# 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, and Laurent Schmalen
> The work of L. Schmid, S. Chimmalgi and L. Schmalen has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 101001899). The work of D. Sulz was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 352 – Project-ID 470903074.L. Schmid and L. Schmalen are with the Communications Engineering Lab (CEL), Karlsruhe Institute of Technology (KIT), Hertzstr. 16, 76187 Karlsruhe, Germany (e-mail: ). C. Chimmalgi is now with Qoherent, Toronto, ON, Canada. D. Sulz is with the Technical University of Munich (TUM), Boltzmannstr. 3, 85748 Garching b. München, Germany (e-mail: ).
## 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.
## I Introduction
Many problems in communications and signal processing reduce to probabilistic inference over noisy observations [proakis_digital_2007]. In most communication scenarios, the randomness of the channel, together with prior knowledge about the transmitter naturally leads to a Bayesian formulation at the receiver: the posterior distribution over the unknown quantities, e.g., the transmitted symbols, given the received signal is characterized and evaluated via its modes, marginals, or the model evidence [bishop_pattern_2006].
A fundamental challenge underlying Bayesian inference is the high dimensionality of the posterior distribution for large system sizes. For discrete-valued unknowns, the posterior can be interpreted as a multidimensional probability table whose order grows linearly with the number of variables while the number of entries grows exponentially. Explicit storage of this probability mass function (PMF) or any operation performed on it quickly becomes infeasible beyond small problem instances, a phenomenon known as the curse of dimensionality.
This challenge has motivated a substantial line of research on approximate inference algorithms, all of which have found application in communications. Message passing algorithms such as belief propagation (BP) [kschischang_factor_2001, worthen_unified_2001, richardson_modern_2008] exploit the local structure of the PMF to perform inference without explicitly computing the global distribution. Variational methods and expectation propagation (EP) [minka_expectation_2013, cespedes_expectation_2014] approximate the posterior within a restricted family of tractable distributions, such as Gaussian or factorized models. Sampling-based techniques [doucet2005monte, farhang2006markov] avoid a closed-form posterior representation altogether and instead approximate expectations and marginals through Monte Carlo estimates. More recently, deep learning has enabled flexible posterior approximations where intractable components of the distribution are parameterized by neural networks, as in variational autoencoders [kingma2013auto, lauinger_blind_2022] and diffusion models [ho2020denoising, fesl2024diffusion].
A complementary perspective is to retain the posterior in its original form and instead seek a compact representation of this high-dimensional PMF. TN representations [Kressner_survey, hackbusch2012tensor] provide such a memory-efficient, data-sparse representation. By decomposing a high-order tensor into a network of connected low-order factors, tensor networks exploit the inherent low-rank structure of the underlying tensor and have proven effective across diverse fields, including supervised learning [stoudenmire2016supervised], quantum many-body physics [haegeman2016unifying], numerical optimization [holtz2012alternating], and plasma physics [kormann2015semi]. Comprehensive surveys can be found in [Kressner_survey, kolda2009tensor].
Early foundational works [cichocki2015tensor_SP, almeida2016overview, sidiropoulos2017tensor] recognized the paradigm shift towards high-dimensional signal representations via multiway arrays (i.e., tensors) and advocated tensor decompositions as a natural modeling tool for signal representations in modern communication systems. Signals in modern communication and sensing systems are inherently multidimensional, including coupled domains such as time, frequency, space, and user indices. Tensor algebra enables these multi-way relationships to be represented and processed jointly, often leading to significantly more compact models than conventional approaches that treat dimensions separately [tokcan2025tensor].
Motivated by this perspective, tensor methods have been applied to various communication problems, including channel estimation in multiple-input multiple-output (MIMO)- orthogonal frequency-division multiplexing (OFDM) [araujo2019tensor] and MIMO - affine frequency-division multiplexing (AFDM) systems [wang2026tensor], as well as parameter estimation in array processing and MIMO radar [chen2021tensor]. They have also been explored for user separation in unsourced massive random access, signal modulation design, and remote multidimensional sensing, as surveyed in [tokcan2025tensor]. These works exploit the tensor structure in the signal or channel domain, representing received signals, propagation channels, or sensing data as low-rank tensors and leveraging this structure for efficient estimation or recovery.
A conceptually distinct and, to our knowledge, largely unexplored direction is to apply tensor methods to the inference domain. Instead of modeling the physical communication signals themselves, this perspective treats the posterior PMF itself as the high-dimensional object to be represented and manipulated in a tensor format. Hence, it focuses the approximation directly on the probabilistic representation used for inference and decision making at the receiver, rather than on the underlying physical signal representation. This viewpoint naturally motivates the use of TN representations to obtain compact and tractable approximations of the posterior distribution.
In this work, we investigate the potential of TNs, specifically tensor-trains [oseledets2011tensor], as a framework for efficient Bayesian inference in discrete-input additive noise models. Our central observation is that the a posteriori probability (APP) mass function of the transmitted symbols often admits a low-rank representation in the TT format when expressed in the logarithmic domain. Exploiting this structure enables computationally tractable inference in regimes where an explicit representation of the PMF would otherwise be prohibitively large.
