The expression "\|x - y\|^2" represents the squared distance between two points, x and y, in a given space. Here, "\| \cdot \|" denotes the norm, which measures the length or size of a vector. Squaring this distance helps eliminate negative values and emphasizes larger distances. In a geometric context, this squared distance can be useful in various applications, such as machine learning and statistics, where minimizing the distance between predicted and actual values is essential. It provides a way to quantify how far apart two points are, aiding in optimization and analysis.