NeuroLifting: Neural Inference on Markov Random Fields at Scale

TL;DR

NeuroLifting introduces a neural lifting paradigm that reparameterizes MRF decision variables via Graph Neural Networks, enabling gradient-based energy minimization in a lifted space. By converting high-order cliques to a graph representation, vectorizing energies with look-up tables, and using a differentiable surrogate loss, it achieves competitive or superior solution quality compared to exact and approximate baselines, with linear scaling for large MRFs. The method is validated on synthetic, UAI 2022, and real-world PCI datasets, showing strong performance on large and dense graphs and robust generalization across problem orders. This approach provides a principled, scalable alternative to traditional inference techniques and opens avenues for applying neural lifting to other large-scale optimization problems.

Abstract

Inference in large-scale Markov Random Fields (MRFs) is a critical yet challenging task, traditionally approached through approximate methods like belief propagation and mean field, or exact methods such as the Toulbar2 solver. These strategies often fail to strike an optimal balance between efficiency and solution quality, particularly as the problem scale increases. This paper introduces NeuroLifting, a novel technique that leverages Graph Neural Networks (GNNs) to reparameterize decision variables in MRFs, facilitating the use of standard gradient descent optimization. By extending traditional lifting techniques into a non-parametric neural network framework, NeuroLifting benefits from the smooth loss landscape of neural networks, enabling efficient and parallelizable optimization. Empirical results demonstrate that, on moderate scales, NeuroLifting performs very close to the exact solver Toulbar2 in terms of solution quality, significantly surpassing existing approximate methods. Notably, on large-scale MRFs, NeuroLifting delivers superior solution quality against all baselines, as well as exhibiting linear computational complexity growth. This work presents a significant advancement in MRF inference, offering a scalable and effective solution for large-scale problems.

Figures (7)

• Figure 1: An overview of NeuroLifting. The energy function of this problem is $E(X) = \theta_{C_1} (x_1, x_2, x_3) + \theta_{C_2} (x_3, x_4, x_5) + \theta_{C_3} (x_2, x_3, x_5, x_6)$. $H_T^{(K)}$ is the output of the model after the $T$-th iteration.
• Figure 2: This illustrates the padding procedure for unary loss terms $\phi(x)$ and clique loss terms $\psi(x_i, x_j, x_k)$, with $|\mathcal{X}|=5$. $x_{max}$ denotes the variable that has the maximum value range. The elements shown in purple represent the energy values in the original $\phi$ and $\psi$. After padding, the dimension of vector $\phi$, as well as each dimension of the energy tensor $\psi(x_i, x_j, x_k)$, will be 5. The padded portion is indicated in orange, with values either $\max(\phi)$ or $\max(\psi)$.
• Figure 3: The loss curves of the Segmentation_14, P_potts_6 and P_potts_8 from pairwise potts synthetic problems.
• Figure 4: The landscape of instance Segmentation_19. From top to the bottom, each column correspond to network layer $\{1, 2, 5, 8\}$. The first row is the landscape range from $[-10, +10]$ for both $\delta$ and $\eta$ direction. The second row is the landscape range from $[-1, +1]$ for both $\delta$ and $\eta$ direction.
• Figure 5: The average loss curves over UAI inference competition 2022 pairwise cases, PCI instances and synthetic instances using GraphSAGE, GCN and GAT as the GNN backbones.

Theorems & Definitions (1)

• Remark 4.1