Specifically, we construct the log-posterior directly in the TT format by building the log-likelihood contributions of all independent observations together with the log-prior using basic TT arithmetic. This construction in the logarithmic domain is exact with low TT ranks. To obtain symbol-wise APPs, the log-posterior must subsequently be exponentiated and marginalized. While marginalization in the linear domain is straightforward in the TT format, exponentiation is the main computational challenge. We address this step using a cross algorithm for TT [Cross_survey], initialized by a truncated Taylor-series. This approximation is controllable: by increasing the maximum TT rank, the resulting approximation can be made arbitrarily accurate, enabling near-optimal inference with complexity adaptable to the available computational resources.
We demonstrate the proposed framework on two well-known problems in communications. First, we consider MIMO detection in the additive white Gaussian noise (AWGN) channel. Second, we consider soft-decision decoding of binary linear block codes over the binary-AWGN (BI-AWGN) channel. For both problems, we derive explicit TT constructions of the log- APP and show that the resulting representations exhibit low ranks. Numerical results demonstrate near-optimal error-rate performance across wide signal-to-noise ratio (SNR) regimes, in many cases outperforming established algorithms such as EP -based detection and BP -based decoding while requiring only modest TT ranks.
We summarize the main contributions of this work:
- We propose a TT framework for Bayesian inference in discrete-input additive white noise models and provide an exact low-rank construction of the log- APP mass function in the TT format for two canonical models: the linear observation model under AWGN, and channel decoding of binary linear block codes over the BI-AWGN channel.
- We develop a practical method for marginal inference that recovers symbol-wise APPs by approximating the exponential of the log-posterior in TT format via a truncated Taylor-series followed by a cross-approximation. We compare a variant of the TT -cross algorithm and a density matrix renormalization group (DMRG)-like cross algorithm and analyze their trade-offs in terms of accuracy and robustness.
- We evaluate the proposed framework on MIMO detection and soft-decision channel decoding, demonstrating near-optimal performance in terms of error rate across all SNR regimes while requiring only modest TT ranks.
The remainder of this paper is organized as follows. Section II introduces the TT format and relevant arithmetics in the TT format. Section III describes the general discrete-input additive white noise observation model and the proposed TT construction of the log- APP, as well as the proposed inference method for exponentiation and marginalization. Sections IV and LABEL:sec:cc specialize the framework to MIMO detection and channel decoding, respectively, providing explicit TT constructions and numerical evaluations. Section LABEL:sec:conclusion concludes the paper.
### Notation
Throughout the paper, uppercase bold letters denote matrices $\bm{X}$ with entries $X_ij$ at row $i$ and column $j$ . Lowercase bold letters denote column vectors $\bm{x}$ , whose $i$ th element is written as $x_i$ . $\lVert·\rVert$ denotes the Euclidean norm, and $d_H(\bm{x},\bm{y})$ denotes the Hamming distance between two vectors $\bm{x},\bm{y}$ . The matrix transpose is denoted by $(·)^⊤$ and $⊗$ is the Kronecker product for matrices. The all-ones column vector of length $n$ is written as $\bm{1}_n$ , and the ${n× n}$ identity matrix is denoted by $\bm{I}_n$ . For a vector $\bm{x}$ , the diagonal matrix with $\bm{x}$ on its main diagonal is denoted by ${diag≤ft(\bm{x}\right)}$ , e.g., ${diag≤ft(\bm{1}_n\right)=\bm{I}_n}$ .
We use uppercase nonbold letters to denote tensors $A$ . For consistency in this context, tensor slices, which may be matrices or vectors, are also denoted by nonbold symbols. In our notation, individual tensor entries are indexed using parentheses, e.g., $A(k_1,\dots,k_N)$ .
We use ${exp≤ft(·\right)}$ to denote the exponential function, and ${\log≤ft(·\right)}$ and ${\log_10≤ft(·\right)}$ to denote the natural logarithm and the base- $10$ logarithm, respectively. For a complex number ${c∈ℂ}$ , ${Re\{c\}}$ ${(Im\{c\})}$ denotes its real (imaginary) part. We use calligraphic letters to denote sets $X$ of cardinality $|X|$ . The indicator function ${\mathbbm{1}_\{x=a\}}$ equals $1$ if ${x=a}$ and $0$ otherwise.
The probability density function of a continuous random variable ${y}$ is denoted by $p_{y}(y)$ or $p(y)$ and the PMF of a discrete random variable $x$ is $P_{x}(x)$ or $P(x)$ . To keep the notation simple, we do not use a special notation for random variables since it is always clear from the context. The Gaussian distribution, characterized by its mean $μ$ and variance $σ^2$ , is written as $N(μ,σ^2)$ . The tail distribution function, or Q-function, of the standard normal distribution is denoted as $Q(·)$ . The circular complex standard Gaussian distribution is denoted by ${CN≤ft(0,1\right)}$ . The noncentral chi-squared distribution with $k$ degrees of freedom and noncentrality parameter $λ$ is denoted by $χ^2_k(λ)$ . In case of ${λ=0}$ , we simply write $χ^2_k$ .
