Complexity of patterns is key information for human brain to differ objects of about the same size and shape. Like other innate human senses, the complexity perception cannot be easily quantified. We propose a transparent and universal machine method for estimating structural (effective) complexity of two-dimensional and three-dimensional patterns that can be straightforwardly generalized onto other classes of objects. It is based on multistep renormalization of the pattern of interest and computing the overlap between neighboring renormalized layers. This way, we can define a single number characterizing the structural complexity of an object. We apply this definition to quantify complexity of various magnetic patterns and demonstrate that not only does it reflect the intuitive feeling of what is “complex” and what is “simple” but also, can be used to accurately detect different phase transitions and gain information about dynamics of nonequilibrium systems. When employed for that, the proposed scheme is much simpler and numerically cheaper than the standard methods based on computing correlation functions or using machine learning techniques.

模式的复杂性是人脑区分大小和形状大致相同的物体的关键信息。与人类其他与生俱来的感官一样，复杂性感知无法轻易量化。我们提出了一种透明且通用的机器方法，用于估计二维和三维模式的结构（有效）复杂性，该方法可以直接推广到其他类别的对象上。它基于感兴趣模式的多步重整化并计算相邻重整化层之间的重叠。这样，我们就可以定义一个表征对象结构复杂性的数字。我们应用这个定义来量化各种磁图案的复杂性，并证明它不仅反映了“复杂”和“简单”的直观感受，而且可以用来准确地检测不同的相变并获得有关非平衡系统的动力学。当用于此目的时，所提出的方案比基于计算相关函数或使用机器学习技术的标准方法更简单并且在数值上更便宜。