A Tensor-Train Framework for Bayesian Inference in High-Dimensional Systems: Applications to MIMO Detection and Channel Decoding

Luca Schmid; Dominik Sulz; Shrinivas Chimmalgi; Laurent Schmalen

cs.ITApr 7, 2026

A Tensor-Train Framework for Bayesian Inference in High-Dimensional Systems: Applications to MIMO Detection and Channel Decoding

Authors: Luca Schmid, Dominik Sulz, Shrinivas Chimmalgi, Laurent Schmalen

Organizations: Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT), Hertzstr. 16, 76187 Karlsruhe, Germany · Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany · Qoherent, Toronto, ON, Canada

Abstract

Bayesian inference in high-dimensional discrete-input additive noise models is a fundamental challenge in communication systems, as the support of the required joint a posteriori probability (APP) mass function grows exponentially with the number of unknown variables. In this work, we propose a tensor-train (TT) framework for tractable, near-optimal Bayesian inference in discrete-input additive noise models. The central insight is that the joint log-APP mass function admits an exact low-rank representation in the TT format, enabling compact storage and efficient computations. To recover symbol-wise APP marginals, we develop a practical inference procedure that approximates the exponential of the log-posterior using a TT-cross algorithm initialized with a truncated Taylor-series. To demonstrate the generality of the approach, we derive explicit low-rank TT constructions for two canonical communication problems: the linear observation model under additive white Gaussian noise (AWGN), applied to multiple-input multiple-output (MIMO) detection, and soft-decision decoding of binary linear block error correcting codes over the binary-input AWGN channel. Numerical results show near-optimal error-rate performance across a wide range of signal-to-noise ratios while requiring only modest TT ranks. These results highlight the potential of tensor-network methods for efficient Bayesian inference in communication systems.

Figures & tables

Panel of Figure 1 — Fig. 1: Graphical representation of the TT decomposition. Each edge can be interpreted as a tensor contraction.

Panel of Figure 2 — Fig. 2: SER over SNR for various MIMO detectors across different ${\tilde{N}\times\tilde{N}}$ system configurations (real-valued dimensions ${N_{\text{T}}=N_{\text{R}}=2\tilde{N}}$ ). The maximum rank r_max in the truncated Taylor-series initialization is set to (from left to right) $10,10,40,20$ .

P(x_{i}=a|\bm{y})\propto\sum\limits_{\mathclap{\begin{subarray}{c}\bm{a}\in\mathcal{A}^{N_{\text{T}}}\\ a_{i}=a\end{subarray}}}\mathrm{e}^{\Lambda(\bm{a})},\quad a\in\mathcal{A},\,i=1,\ldots,N_{\text{T}},

Algorithm 1 TTDet for MIMO detection

	$r_{\text{max}}$ after TT -cross		Memory savings w.r.t. $M^{\tilde{N}}$
$E_{\text{b}}/N_{0}$	mean	median	mean	median
$-11\,$ dB	$210$	$209$	$10^{13}$	$10^{13}$
$-5\,$ dB	$98$	$86$	$10^{14}$	$10^{14}$

TABLE I: Memory complexity of the TTDet-sample algorithm for

{\tilde{N}=32}

and

4

- QAM

Bayesian Inference

Bayesian inference updates probability distributions over unknowns with evidence to quantify uncertainty and guide prediction and decisions.

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Joint Bayesian Parameter and Model Order Estimation for Low-Rank Probability Mass Tensors

Joseph K. Chege, Arie Yeredor, Martin Haardt

Communications Research Laboratory, Ilmenau University of Technology, Ilmenau, Germany · School of Electrical and Computer Engineering, Tel Aviv University, Tel Aviv, Israel

Bayesian Inference Tensor Decomposition Low-Rank Approximation

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Obtaining a reliable estimate of the joint probability mass function (PMF) of a set of random variables from observed data is a significant objective in statistical signal processing and machine learning. Modelling the joint PMF as a tensor that admits a low-rank canonical polyadic decomposition (CPD) has enabled the development of efficient PMF estimation algorithms. However, these algorithms require the rank (model order) of the tensor to be specified beforehand. In real-world applications, the true rank is unknown. Therefore, an appropriate rank is usually selected from a candidate set either by observing validation errors or by computing various likelihood-based information criteria, a procedure that could be costly in terms of computational time or hardware resources, or could result in mismatched models which affect the model accuracy. This paper presents a novel Bayesian framework for estimating the low-rank components of a joint PMF tensor and simultaneously inferring its rank from the observed data. We specify a Bayesian PMF estimation model and employ appropriate prior distributions for the model parameters, allowing the rank to be inferred without cross-validation.We then derive a deterministic solution based on variational inference (VI) to approximate the posterior distributions of various model parameters. Numerical experiments involving both synthetic data and real classification and item recommendation data illustrate the advantages of our VI-based method in terms of estimation accuracy, automatic rank detection, and computational efficiency.

Abstract

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