## II The Tensor-Train Format
Tensors are a multidimensional generalization of matrices and are widely used in many applications [kolda2009tensor]. Due to their exponential scaling in memory requirements, tensors are typically approximated in a decomposed form. Classical tensor formats are the canonical polyadic (CP) decomposition [hitchcock1927_CP], Tucker tensors [tucker1966some], tensor-trains (TTs) [oseledets2011tensor] and the more general class of tree tensor networks [hackbusch2012tensor]. In this work, we consider TTs, which are also known as matrix product states in other fields [MPS_paper].
Consider a tensor ${A∈{ℝ}^n_1×\dots× n_N}$ of order $N$ with the physical dimensions $n_i, i=1,…,N$ . The memory requirement of the full tensor $A$ in explicit form scales exponentially with $N$ . The decomposition of $A$ in the TT format is given by
$$
A(k_1,\dots,k_N)=G_1(k_1)G_2(k_2)⋯ G_N(k_N),
$$
with ${G_i(k_i)∈{ℝ}^r_i-1× r_i}$ , ${i=1,\dots,N}$ , and ${r_0=r_N=1}$ . Equivalently, the TT decomposition can be expressed in index notation as
| | $\displaystyle A(k_1,\dots,k_N)$ | |
| --- | --- | --- |
Fig. 1 visualizes the TT decomposition of a tensor $A$ into a sequence of core tensors $G_i$ . For indexing the core tensors ${G_i∈{ℝ}^r_i-1× n_i× r_i}$ of order $3$ , we follow the standard nomenclature in TN literature [oseledets2011tensor]: The second dimension representing the physical index $n_i$ , is oriented along the $z$ -axis (depth). The first and third dimensions correspond to the auxiliary rank indices $r_i-1$ and $r_i$ , which encode the connection between adjacent cores, respectively, as illustrated in Fig. 1. The matrices $G_i(k_j)=:G_i(:,k_j,:)$ for $k_j=1,…,n_i$ are often called slices of the tensor $G_i$ .
The maximum rank of a TT is defined as ${r_max=\max_i=0,… Nr_i}$ . While any tensor theoretically admits a TT representation, the format only yields significant gains in storage and computational efficiency when the ranks of the underlying tensor, or an approximation thereof, remain small. For a more intuitive grasp of the TT format, we refer the reader to Appendix LABEL:appendix:ex, where we provide an illustrative example.
### II-A Arithmetics in the Tensor-Train Format
Many basic operations, such as addition and element-wise multiplication, can directly be performed in the TT format, without forming the full tensor. We briefly summarize the most basic operations, which can be originally found in [oseledets2011tensor].
Consider two TTs ${A=A_1(k_1)⋯ A_N(k_N)}$ and ${B=B_1(k_1)⋯ B_N(k_N)}$ and denote by ${(r_i^A)_i=0^N}$ and ${(r_i^B)_i=0^N}$ their respective ranks.
#### II-A 1 Addition
The sum ${C=A+B}$ can be computed as
| | $\displaystyle C_i(k_i)$ | $\displaystyle=\begin{pmatrix}A_i(k_i)&0\ 0&B_i(k_i)\end{pmatrix},\$ | $\displaystyle i=2,\dots,N-1,$ | |
| --- | --- | --- | --- | --- |
This strategy requires no arithmetic operations but increases the ranks of $C$ to ${r_i^C=r_i^A+r_i^B,}$ ${i=1,\dots,N-1}$ . Typically, a rank truncation is performed after the addition, which finds a rank-reduced approximation, see Sec. II-A 6.
#### II-A 2 Tensor-matrix multiplication
Let ${A∈{ℝ}^n_1×\dots× n_N}$ be a full tensor and $\bm{U}∈{ℝ}^m× n_i$ a matrix. The tensor-matrix multiplication $A×_i\bm{U}$ is defined by [LMV2000_HOSVD]
| | $\displaystyle(A×_i\bm{U})(k_1,\dots,k_i,\dots,k_N)$ | |
| --- | --- | --- |
for $k_i=1,\dots,m$ . This operation can be interpreted as the contraction of the matrix $\bm{U}$ with the tensor $A$ in the $i$ th mode. In the following, we slightly abuse the notation $×_i$ to denote the tensor-matrix multiplication in the TT format. Specifically, ${A×_i\bm{U}}$ is defined as the TT obtained from $A$ by replacing its $i$ th core $G_i$ with the tensor ${G_i×_2\bm{U}}$ . $G_1$ $G_2$ $G_3$ $G_4$ $G_N$ $r_1$ $r_2$ $r_3$ $r_4$ $r_N-1$ $r_0$ $r_N$ $n_1$ $n_2$ $n_3$ $n_4$ $n_N$
Figure 1: Graphical representation of the TT decomposition. Each edge can be interpreted as a tensor contraction.
