Compressed Bayesian Tensor Regression

• Compressed Bayesian tensor regression is a family of methods that compress high-dimensional tensors using low-rank factorizations to reduce parameters while preserving multiway structure.
• These techniques apply CP/PARAFAC, Tucker decompositions, and random projections to facilitate posterior inference with Gibbs sampling or MCMC algorithms.
• The approach addresses scalar-on-tensor, tensor-on-tensor, and mixed-effects models, enhancing prediction, uncertainty quantification, and dimension recovery.

Compressed Bayesian tensor regression denotes a family of Bayesian regression methods for tensor-valued data in which high-dimensional coefficient structures, covariates, or covariance components are replaced by lower-dimensional representations that preserve multiway structure while enabling posterior inference. In the cited literature, compression is achieved through low-rank CP/PARAFAC factorizations, Tucker decompositions, generalized tensor random projections, compressed random-effects covariance parameterizations, and sparse Kronecker product decompositions, depending on whether the setting is scalar-on-tensor, tensor-on-tensor, mixed-effects, or mixed-response regression (Guhaniyogi et al., 2015). Across these formulations, the common objective is to avoid naive vectorization of tensors, reduce the effective parameter dimension, and retain uncertainty quantification through Gibbs sampling or related MCMC schemes (Lock, 2017).

1. Scope and model classes

Compressed Bayesian tensor regression appears in several closely related regression settings. In scalar-on-tensor regression, a scalar response is modeled against a tensor predictor through an inner product with a coefficient tensor, as in

$$y_i \mid \mathcal X_i,\boldsymbol{\beta},\sigma^2 \sim \mathcal{N}\!\Bigl(\langle \boldsymbol{\beta},\,\mathcal X_i\rangle,\;\sigma^2\Bigr),$$

with $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$ and $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$ (Guhaniyogi et al., 2015). A related form retains the same scalar-on-tensor structure but first replaces the original tensor covariates by compressed tensors $\widetilde{\mathcal X}_j$, yielding

$$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$

with $m_1\cdots m_M\ll d_1\cdots d_K$ after projection (Casarin et al., 2 Oct 2025).

In tensor-on-tensor regression, both predictor and response are tensors. One formulation assumes

$$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$

where $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$, $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$, and $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$ is the regression coefficient tensor (Lock, 2017). This framework generalizes prediction of a scalar outcome from a tensor, a matrix from a matrix, or a tensor from a scalar (Lock, 2017).

Mixed-effects extensions incorporate repeated measures and random effects. CoMET assumes, for subject $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$0 and occasion $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$1,

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$2

with tensor-valued fixed-effect and random-effect covariates and tensor-normal random slopes (Sarkar et al., 22 Feb 2026). A different mixed-effects construction treats the response itself as tensor-valued and introduces subject- and region-specific random effects to study both activation and connectivity in fMRI (Spencer et al., 2019).

A further generalization handles mixed-type multivariate responses. In the BSKPD framework, each response component satisfies

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$3

with Gaussian, Poisson, and Bernoulli outcomes embedded in a common exponential-family form (Wang et al., 20 May 2025).

2. Compression mechanisms

A central distinction within the literature is where compression is applied. In one major line of work, compression is imposed directly on the regression coefficient tensor through low-rank structure. For scalar-on-tensor regression, the coefficient tensor is written in rank-$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$4 CP form,

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$5

reducing the parameter count from $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$6 to $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$7 (Guhaniyogi et al., 2015). In tensor-on-tensor regression, the same idea appears as a reduced CP-rank coefficient tensor

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$8

with total free parameters

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$9

often orders of magnitude smaller than $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$0 (Lock, 2017).

A second line replaces CP by Tucker compression. In Bayesian tensor-on-tensor regression with efficient computation, the coefficient tensor is parameterized as

$$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$1

where the core tensor $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$2 and factor matrices $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$3, $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$4 determine mode-specific multilinear ranks $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$5 (Wang et al., 2022). This permits modewise rank selection rather than a single CP rank.

A third mechanism compresses the predictors rather than only the coefficients. CBTR introduces a Generalized Tensor Random Projection in which a tensor $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$6 is mapped to $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$7 through mode-wise projections, tensor-wise projections, or a combination of the two (Casarin et al., 2 Oct 2025). The paper explicitly identifies classic random projection on $$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$8, separate mode-wise projection on each mode, and tensor-wise sketching that reduces the tensor order as special cases (Casarin et al., 2 Oct 2025).

A fourth mechanism compresses covariance structure in mixed-effects models. In CoMET, each mode-$$\boldsymbol{\beta}\in\mathbb R^{p_1\times\cdots\times p_D}$$9 random-effects covariance factor is approximated using random matrices $\widetilde{\mathcal X}_j$0 with $\widetilde{\mathcal X}_j$1, producing compressed random-effects covariates $\widetilde{\mathcal X}_j$2 and low-dimensional covariance parameters $\widetilde{\mathcal X}_j$3 (Sarkar et al., 22 Feb 2026). The resulting covariance parameter is low-dimensional enough to admit simple Gaussian priors (Sarkar et al., 22 Feb 2026).

