The central question in representation learning is what constitutes a good or meaningful representation. In this work we argue that if we consider data with inherent cluster structures, where clusters can be characterized through different means and covariances, those data structures should be represented in the embedding as well. While Autoencoders (AE) are widely used in practice for unsupervised representation learning, they do not fulfil the above condition on the embedding as they obtain a single representation of the data. To overcome this we propose a meta-algorithm that can be used to extend an arbitrary AE architecture to a tensorized version (TAE) that allows for learning cluster-specific embeddings while simultaneously learning the cluster assignment. For the linear setting we prove that TAE can recover the principle components of the different clusters in contrast to principle component of the entire data recovered by a standard AE. We validated this on planted models and for general, non-linear and convolutional AEs we empirically illustrate that tensorizing the AE is beneficial in clustering and de-noising tasks.

Comparison-based learning addresses the problem of learning when, instead of explicit features or pairwise similarities, one only has access to comparisons of the form: \emph{Object $A$ is more similar to $B$ than to $C$.} Recently, it has been shown that, in Hierarchical Clustering, single and complete linkage can be directly implemented using only such comparisons while several algorithms have been proposed to emulate the behaviour of average linkage. Hence, finding hierarchies (or dendrograms) using only comparisons is a well understood problem. However, evaluating their meaningfulness when no ground-truth nor explicit similarities are available remains an open question. In this paper, we bridge this gap by proposing a new revenue function that allows one to measure the goodness of dendrograms using only comparisons. We show that this function is closely related to Dasgupta's cost for hierarchical clustering that uses pairwise similarities. On the theoretical side, we use the proposed revenue function to resolve the open problem of whether one can approximately recover a latent hierarchy using few triplet comparisons. On the practical side, we present principled algorithms for comparison-based hierarchical clustering based on the maximisation of the revenue and we empirically compare them with existing methods.

近年来，监督学习环境的几个结果表明，古典统计学习 - 理论措施，如VC维度，不充分解释深度学习模型的性能，促使在无限宽度和迭代制度中的工作摆动。但是，对于超出监督环境之外的神经网络成功几乎没有理论解释。在本文中，我们认为，在一些分布假设下，经典学习 - 理论措施可以充分解释转导造型中的图形神经网络的概括。特别是，我们通过分析节点分类问题图卷积网络的概括性特性，对神经网络的性能进行严格分析神经网络。虽然VC维度确实导致该设置中的琐碎泛化误差界限，但我们表明转导变速器复杂性可以解释用于随机块模型的图形卷积网络的泛化特性。我们进一步使用基于转换的Rademacher复杂性的泛化误差界限来展示图形卷积和网络架构在实现较小的泛化误差方面的作用，并在图形结构可以帮助学习时提供洞察。本文的调查结果可以重新新的兴趣在学习理论措施方面对神经网络的概括，尽管在特定问题中。

尽管预测方法的相关性越来越高，但这些算法的因果影响仍然很大程度上是未开发的。这与考虑到，即使在简化因果充足之类的假设下，模型的统计风险也可能与其\ Textit {因果风险}有显着差异。在这里，我们研究了*因果概括* - 从观察到介入分布的概括 - 预测。我们的目标是找到问题的答案：自回归（var）模型在预测统计协会方面的疗效如何与其在干预措施下预测的能力相比？为此，我们介绍了*因果学习理论*预测的框架。使用此框架，我们获得了统计和因果风险之间差异的表征，这有助于识别它们之间的分歧源。在因果充足之下，因果概括的因果概括金额与额外的结构（限制介入介入分配）。该结构允许我们获得统一的收敛界面对VAR模型类的因果概括性。据我们所知，这是第一个为时序设置中因果概念提供理论保障的工作。

由于其复杂结构和顶点对应，在各种应用中遇到了网络价值的数据并在学习中提出挑战。这些问题的典型示例包括蛋白质结构和社交网络的分类或分组。已经提出了各种方法，从图形内核到图形神经网络，从而在图形分类问题中取得了一些成功。然而，大多数方法都有有限的理论典范化，其超越分类的适用性仍未开发。在这项工作中，我们提出了用于群集多个图形的方法，而没有顶点对应，这是由最近关于估计与图形的无限顶点限制对应的石墨函数的文献的启发。我们提出了一种基于排序和平滑的石墨估计的新颖曲线距离。使用所提出的图形距离，我们呈现了两个聚类算法，并表明他们实现了最先进的结果。我们在图形度上的Lipschitz假设下的两个算法下的统计一致性。我们进一步研究了建议的距离的适用性，用于图形两样测试问题。