#### II-A 3 Marginalization
The marginalization in the $i$ th component of a tensor $A$ is the summation over all dimensions except $i$ , which is mathematically equivalent to
| | $\displaystyle A\bigtimes_\begin{subarray{c}j=1\ j≠ i\end{subarray}}^N1_n_{j}^⊤, 1_n_{j}=(1,\dots,1)^⊤∈{ℝ}^n_j,$ | |
| --- | --- | --- |
where
| | $\displaystyle A\bigtimes_j=1^N\bm{U}_j=A×_1\bm{U}_1\dots×_N\bm{U}_N.$ | |
| --- | --- | --- |
#### II-A 4 Element-wise multiplication
The element-wise multiplication or Hadamard product ${C=A∘ B}$ is computed by
| | $\displaystyle C_i(k_i)=A_i(k_i)⊗ B_i(k_i), i=1,\dots,N.$ | |
| --- | --- | --- |
The Kronecker product for matrices $⊗$ causes a multiplication of the ranks of the two tensors $A$ and $B$ in the TT format, i.e., ${r_i^C=r_i^A· r_i^B}$ for $i=0,\dots,N$ .
#### II-A 5 Scalar multiplication
The product ${λ A}$ of a tensor $A$ in the TT format with a scalar ${λ∈{ℝ}}$ is obtained by multiplying $λ$ to one of the core tensors of $A$ . It does not matter to which of the $N$ core tensors the multiplication is applied, and the rank is not increased.
#### II-A 6 Truncation
Many operations, such as addition or element-wise multiplication of tensors in the TT format, significantly increase the ranks. To keep computations and memory feasible, these operations are typically followed by a rank truncation. Common truncation algorithms use consecutive singular value decompositions of the core tensors. By discarding singular values smaller than a tolerance $ϑ>0$ , the tensor is typically compressed to a TT representation of lower ranks. This procedure ensures that the approximation error remains bounded and is directly controlled by the choice of $ϑ$ , as the error grows linearly with $ϑ$ [hackbusch2012tensor, oseledets2011tensor]. Since rank truncation is a standard operation in the context of tensor networks, we refer to [oseledets2011tensor] for a detailed description.
#### II-A 7 TT-cross algorithm
Another key component of the proposed method is the element-wise application of a nonlinear function ${f(A)}$ , where $A$ is a tensor in TT format. We consider TT -cross algorithms [oseledets_TT_cross, ghahremani2024deim] to evaluate $f(A)$ directly in the TT format. The TT -cross can be interpreted as an interpolation scheme for multivariate functions on a tensor grid. This method allows evaluations without forming the full tensor. Since a detailed description of the algorithm would obscure the main ideas of this work, we refer to Appendix LABEL:app:cross and [oseledets_TT_cross] for a more detailed discussion and to [Cross_survey] for a broader survey of tensor cross algorithms.
**Remark**
*Our considerations can be extended to the more general class of tree tensor networks (TTN), which were first introduced in quantum chemistry [wang2003multilayer]. In mathematical literature, they were considered first as hierarchical Tucker tensors in the binary case [hackbusch2012tensor, hackbusch_first_HT_paper] and in [falco2015geometric] for the general case. We note that both binary and general TTNs include TTs as a special case. While TTNs often allow representations with lower ranks, their implementation is considerably more involved.*
## III Detection in Additive Noise Models
### III-A System Model
We consider the real-valued All considerations can trivially be extended to complex-valued models. For the sake of exposition, we only consider the real-valued case. observation model with additive white noise
$$
\bm{y}=f(\bm{x})+\bm{n}, \bm{x}∈A^N_T, \bm{n}∈{ℝ}^N_R \tag{1}
$$
where $n_1,…,n_N_{R}$ are independent and identically distributed (i.i.d.) noise samples from an exponential family distribution. The transmit sequence $\bm{x}$ consists of i.i.d. symbols $x_i$ , ${i=1,…,N_T}$ , each drawn from a discrete alphabet ${A⊂{ℝ}}$ of cardinality ${|A|=L}$ with probability ${P(x_i)}$ . In the context of Bayesian inference, we are interested in the symbol-wise APPs
$$
\displaystyle P(x_i=a|\bm{y})=∑\limits_\mathclap{\begin{subarray{c}\bm{a}∈A^N_T\\
a_i=a\end{subarray}}}P(\bm{x}=\bm{a}|\bm{y}), a∈A, i=1,…,N_T. \tag{2}
$$
The symbol-wise maximum a posteriori (MAP) detector, which minimizes the symbol error probability, is defined as
$$
\displaystyle\hat{x}_i,MAP:=\arg\max\limits_a∈AP(x_i=a|\bm{y}). \tag{3}
$$
Using Bayes’ theorem, the APP distribution can be written as
| | $\displaystyle P(\bm{x}|\bm{y})∝ p(\bm{y}|\bm{x})P(\bm{x})∝∏\limits_j=1^N_Re^\ell(y_j|\bm{x)}∏\limits_i=1^N_TP(x_i)$ | |
| --- | --- | --- |
where ${\ell(y_j|\bm{x})}$ denotes the log-likelihood of the observation $y_j$ , and the proportionality $∝$ means that two terms differ only in a factor independent of $\bm{x}$ . Equivalently, the APP can be expressed in the logarithmic domain as
$$
\log P(\bm{x}|\bm{y})=C+∑\limits_j=1^N_R\ell(y_j|\bm{x})+∑\limits_i=1^N_T\log P(x_i)=:C+Λ(\bm{x}). \tag{4}
$$
Since $C$ is a constant with respect to $\bm{x}$ , we can neglect it in the context of detection and instead consider the unnormalized log- APP metric ${Λ(\bm{x})}$ in the following.
