首页|Universal Scaling Laws in Quantum-Probabilistic Machine Learning by Tensor Network:Toward Interpreting Representation and Generalization Powers
Universal Scaling Laws in Quantum-Probabilistic Machine Learning by Tensor Network:Toward Interpreting Representation and Generalization Powers
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The interpretation of representations and generalization powers has been a long-standing challenge in the fields of machine learning(ML)and artificial intelligence.This study contributes to understanding the emergence of universal scaling laws in quantum-probabilistic ML.We consider 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,that is,L～kM+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.This study reveals that,while gaining information through training,the linear-scaling law is suppressed by a negative quadratic correction,leading to L(～)βM-αM2+const.The scaling coefficients exhibit logarithmic relationships with the number of training samples and quantum channels The emergence of a quadratic correction term in the NLL for the testing(training)set can be regarded as evidence of the generalization(representation)power of the GTN.Over-parameterization can be identified by the deviation in the values of α between the training and testing sets while increasing x.We further investigate how orthogonality in the quantum-feature map relates to the satisfaction of quantum-probabilistic interpretation and the representation and generalization powers of the GTN.Unveiling 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.
Sheng-Chen Bai、Shi-Ju Ran
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Center for Quantum Physics and Intelligent Sciences,Department of Physics,Capital Normal University,Beijing 100048,China
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