Tensor cross interpolation for global discrete optimization with application to Bayesian network inference

TL;DR

The paper tackles global discrete optimization by integrating Tensor-Train (TT) cross interpolation with a greedy, fiber-based sampling strategy to maximize a likelihood tensor over network configurations in a Bayesian epidemic setting. The network likelihood is encoded as a high-order tensor L(hat{g}) over adjacency configurations (with d = 1/2 N(N−1) binary entries), and TT cross interpolation with tempering and caching is used to efficiently search for the global maximizer without exhaustive evaluation. Numerical experiments on an ε-SIS chain demonstrate accurate network recovery at modest TT ranks, highlighting the method’s potential for discrete hyperparameter tuning and other high-dimensional optimization tasks. The approach offers a flexible, scalable alternative for global optimization in discrete stochastic systems and can be extended to other domains requiring efficient exploration of large combinatorial spaces.

Abstract

Global discrete optimization is notoriously difficult due to the lack of gradient information and the curse of dimensionality, making exhaustive search infeasible. Tensor cross approximation is an efficient technique to approximate multivariate tensors (and discretized functions) by tensor product decompositions based on a small number of tensor elements, evaluated on adaptively selected fibers of the tensor, that intersect on submatrices of (nearly) maximum volume. The submatrices of maximum volume are empirically known to contain large elements, hence the entries selected for cross interpolation can also be good candidates for the globally maximal element within the tensor. In this paper we consider evolution of epidemics on networks, and infer the contact network from observations of network nodal states over time. By numerical experiments we demonstrate that the contact network can be inferred accurately by finding the global maximum of the likelihood using tensor cross interpolation. The proposed tensor product approach is flexible and can be applied to global discrete optimization for other problems, e.g. discrete hyperparameter tuning.

Figures (4)

• Figure 1: Markov chain and transitions between network states for the $\varepsilon$-SIS infection process. (a) In this example, the contact network is a chain of $N=4$ people. (b) Transition rates in the Markov process depend not just on the size of infected and susceptible group, but on the exact state of the network. Note that the infection rate for node $n$ is $I_n \beta + \varepsilon$, where $I_n$ is the number of infected neighbours of node $n$, $\beta$ is per contact infection rate, and $\varepsilon$ is the self--infection rate, responsible for contacts with potential threats outside the network. (c) All network states, with recovery transitions shown with green arrows, and infection transition shown with red arrows with a circled number $k$ indicating the infection rate $k\beta+\varepsilon.$
• Figure 2: One step of the matrix cross interpolation algorithm
• Figure 3: Tensor cross interpolation algorithm visualised. Top: one step of tensor cross interpolation updates searches for a new pivot $(\hat{g}_{\leqslant k}^\star,\hat{g}_{>k}^\star)$ in the submatrix $\mathsf{L}(\mathcal{I}_{\leqslant k-1}\hat{g}_k,\hat{g}_{k+1}\mathcal{I}_{>k+1})$ and adds a new cross to the index set $(\mathcal{I}_{\leqslant k}, \mathcal{I}_{>k}).$ Bottom: the process repeats for $k=1,\ldots,d-1$ (left--to--right sweep) and back for $k=d-1,\ldots,1$ (right--to--left sweep).
• Figure 4: Inferring linear chain network with $N=9$ people from $\varepsilon$--SIS epidemic process with $\beta=1,$$\gamma=0.5$ and $\varepsilon=0.01$: (a) the ground truth network $\mathcal{G}_\star$ in its initial state; (b) and (c) convergence of the network $\mathcal{G}$ towards $\mathcal{G}_\star$ in the TT cross interpolation algorithm; average (solid lines) $\pm$ one standard deviation (shaded areas) over $N_s=42$ datasets; shown for temperatures $\tau=1, 10, 100.$ Vertical dashed lines in (b) and (c) indicate full iterations of the TT cross interpolation algorithm. Intervals around the dashed lines in (c) indicate the average time $\pm$ 1 standard deviation. (d) distribution of relative errors for the tensor approximation, based on $N_s=39$ datasets; (e) distribution of errors for the network inference, based on $N_s=39$ datasets.