### III-B Main Contribution
Our proposed approach represents the log-likelihood terms $\ell(y_j|\bm{x})$ and the log-prior $\log P(\bm{x})$ exactly in the TT format with low ranks. In this framework, the tensor structure serves as a probability lookup table, where each entry of the tensor in ${ℝ}^L×\dots× L$ corresponds to an evaluation of the discrete APP mass function. If the problem admits a low-rank TT decomposition or approximation, this representation enables efficient Bayesian inference in high-dimensional settings. We construct the APP in the logarithmic domain according to (4), as this typically yields much lower TT ranks compared to the linear domain. The symbol-wise APPs are then obtained by exponentiation and marginalization, with all operations carried out directly in the TT format, as described in Sec. II-A.
#### III-B 1 Construction of ${\log P(\bm{x})=∑_i=1^N_T\log P(x_i)}$
The prior can be exactly represented by a rank- $2$ TT, where the rank is independent of $N_R$ and $N_T$ . We denote the vector of log-priors for any symbol $x_i$ by $\bm{v}=≤ft(\log P(x_i=a_1),…,\log P(x_i=a_L)\right)^⊤∈{ℝ}^L$ , where $\{a_1,…,a_L\}=A$ . The TT construction has the first and last core
| | $\displaystyle G_1$ | $\displaystyle=\begin{pmatrix}1_L&\bm{v}\end{pmatrix}∈{ℝ}^1× L× 2,$ | |
| --- | --- | --- | --- |
The remaining cores $G_i∈{ℝ}^2× L× 2$ , for $i=2,\dots,N_T-1$ , can be constructed by
| | $\displaystyle G_i(:,j,:)=\begin{pmatrix}1&v_j\ 0&1\end{pmatrix}, for j=1,\dots,L.$ | |
| --- | --- | --- |
#### III-B 2 Sum over log-likelihoods
For a low-rank TT representation of the sum over the log-likelihood terms in (4), we consider two strategies: constructing the full sum directly in the TT format, or constructing each log-likelihood term $\ell(y_j|\bm{x})$ , $j=1,\dots,N_R$ , separately and summing the resulting $N_R$ TT s. Both strategies are mathematically equivalent; the choice is problem-dependent and can be based on the simplicity of the construction, respectively.
If each log-likelihood term $\ell(y_j|\bm{x})$ , ${i=1,…,N_R,}$ is constructed individually in the TT format of rank at most ${r}$ , summing the $N_r$ log-likelihood terms also sums the ranks, yielding a result of rank at most $rN_R$ . The rank of the directly constructed full sum typically scales with $N_R$ as well. In either case, a truncation with tolerance $ϑ>0$ may recompress the resulting tensor to a moderate rank. Note that the above rank estimates are only upper bounds.
Finally, we add the log-prior $\log P(\bm{x})$ to the joint log-likelihood, completing the TT construction of the full log- APP in (4).
#### III-B 3 Exponentiation
Let $A$ denote the TT representation of the log- APP metric ${Λ(\bm{x})}$ in (4), constructed as described above. Our goal is to compute $\exp(A)$ , however, since $\exp(·)$ is nonlinear, the ranks of an exact representation of $\exp(A)$ can become prohibitively large. Therefore, we approximate $\exp(A)$ using a TT -cross approach (cf. Sec. II-A 7). We consider two TT -cross algorithms, a classical TT -cross variant and a DMRG -like cross algorithm. For the numerical evaluation we use the implementations multifuncrs and funcrs from the TT -toolbox [oseledets2016_toolbox], respectively.
To improve the quality of the approximation, both methods support an initialization step for the TT -cross algorithm. We use a truncated Taylor series
$$
\exp(A)≈∑\limits_k=0^p\frac{A^k}{k!},
$$
for initialization, where $A^k$ denotes the element-wise power. For this, we use the tt_exp implementation from the TT -toolbox [oseledets2016_toolbox]. Typically, $p≈ 10$ provides a reliable initialization. The tt_exp function also accepts a maximum rank parameter r_max to keep the computation tractable.
The two cross functions differ in their underlying algorithmic strategy. The multifuncrs routine follows a more classical TT -cross interpolation approach, reconstructing the tensor from adaptively sampled entries. In contrast, funcrs employs a DMRG -like sweeping algorithm that updates local TT cores through low-rank projections, while still falling into the class of cross approximations. The latter typically achieves higher accuracy at the cost of increased computational complexity. A central ingredient in both cross functions is the use of random sampling in each core. This enables the addition of new directions to avoid getting stuck in local minima. Consequently, both functions are non-deterministic.
#### III-B 4 Marginalization
To obtain the symbol-wise APPs in (2), the marginalization over all dimensions except for one needs to be computed. This can efficiently be done in the TT format as described in Sec. III-B 4. The result of the marginalization is a tensor of dimension 1, i.e., a vector that represents the entries proportional to ${P(x_i=a|\bm{y})}$ , $a∈A$ .
