Self-supervised representation learning is heavily dependent on data
augmentations to specify the invariances encoded in representations. Previous
work has shown that applying diverse data augmentations is crucial to
downstream performance, but augmentation techniques remain under-explored. In
this work, we propose a new family of local transformations based on Gaussian
random fields to generate image augmentations for self-supervised
representation learning. These transformations generalize the well-established
affine and color transformations (translation, rotation, color jitter, etc.)
and greatly increase the space of augmentations by allowing transformation
parameter values to vary from pixel to pixel. The parameters are treated as
continuous functions of spatial coordinates, and modeled as independent
Gaussian random fields. Empirical results show the effectiveness of the new
transformations for self-supervised representation learning. Specifically, we
achieve a 1.7% top-1 accuracy improvement over baseline on ImageNet downstream
classification, and a 3.6% improvement on out-of-distribution iNaturalist
downstream classification. However, due to the flexibility of the new
transformations, learned representations are sensitive to hyperparameters.
While mild transformations improve representations, we observe that strong
transformations can degrade the structure of an image, indicating that
balancing the diversity and strength of augmentations is important for
improving generalization of learned representations.

基于高斯随机场的新型局部变换用于自监督表征学习，并通过改进的数据增强方法，在 ImageNet 和 iNaturalist 数据集上实现了准确性的提升。

自我监督表示学习的随机场增强

Random Field Augmentations for Self-Supervised Representation Learning

Many machine learning applications require learning a function with a small
worst-case error over the entire input domain, that is, the $L_\infty$-error,
whereas most existing theoretical works only guarantee recovery in average
errors such as the $L_2$-error. $L_\infty$-recovery from polynomial samples is
even impossible for seemingly simple function classes such as constant-norm
infinite-width two-layer neural nets. This paper makes some initial steps
beyond the impossibility results by leveraging the randomness in the
ground-truth functions. We prove a polynomial sample complexity bound for
random ground-truth functions drawn from Gaussian random fields. Our key
technical novelty is to prove that the degree-$k$ spherical harmonics
components of a function from Gaussian random field cannot be spiky in that
their $L_\infty$/$L_2$ ratios are upperbounded by $O(d \sqrt{\ln k})$ with high
probability. In contrast, the worst-case $L_\infty$/$L_2$ ratio for degree-$k$
spherical harmonics is on the order of $\Omega(\min\{d^{k/2},k^{d/2}\})$.

通过利用高斯随机场中真实函数的随机性，证明了一些难以置信的结果，为高斯随机场中随机真实函数的多项式样本复杂度提供了一个有界上限。聚焦于球面谐波上，利用 L_inf 和 L_2 比率控制证明 L_inf-L_2 问题的解决方案。

Gaussian 随机场的非线性函数 L∞恢复： 多项式样本复杂度上界

Toward $L_\infty$-recovery of Nonlinear Functions: A Polynomial Sample  Complexity Bound for Gaussian Random Fields

Gaussian random fields (GRF) are a fundamental stochastic model for
spatiotemporal data analysis. An essential ingredient of GRF is the covariance
function that characterizes the joint Gaussian distribution of the field.
Commonly used covariance functions give rise to fully dense and unstructured
covariance matrices, for which required calculations are notoriously expensive
to carry out for large data. In this work, we propose a construction of
covariance functions that result in matrices with a hierarchical structure.
Empowered by matrix algorithms that scale linearly with the matrix dimension,
the hierarchical structure is proved to be efficient for a variety of random
field computations, including sampling, kriging, and likelihood evaluation.
Specifically, with $n$ scattered sites, sampling and likelihood evaluation has
an $O(n)$ cost and kriging has an $O(\log n)$ cost after preprocessing,
particularly favorable for the kriging of an extremely large number of sites
(e.g., predicting on more sites than observed). We demonstrate comprehensive
numerical experiments to show the use of the constructed covariance functions
and their appealing computation time. Numerical examples on a laptop include
simulated data of size up to one million, as well as a climate data product
with over two million observations.

本文提出了一种具有层次结构的协方差函数的构造，这种协方差函数在各种随机场计算中，包括采样、克里金以及似然估计中都表现出高效的特点，特别适用于极大数量站点的克里金预测。

高斯随机场线性代价协方差函数

Linear-Cost Covariance Functions for Gaussian Random Fields

Given a distribution $\rho$ on persistence diagrams and observations
$X_1,...X_n \stackrel{iid}{\sim} \rho$ we introduce an algorithm in this paper
that estimates a Fr\'echet mean from the set of diagrams $X_1,...X_n$. If the
underlying measure $\rho$ is a combination of Dirac masses $\rho = \frac{1}{m}
\sum_{i=1}^m \delta_{Z_i}$ then we prove the algorithm converges to a local
minimum and a law of large numbers result for a Fr\'echet mean computed by the
algorithm given observations drawn iid from $\rho$. We illustrate the
convergence of an empirical mean computed by the algorithm to a population mean
by simulations from Gaussian random fields.

该论文介绍了一种可以从任意分布的坚持图像计算出 Frechet mean 的算法，并证明了当分布是 Dirac masses 的组合时，算法收敛于局部最小值，并给出了一个取自该分布的观测序列的 Frechet mean 的大数定理结果。同时，作者还用高斯随机场的模拟表明算法计算出的经验 mean 会收敛到总体 mean。