Universal scaling laws in quantum-probabilistic machine learning by tensor network towards interpreting representation and generalization powers
This work addresses a foundational issue in ML/AI by providing theoretical insights into quantum-probabilistic models, potentially advancing interpretable white-box ML schemes, though it appears incremental as it builds on existing tensor network concepts.
The paper tackles the problem of interpreting representation and generalization in machine learning by uncovering universal scaling laws in quantum-probabilistic ML, showing that negative logarithmic likelihood scales linearly with features in untrained models and gains a quadratic correction after training, with coefficients linked to sample size and quantum channels.
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 $L \simeq βM - αM^2 + const$. The scaling coefficients exhibit logarithmic relationships with the number of training samples and the number of quantum channels $χ$. 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 $α$ between training and testing sets while increasing $χ$. 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.