### III-C Computational Complexity
The computational complexity is dominated by the TT -cross algorithm and the truncation function, both of which rely on successive singular value decompositions of the underlying core tensors. If the core tensors are of size $r× n× r$ , the computational complexity of a single SVD scales with $O(r^3)$ , where $r≤ r_max$ . One full sweep through the TT scales with $O(Nr^3)$ . Such sweeps are performed iteratively, for instance, to evaluate the terms of the truncated Taylor series.
**Remark**
*Instead of computing the exponential of $A$ followed by a marginalization, an alternative approach is to find the maximal element of the full tensor $A$ in the log-domain, which corresponds to MAP sequence estimation. A promising candidate for solving this maximization problem is the TT Optimizer (TTOpt) [sozykin2022ttopt], which provides a gradient-free and efficient optimization approach in the TT format. We leave this maximization-based approach for future investigation and focus here on the marginalization, i.e., the symbol-wise detection problem.*
In the following, we consider two inference problems relevant to the field of communications: MIMO detection and channel decoding for AWGN channels. We provide explicit log- APP constructions in the TT format and thereby demonstrate that these problems inherently admit low-rank representations.
## IV Example: MIMO Detection
MIMO systems are a key technology in many current and emerging wireless communication systems due to their high spectral efficiency and throughput [yang_fifty_2015]. In this context, efficient symbol detection is a crucial computational bottleneck in realizing these theoretical performance gains in practical systems [cespedes_expectation_2014].
### IV-A MIMO Channel Model
We consider a MIMO system with $\tilde{N}_T$ antennas at the transmitter and $\tilde{N}_R$ antennas at the receiver. The transmit symbols ${\tilde{x}_i}$ , ${i=1,…,\tilde{N}_T}$ are independently and uniformly drawn from an $M$ - quadrature amplitude modulation (QAM) constellation ${M⊂ℂ}$ . The channel output ${\tilde{\bm{y}}∈ℂ^\tilde{N_R}}$ is given by ${\tilde{\bm{y}}=\tilde{\bm{H}}\tilde{\bm{x}}+\tilde{\bm{n}}}$ , where $\tilde{\bm{n}}$ is circular complex AWGN. We assume a Rayleigh-fading MIMO channel, i.e., each element of $\tilde{\bm{H}}$ is independently sampled from a circular complex standard Gaussian distribution $CN(0,1)$ . We define the receiver SNR of each transmission as
$$
SNR:=10\log_10≤ft(\frac{\|\tilde{\bm{H}}\tilde{\bm{x}}\|^2}{\|\tilde{\bm{n}}\|^2}\right) in dB.
$$
The complex-valued MIMO system can be decomposed into an equivalent real-valued representation
$$
\bm{H}=\begin{pmatrix}Re\{\tilde{\bm{H}}\}&-Im\{\tilde{\bm{H}}\}\\
Im\{\tilde{\bm{H}}\}&Re\{\tilde{\bm{H}}\}\end{pmatrix}∈ℝ^N_R× N_\text{T}, \begin{matrix}N_T=2\tilde{N}_T,\\
N_R=2\tilde{N}_R,\end{matrix}
$$
to match the model in (1) with $\bm{y}=(Re\{\tilde{\bm{y}}\},Im\{\tilde{\bm{y}}\})^⊤$ , $\bm{x}=(Re\{\tilde{\bm{x}}\},Im\{\tilde{\bm{x}}\})^⊤$ , $f(\bm{x})=\bm{H}\bm{x}$ , $\bm{n}=(Re\{\tilde{\bm{n}}\},Im\{\tilde{\bm{n}}\})^⊤$ , and $A=\{± 1,± 3,…\}$ with $L=|A|=√{M}$ .
In general, MIMO detection refers to the task of inferring the transmit vector $\bm{x}$ from the channel observation $\bm{y}$ [yang_fifty_2015], and is a proven nondeterministic polynomial-time (NP)-hard problem [verdu_computational_1989]. In this work, we assume perfect channel state information (CSI) at the receiver, i.e., knowledge of $\bm{H}$ and the SNR.
In the following, we consider symbol-wise MAP MIMO detection, which minimizes the symbol error probability, and requires the computation of the marginal APPs ${P(x_i|\bm{y})}$ , cf. (2). This marginalization involves the summation over an exponentially large set of transmit vectors $\bm{x}$ and is therefore #P-hard in general.
