.png){kind=icon}
.png){kind=icon}
Return
Representation Theorems for Matrix Product States
.png){kind=icon}
PRE
AI
E
D
DOI：10.1137/24M172010X.png)
.png){kind=icon}
Original request for help.png){kind=icon}
Share.png){kind=icon}
Save
Abstract
En 中文
In this work, we investigate the universal representational capacity of matrix product states (MPSs) in the context of Boolean and continuous functions. We show that MPSs can precisely realize arbitrary Boolean functions by presenting a constructive method for designing MPS representations of Boolean gates. Furthermore, we prove that the function space of MPSs, equipped with scale-invariant sigmoidal activation functions, is dense in the space of continuous functions on compact subsets of \BbbRn. By examining the connection between MPSs and neural networks (NNs), we establish that an MPS with scale-invariant sigmoidal activation is equivalent to a one-hidden-layer NN equipped with kernel functions. For specific MPS models, we construct equivalent NNs and show that nonlinear kernels, such as polynomial kernels coupling input components, arise naturally in these representations. Finally, we analyze the realization of Gaussian processes via infinitely wide MPSs, leveraging their equivalence to NNs. Our analysis of MPSs highlights the critical role of the kernel function in MPSs, as it encodes the correlations between different components of the input data. Its careful design directly influences the model's representational capacity.
Keywords:
tensor networks
universal approximation theorem
neural networks
Hahn-Banach theorem
Journal
S
IF:
2.6
Papers:
17
Citations:
0
Organization
Cited Papers