Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning
Towards a topological-geometrical theory of group equivariant non-expansive operators for data analysis and machine learning
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DOI：
10.1038/s42256-019-0087-3
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发表时间：
2019-09-01
影响因子：
23.8
通讯作者：
Quercioli, Nicola
中科院分区：
文献类型：
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作者：
Bergomi, Mattia G.;Frosini, Patrizio;Quercioli, Nicola
我们为机器学习中的组和集合等方差提供了一个通用的数学框架。我们定义群等变非扩张算子（GENEOs）作为与变换群相关联的函数空间之间的映射。我们研究的拓扑和度量属性的空间的GENEOs评估其逼近能力，并设置的基础上，一般的战略，以初始化和组成运营商。我们定义了函数空间、等方差群和非扩张算子集的伪度量。我们证明，在适当的假设下，空间的GENEOs是紧的和凸的。这些结果从机器学习的角度提供了基本保证。通过考虑等距-等变非扩张算子，我们描述了一个简单的策略来选择和采样算子。此后，我们展示了如何选择和采样运营商可以用来执行经典的度量学习和注入人工神经网络的知识。控制深度神经网络中的信息流和表示是使网络变得可理解的基础。Bergomi等人引入了一个数学框架，其中表示数据的可能运算符的空间通过使用对称性来约束。这种受限空间仍然适用于机器学习：可以有效地计算、近似和参数化算子以进行优化。
We provide a general mathematical framework for group and set equivariance in machine learning. We define group equivariant non-expansive operators (GENEOs) as maps between function spaces associated with groups of transformations. We study the topological and metric properties of the space of GENEOs to evaluate their approximating power and set the basis for general strategies to initialize and compose operators. We define suitable pseudo-metrics for the function spaces, the equivariance groups and the set of non-expansive operators. We prove that, under suitable assumptions, the space of GENEOs is compact and convex. These results provide fundamental guarantees in a machine learning perspective. By considering isometry-equivariant non-expansive operators, we describe a simple strategy to select and sample operators. Thereafter, we show how selected and sampled operators can be used both to perform classical metric learning and to inject knowledge in artificial neural networks. Controlling the flow and representation of information in deep neural networks is fundamental to making networks intelligible. Bergomi et al introduce a mathematical framework in which the space of possible operators representing the data is constrained by using symmetries. This constrained space is still suitable for machine learning: operators can be efficiently computed, approximated and parameterized for optimization.