Interpreting the representation and generalization powers has been a long-standing issue in the field of machine learning (ML) and artificial intelligence. This work contributes to uncovering the emergence of universal scaling laws in quantum-probabilistic ML. We take the generative tensor network (GTN) in the form of a matrix product state as an example and show that with an untrained GTN (such as a random TN state), the negative logarithmic likelihood (NLL) $L$ generally increases linearly with the number of features $M$, i.e., $L \simeq k M + const$. This is a consequence of the so-called ``catastrophe of orthogonality,'' which states that quantum many-body states tend to become exponentially orthogonal to each other as $M$ increases. We reveal that while gaining information through training, the linear scaling law is suppressed by a negative quadratic correction, leading to $L \simeq \beta M - \alpha M^2 + const$. The scaling coefficients exhibit logarithmic relationships with the number of training samples and the number of quantum channels $\chi$. The emergence of the quadratic correction term in NLL for the testing (training) set can be regarded as evidence of the generalization (representation) power of GTN. Over-parameterization can be identified by the deviation in the values of $\alpha$ between training and testing sets while increasing $\chi$. We further investigate how orthogonality in the quantum feature map relates to the satisfaction of quantum probabilistic interpretation, as well as to the representation and generalization powers of GTN. The unveiling of universal scaling laws in quantum-probabilistic ML would be a valuable step toward establishing a white-box ML scheme interpreted within the quantum probabilistic framework.

该研究解决了机器学习领域中长期存在的表示与泛化能力解读难题。通过分析生成性张量网络（GTN）的负对数似然（NLL）线性增长规律，揭示了训练过程中的信息积累和缩放系数与训练样本及量子通道数量之间的关系。这些发现表明量子概率机器学习中的普适缩放规律为建立可解释的机器学习方案提供了重要一步。

通过张量网络研究量子概率机器学习中的普适缩放规律以解读表示与泛化能力