Tensor Network-Constrained Kernel Machines Converge to Gaussian Processes

Core Concepts

Tensor Network-constrained kernel machines converge to Gaussian Processes when specifying i.i.d. priors over their parameters. The convergence rate of Tensor Train models is faster than Canonical Polyadic Decomposition models for the same number of parameters.

Abstract

The paper establishes a connection between Tensor Network (TN)-constrained kernel machines and Gaussian Processes (GPs). It proves that the outputs of Canonical Polyadic Decomposition (CPD) and Tensor Train (TT)-constrained kernel machines converge to a fully characterized GP when placing i.i.d. priors over their parameters.
The key insights are:

CPD and TT-constrained kernel machines converge to a GP in the limit of large TN ranks. The target GP is fully characterized by the basis functions and prior covariances.

TT-constrained models converge to the GP at a polynomially faster rate compared to CPD models when dealing with higher-dimensional inputs, for the same number of model parameters. This is due to the hierarchical structure of TT.

The priors specified in the theorems provide a sensible initialization strategy for gradient-based optimization of TN-constrained kernel machines, which addresses the vanishing/exploding gradients issue.

In the finite rank regime, both CPD and TT models can craft nonlinear features from the provided basis functions, potentially learning more latent patterns in the data compared to the fixed GP representation.

The authors empirically demonstrate the GP convergence and GP behavior of these models through numerical experiments on regression tasks.

Stats

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Key Insights Distilled From

Tensor Network-Constrained Kernel Machines as Gaussian Processes

by Frederiek We... at arxiv.org 03-29-2024

https://arxiv.org/pdf/2403.19500.pdf

Deeper Inquiries

How do the insights from this work extend to other tensor network architectures beyond CPD and TT, such as the Multi-Scale Entanglement Renormalization Ansatz (MERA)

The insights from this work can be extended to other tensor network architectures beyond CPD and TT, such as the Multi-Scale Entanglement Renormalization Ansatz (MERA). MERA is a tensor network structure commonly used in quantum many-body systems to represent wave functions. Just like CPD and TT, MERA can be used to compress and represent high-dimensional data efficiently. By applying similar principles of tensor decomposition and constraining the model weights to be low-rank tensors, it is possible to establish connections between MERA-constrained models and Gaussian Processes (GPs). The convergence properties and GP behavior observed in CPD and TT models can potentially be extended to MERA-based models, providing a deeper understanding of their relationship to GPs and their performance characteristics in various applications.

What are the implications of the found connection between TN-constrained kernel machines and GPs for model selection and hyperparameter tuning in practice

The found connection between TN-constrained kernel machines and GPs has significant implications for model selection and hyperparameter tuning in practice.

Model Selection: The insights from this work suggest that TN-constrained kernel machines can exhibit GP behavior with a reduced number of model parameters, particularly in higher-dimensional settings. This knowledge can guide practitioners in choosing the appropriate tensor network architecture (such as CPD or TT) based on the desired trade-off between model complexity and GP-like behavior. For example, if the goal is to achieve faster convergence to GP behavior, TT-based models may be preferred over CPD-based models.

Hyperparameter Tuning: Understanding the convergence rates and GP behavior of TN-constrained models can inform hyperparameter tuning strategies. Practitioners can leverage this knowledge to optimize the rank hyperparameters of CPD and TT models to achieve the desired level of GP behavior while balancing computational efficiency. By adjusting the ranks of the tensor decomposition, model performance can be fine-tuned to match the characteristics of a GP, leading to improved generalization and scalability.

Overall, the connection between TN-constrained kernel machines and GPs provides valuable insights for practitioners to make informed decisions during model selection and hyperparameter tuning processes.

Can the ideas presented here be leveraged to develop new GP approximation methods that combine the flexibility of deep learning models with the closed-form inference of GPs

The ideas presented in this work can be leveraged to develop new GP approximation methods that combine the flexibility of deep learning models with the closed-form inference of GPs. By exploring different tensor network architectures and their convergence properties to GPs, researchers can design novel GP approximation techniques that offer a balance between expressiveness and computational efficiency.
One potential approach could involve incorporating elements from TN-constrained kernel machines, such as CPD and TT, into existing GP approximation frameworks. By constraining the model weights to low-rank tensor structures and leveraging the insights on convergence rates and GP behavior, new GP approximation methods can be developed to capture complex patterns in data while benefiting from the analytical tractability of GPs.
Furthermore, researchers can explore hybrid models that combine deep learning architectures with GP priors derived from tensor network constraints. This fusion of techniques could lead to the development of hybrid models that inherit the strengths of both approaches, enabling efficient inference, improved generalization, and enhanced interpretability in complex data modeling tasks.