The issue of fairness in AI has received an increasing amount of attention in recent years. The problem can be approached by looking at different protected attributes (e.g., ethnicity, gender, etc) independently, but fairness for individual protected attributes does not imply intersectional fairness. In this work, we frame the problem of intersectional fairness within a geometrical setting. We project our data onto a hypercube, and split the analysis of fairness by levels, where each level encodes the number of protected attributes we are intersecting over. We prove mathematically that, while fairness does not propagate "down" the levels, it does propagate "up" the levels. This means that ensuring fairness for all subgroups at the lowest intersectional level (e.g., black women, white women, black men and white men), will necessarily result in fairness for all the above levels, including each of the protected attributes (e.g., ethnicity and gender) taken independently. We also derive a formula describing the variance of the set of estimated success rates on each level, under the assumption of perfect fairness. Using this theoretical finding as a benchmark, we define a family of metrics which capture overall intersectional bias. Finally, we propose that fairness can be metaphorically thought of as a "fractal" problem. In fractals, patterns at the smallest scale repeat at a larger scale. We see from this example that tackling the problem at the lowest possible level, in a bottom-up manner, leads to the natural emergence of fair AI. We suggest that trustworthiness is necessarily an emergent, fractal and relational property of the AI system.

翻译：近年来，人工智能中的公平性问题日益受到关注。这一问题可以通过单独考察不同受保护属性（如种族、性别等）来解决，但单个受保护属性的公平性并不能保证交叉公平性。在本文中，我们将交叉公平性问题置于几何框架下进行阐述。我们将数据投影到超立方体上，并按层级划分公平性分析，其中每个层级编码了我们所交叉的受保护属性数量。我们从数学上证明，虽然公平性不会向"下"层级传播，但确实会向"上"层级传播。这意味着，确保最低交叉层级所有子组（如黑人女性、白人女性、黑人男性、白人男性）的公平性，将必然导致所有更高层级（包括单独考察的每个受保护属性，如种族和性别）也具有公平性。我们还推导出一个公式，在完美公平性假设下，描述了每个层级上估计成功率集合的方差。利用这一理论发现作为基准，我们定义了一组用于捕捉总体交叉偏差的度量指标。最后，我们认为公平性可以被隐喻性地视为一个"分形"问题。分形中，最小尺度上的模式会在更大尺度上重复。从这一范例中我们看到，以自下而上的方式在尽可能低的层级解决问题，能够自然催生公平的人工智能。我们提出，可信赖性必然是人工智能系统的一种涌现性、分形性和关系性属性。