Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers

TL;DR

This paper uncovers universal scaling laws in quantum-probabilistic machine learning, revealing how the negative log-likelihood scales with features and training samples, and linking these laws to the model's representation and generalization capabilities.

Contribution

It introduces the concept of universal scaling laws in quantum-probabilistic ML, connecting the scaling behavior to the model's representation and generalization powers, and analyzing the role of orthogonality.

Findings

Negative log-likelihood scales linearly with features in untrained models.

Quadratic correction in NLL indicates generalization and representation power.

Over-parameterization is detectable through deviations in scaling coefficients.

Abstract

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.