with
noise-corrupted data.

使用测地线进行协方差矩阵建模，通过最小化样本协方差矩阵到测地线的距离解决代表性协方差矩阵的选择问题，并展示了其在有噪声数据下解决方案的精度较高。

几何参数化协方差估计

Geodesically parameterized covariance estimation

Deep generative models learn a mapping from a low dimensional latent space to
a high-dimensional data space. Under certain regularity conditions, these
models parameterize nonlinear manifolds in the data space. In this paper, we
investigate the Riemannian geometry of these generated manifolds. First, we
develop efficient algorithms for computing geodesic curves, which provide an
intrinsic notion of distance between points on the manifold. Second, we develop
an algorithm for parallel translation of a tangent vector along a path on the
manifold. We show how parallel translation can be used to generate analogies,
i.e., to transport a change in one data point into a semantically similar
change of another data point. Our experiments on real image data show that the
manifolds learned by deep generative models, while nonlinear, are surprisingly
close to zero curvature. The practical implication is that linear paths in the
latent space closely approximate geodesics on the generated manifold. However,
further investigation into this phenomenon is warranted, to identify if there
are other architectures or datasets where curvature plays a more prominent
role. We believe that exploring the Riemannian geometry of deep generative
models, using the tools developed in this paper, will be an important step in
understanding the high-dimensional, nonlinear spaces these models learn.

研究了深度生成模型所学习的流形的黎曼几何，并提出了计算测地线和沿流形路径平行传递切向量的算法，发现这些模型学习的流形近似于零曲率，并探讨了这种现象的实际影响。