**Remark**
*The following considerations regarding symbol-wise MAP detection for MIMO channels apply to any real-valued linear observation model with AWGN
$$
\bm{y}=\bm{H}\bm{x}+\bm{n}, \bm{n}∼N(\bm{0},σ^2\bm{I}_N_{R}), \tag{5}
$$
where $\bm{y}$ is the received signal, $\bm{H}=:(\bm{h}_1,…,\bm{h}_N_{R})^⊤∈{ℝ}^N_R× N_\text{T}$ is the observation matrix, and $\bm{x}$ is the discrete-valued vector of unknowns.*
### IV-B TT Construction of the Log-Likelihood Terms
Our objective is to express the unnormalized log- APP metric ${Λ(\bm{x})}$ in (4) in the TT format. The construction of the log-priors ${\log P(\bm{x})}$ , which is described in Sec. III-B 1, can be omitted in case of uniform priors, as assumed here. This section focuses on the specific construction of the log-likelihood terms $\ell(y_j|\bm{x})=-(y_j-h_j^⊤\bm{x})^2/(2σ^2)$ , ${j=1,…,N_R}$ for the linear observation model (5), such as the MIMO channel model. $-10$ $-8$ $-6$ $-4$ $-2$ $0$ $2$ $4$ $6$ $8$ $10$ $10^-3$ $10^-2$ $10^-1$ $4$ -QAM ${\tilde{N}=16}$ $4$ -QAM ${\tilde{N}=32}$ $16$ -QAM ${\tilde{N}=8}$ $16$ -QAM ${\tilde{N}=16}$ SNR (dB) SER LMMSE EP [cespedes_expectation_2014] TTDet-sample (new) TTDet-sweep (new) Sphere decoder
Figure 2: SER over SNR for various MIMO detectors across different ${\tilde{N}×\tilde{N}}$ system configurations (real-valued dimensions ${N_T=N_R=2\tilde{N}}$ ). The maximum rank r_max in the truncated Taylor-series initialization is set to (from left to right) $10,10,40,20$ .
#### IV-B 1 Construction of $y_j$
The $j$ th channel observation ${y_j∈ℝ}$ can be trivially represented as a rank- $1$ TT. We obtain an exact TT representation by setting ${G_1=y_j\bm{1}_L}$ and all core tensors ${G_j=1_L}$ for $j=2,\dots,N_R$ . Note that this is equivalent to the representation
$$
y_j\bigtimes_j=1^N1_L∈ℝ^L×\dots× L.
$$
#### IV-B 2 Construction of $h_j^⊤\bm{x}$
Let ${\bm{v}=(a_1,…,a_L)^⊤}$ be the alphabet vector with ${A=\{a_1,…,a_L\}}$ . Further, let ${\bm{h}_j∈{ℝ}^N_T}$ and define ${\bm{U}=(\bm{v},\bm{1}_L)∈{ℝ}^L× 2}$ . Using ${\bm{h}^⊤_j=(h_j^1,\dots,h_j^N_T)}$ , $h_j^⊤\bm{x}$ can be represented in the TT format of rank $3$ with the core tensors
| | $\displaystyle G_1$ | $\displaystyle=\begin{pmatrix}0&h_j^N_T&0\ 1&0&0\end{pmatrix}×_2\bm{U}∈{ℝ}^1× L× 3,$ | |
| --- | --- | --- | --- |
For $i=2,\dots,N_T-1$ we first define the slices
| | $\displaystyle\widetilde{G}_i(:,1,:)=\begin{pmatrix}0&h_j^N_T-i+1&0\ 0&0&h_j^N_T-i+1\ 0&0&h_j^N_T-i+1\end{pmatrix}, \widetilde{G}_i(:,2,:)=\bm{I}_3,$ | |
| --- | --- | --- |
and then set ${G_i=\widetilde{G}_i×_2\bm{U}∈{ℝ}^3× L× 3}$ . Although the reversed ordering of $h_j^N_T,…,h_j^1$ within $G_1,…,G_N_{T}$ may appear unconventional, this construction is, to the best of our knowledge, the simplest. We note, however, that alternative constructions exist that represent the same tensor.
The addition ${y_j-h_j^⊤\bm{x}}$ is of rank at most $4$ , recall Sec. II-A. By performing the element-wise and scalar multiplication from Sec. II-A, we can exactly represent each of the $N_R$ log-likelihood terms ${\ell(y_j|\bm{x})=-(y_j-h_j^⊤\bm{x})^2/(2σ^2)}$ in TT format with rank ${16}$ or lower. For a possible rank reduction, we perform a truncation with given tolerance $ϑ>0$ after constructing each log-likelihood term.
Algorithm 1 summarizes the proposed TT -based algorithm for symbol-wise MIMO detection. Note that all constructions and operations in Algorithm 1 are performed in the TT format, except for the hard decision in line 4, which operates on the explicit vector representation of the symbol-wise posterior marginals. We refer to this algorithm as TTDet, and distinguish its two variants by the cross approximation method applied for the exponentiation in TT format in line 3: TTDet-sample, based on the more classical TT -cross algorithm, and TTDet-sweep, based on the DMRG -like cross algorithm.
