For the first time, this position paper introduces a fundamental link between tensor networks (TNs) and Green AI, highlighting their synergistic potential to enhance both the inclusivity and sustainability of AI research. We argue that TNs are valuable for Green AI due to their strong mathematical backbone and inherent logarithmic compression potential. We undertake a comprehensive review of the ongoing discussions on Green AI, emphasizing the importance of sustainability and inclusivity in AI research to demonstrate the significance of establishing the link between Green AI and TNs. To support our position, we first provide a comprehensive overview of efficiency metrics proposed in Green AI literature and then evaluate examples of TNs in the fields of kernel machines and deep learning using the proposed efficiency metrics. This position paper aims to incentivize meaningful, constructive discussions by bridging fundamental principles of Green AI and TNs. We advocate for researchers to seriously evaluate the integration of TNs into their research projects, and in alignment with the link established in this paper, we support prior calls encouraging researchers to treat Green AI principles as a research priority.

This paper presents a method for approximate Gaussian process (GP) regression with tensor networks (TNs). A parametric approximation of a GP uses a linear combination of basis functions, where the accuracy of the approximation depends on the total number of basis functions M. We develop an approach that allows us to use an exponential amount of basis functions without the corresponding exponential computational complexity. The key idea to enable this is using low-rank TNs. We first find a suitable low-dimensional subspace from the data, described by a low-rank TN. In this low-dimensional subspace, we then infer the weights of our model by solving a Bayesian inference problem. Finally, we project the resulting weights back to the original space to make GP predictions. The benefit of our approach comes from the projection to a smaller subspace: It modifies the shape of the basis functions in a way that it sees fit based on the given data, and it allows for efficient computations in the smaller subspace. In an experiment with an 18-dimensional benchmark data set, we show the applicability of our method to an inverse dynamics problem.

This paper proposes a Bayesian Volterra tensor network (TN) to solve high-order discrete nonlinear multiple-input multiple-output (MIMO) Volterra system identification problems. Using a low-rank tensor network to compress all Volterra kernels at once, we avoid the exponential growth of monomials with respect to the order of the Volterra kernel. Our contribution is to introduce a Bayesian framework for the low-rank Volterra TN. Compared to the least squares solution for Volterra TNs, we include prior assumptions explicitly in the model. In particular, we show for the first time how a zero-mean prior with diagonal covariance matrix corresponds to implementing a Tikhonov regularization for the MIMO Volterra TN. Furthermore, adopting a Bayesian viewpoint enables simulations with Bayesian uncertainty bounds based on noise and prior assumptions. In addition, we demonstrate via numerical experiments how Tikhonov regularization prevents overfitting in the case of higher-rank TNs.