The machine learning of lattice operators has three possible bottlenecks. From a statistical standpoint, it is necessary to design a constrained class of operators based on prior information with low bias, and low complexity relative to the sample size. From a computational perspective, there should be an efficient algorithm to minimize an empirical error over the class. From an understanding point of view, the properties of the learned operator need to be derived, so its behavior can be theoretically understood. The statistical bottleneck can be overcome due to the rich literature about the representation of lattice operators, but there is no general learning algorithm for them. In this paper, we discuss a learning paradigm in which, by overparametrizing a class via elements in a lattice, an algorithm for minimizing functions in a lattice is applied to learn. We present the stochastic lattice gradient descent algorithm as a general algorithm to learn on constrained classes of operators as long as a lattice overparametrization of it is fixed, and we discuss previous works which are proves of concept. Moreover, if there are algorithms to compute the basis of an operator from its overparametrization, then its properties can be deduced and the understanding bottleneck is also overcome. This learning paradigm has three properties that modern methods based on neural networks lack: control, transparency and interpretability. Nowadays, there is an increasing demand for methods with these characteristics, and we believe that mathematical morphology is in a unique position to supply them. The lattice overparametrization paradigm could be a missing piece for it to achieve its full potential within modern machine learning.

翻译：格算子的机器学习存在三个潜在瓶颈。从统计学角度看，需要基于先验信息设计一个兼具低偏差、低复杂度且与样本量匹配的约束算子类；从计算层面而言，需要存在高效算法能在该类上最小化经验误差；从理论理解维度出发，需推导学习后算子的性质以揭示其行为机理。尽管关于格算子表征的丰富文献已解决了统计瓶颈，但尚缺乏针对这类算子的通用学习算法。本文讨论一种学习范式：通过利用格中元素对算子类进行过参数化，将格函数最小化算法应用于学习过程。我们提出随机格梯度下降算法作为通用学习框架——只要固定格的过参数化表示，该算法就能在约束算子类上进行学习，并引用前期概念验证工作进行佐证。此外，若存在从过参数化表示中提取算子基的算法，则可推导算子性质，从而突破理论理解瓶颈。该学习范式具备现代神经网络方法所欠缺的三大特性：可控性、透明性与可解释性。当前学界对具备这些特性的方法需求日益增长，我们认为数学形态学具有提供此类方案的独特优势。格过参数化范式或将成为其在现代机器学习中充分发挥潜力的关键拼图。