Initialization: r_max, $ϑ$
Data: Observation $\bm{y}∈ℝ^N_R$ , CSI $≤ft\{\bm{H}∈ℝ^N_R× N_\text{T},σ^2\right\}$
// TT construction
Construct $\ell(y_j|\bm{x}), j=1,…,N_R$
// Sec. IV-B
Compute $Λ(\bm{x})=∑_j=1^N_R\ell(y_j|\bm{x})$
// Sec. III-B 2
// TT exponentiation & marginalization
Compute symbol-wise APPs
// Sec. III-B 3 -B4
-1em
$$
P(x_i=a|\bm{y})∝∑\limits_\mathclap{\begin{subarray{c}\bm{a}∈A^N_T\\
a_i=a\end{subarray}}}e^Λ(\bm{a)}, a∈A, i=1,…,N_T,
$$
// Switch from TT to explicit vector format
MAP hard decision ${\hat{x}_i=\arg\max\limits_a∈AP(x_i=a|\bm{y})}$
Result: Detection result $\hat{\bm{x}}←(\hat{x}_1,…,\hat{x}_N_{T})^⊤$
Algorithm 1 TTDet for MIMO detection
### IV-C Numerical Evaluation
We numerically evaluate the performance of the TTDet algorithm for different MIMO settings All numerical results in this paper were obtained using the MATLAB TT-Toolbox [oseledets2016_toolbox] and the MATLAB Tensor Toolbox [Tensor_Toolbo_Kolda]. We provide the source code for all simulations in [schmid2026github].. We consider $4$ - and $16$ - QAM constellations and quadratic MIMO channel matrices with ${\tilde{N}_T=\tilde{N}_R=:\tilde{N}=8,16,32}$ . For these dimensions, a full tensor representation of the APP distribution ${P(\bm{x}|\bm{y}})$ is infeasible, as it would require storage of ${M^\tilde{N}}$ elements. As a baseline, we consider three well-established MIMO detection algorithms:
- The linear minimum mean squared error (LMMSE) detector, and the EP detector [cespedes_expectation_2014] with $10$ iterations. Both algorithms rely on a Gaussian approximation of the APP distribution, followed by a symbol-wise nearest-neighbor decision.
- The sphere decoder [yang_fifty_2015], a tree-search based MIMO detection algorithm. We employ the sphere detector without early termination, such that it achieves optimal performance in the maximum likelihood sequence estimation (MLSE) sense.
Fig. 2 shows the symbol error rate (SER) over the SNR for the considered MIMO detectors. To estimate the SER, we perform Monte Carlo simulations in which the transmit vector $\bm{x}$ and the channel matrix $\bm{H}$ are randomly sampled for each new transmission, as described in Sec. IV-A. All detection algorithms are then evaluated on the same data samples. For each SNR value, the simulation is continued until the sphere decoder records $100$ block errors, i.e., transmissions in which one or more symbol errors occur.
For the $4$ - QAM scenarios, the proposed TTDet algorithm achieves near-optimal performance, closely approaching the MLSE -optimal performance of the sphere decoder. It significantly outperforms the LMMSE detector, and yields a $0.7$ dB gain over the EP detector at a target ${SER=10^-2}$ . The TTDet-sample variant behaves more robustly in the low- SNR regime, where it slightly outperforms the TTDet-sweep variant and the sphere decoder in terms of SER Note that the sphere detector is optimal in the MLSE sense, but not necessarily in terms of SER, which explains the slightly lower SER of the TTDet-sample and EP detector for low SNR values..
We emphasize that TTs with a maximum rank of ${\texttt{r\_max}=10}$ are sufficient to initialize the TT -cross algorithm and achieve this quasi-optimal performance, indicating the low-rank nature of this problem. Note that the DMRG -like TT -cross algorithm may require substantially higher intermediate ranks than ${\texttt{r\_max}=10}$ . Table I reports the maximum TT rank $r_max$ after exponentiation at the output of the TT -cross algorithm (TTDet-sample variant) for the $4$ - QAM scenario with ${\tilde{N}=32}$ . Additionally, we analyze the memory savings relative to the explicit tensor representation, where the latter requires $M^\tilde{N}$ floating-point numbers to store. In our simulations, this corresponds to a memory reduction in the order of $10^13$ and $10^14$ at $SNR=-11$ dB and $-5$ dB, respectively.
TABLE I: Memory complexity of the TTDet-sample algorithm for ${\tilde{N}=32}$ and $4$ - QAM
| | $r_max$ after TT -cross | Memory savings w.r.t. $M^\tilde{N}$ | | |
| --- | --- | --- | --- | --- |
| $E_b/N_0$ | mean | median | mean | median |
| $-11$ dB | $210$ | $209$ | $10^13$ | $10^13$ |
| $-5$ dB | $98$ | $86$ | $10^14$ | $10^14$ |
Finally, we consider the SER performance for the $16$ - QAM scenarios in Fig. 2. The TTDet algorithm significantly outperforms the LMMSE detector and achieves a $2$ dB gain over the EP detector. In the high- SNR regime, the TTDec algorithm exhibits a performance degradation relative to the sphere decoder. Here, the TTDet-sweep variant slightly outperforms the TTDet-sample variant. This suggests that in this regime, the TT -cross algorithm may fail to approximate the exponential sufficiently accurate, which indicates that the Taylor-series initialization with ${\texttt{r\_max}=20}$ and $40$ , respectively, is insufficient to achieve optimal performance or higher intermediate ranks in the TT -cross are required.
Fig. LABEL:fig:rank_sweep_mimo illustrates the dependency of the SER performance on the maximum rank r_max of the truncated Taylor series initialization for ${\tilde{N}=8}$ . At low SNR, a small rank ${\texttt{r\_max}=6}$ suffices to achieve quasi-optimal performance. In the high-SNR regime, we can observe a clear convergence behavior as r_max increases. This indicates that higher ranks are required to close the remaining gap to optimality.