A fifth form of compression uses a sparse Kronecker product decomposition. In BSKPD, each third-order coefficient tensor $\widetilde{\mathcal X}_j$4 is represented through an unfolding

$\widetilde{\mathcal X}_j$5

which yields

$\widetilde{\mathcal X}_j$6

(Wang et al., 20 May 2025). This preserves multilinear structure while decomposing the coefficient tensor into lower-dimensional factors.

These constructions indicate that “compression” in this area is not confined to one algebraic device. In the cited work, it can mean rank restriction, sketching, covariance projection, or structured factorization, provided that the resulting model remains Bayesian and tensor-aware.

3. Prior design and Bayesian regularization

Bayesian compression is coupled with prior distributions that stabilize estimation and often induce adaptive sparsity. In the CP-based scalar-on-tensor model, Guhaniyogi et al. introduce the M–DGDP prior. Component scales satisfy $\widetilde{\mathcal X}_j$7, with $\widetilde{\mathcal X}_j$8 and $\widetilde{\mathcal X}_j$9, while margin vectors satisfy

$$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$0

with exponential local scales $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$1 and Gamma hyperpriors on $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$2 (Guhaniyogi et al., 2015). Small $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$3 values effectively switch off redundant CP components (Guhaniyogi et al., 2015).

In Lock’s tensor-on-tensor framework, the ridge-penalized least-squares criterion

$$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$4

is shown to be the mode of a Bayesian posterior under a Gaussian likelihood and a Gaussian prior on $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$5 supported on rank $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$6 (Lock, 2017). Here the compression is low rank and the regularization is ridge-like, rather than sparse in the usual global-local sense.

The Tucker-based BayTensor model employs conjugate Gaussian-inverse-gamma priors on the core tensor, factor matrices, and noise variance: $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$7, $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$8, $$y_j = \mu + \bigl\langle\mathcal{B},\,\widetilde{\mathcal{X}_j}\bigr\rangle +\sigma\,\varepsilon_j, \quad \varepsilon_j\overset{\mathrm{iid}{\sim}}N(0,1),$$9, and $m_1\cdots m_M\ll d_1\cdots d_K$0, with diagonal covariance matrices and a prior $m_1\cdots m_M\ll d_1\cdots d_K$1 on the model dimensions $m_1\cdots m_M\ll d_1\cdots d_K$2 (Wang et al., 2022). This separates parameter shrinkage from model-dimension uncertainty.

CoMET combines low-rank CP structure for fixed effects with Horseshoe priors. Each entry $m_1\cdots m_M\ll d_1\cdots d_K$3 receives

$m_1\cdots m_M\ll d_1\cdots d_K$4

with local half-Cauchy scales $m_1\cdots m_M\ll d_1\cdots d_K$5, component-level half-Cauchy scales $m_1\cdots m_M\ll d_1\cdots d_K$6, and inverse-gamma prior on $m_1\cdots m_M\ll d_1\cdots d_K$7 (Sarkar et al., 22 Feb 2026). The paper states that this supports both parameter reduction and sparsity and enables fixed-effects selection (Sarkar et al., 22 Feb 2026).

Spencer et al. use a multiway stick-breaking shrinkage prior for tensor-response regression. With

$m_1\cdots m_M\ll d_1\cdots d_K$8

the component weights satisfy an ordered shrinkage pattern $m_1\cdots m_M\ll d_1\cdots d_K$9, while local exponential mixtures induce generalized double-Pareto-type voxelwise shrinkage (Spencer et al., 2019).

BSKPD adopts Three-Parameter Beta-Normal shrinkage priors for coefficient-factor matrices and auxiliary covariates, together with an inverse-Wishart prior for the residual covariance $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$0 (Wang et al., 20 May 2025). This construction is explicitly designed for voxel-level sparsity and interpretability in mixed-type response models (Wang et al., 20 May 2025).

CBTR, after projection, returns to a PARAFAC coefficient representation and a hierarchical global-local shrinkage prior inherited from Guhaniyogi et al. The margin vectors $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$1 follow Gaussian priors with component-wise and global scales, local exponential scales, Gamma hyperpriors, inverse-gamma prior on $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$2, and Dirichlet prior on the component weights $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$3 (Casarin et al., 2 Oct 2025).

4. Posterior computation and optimization

The computational core of compressed Bayesian tensor regression is typically a block Gibbs sampler exploiting conjugate or conditionally Gaussian updates. In the rank-restricted tensor-on-tensor model, under the CP parameterization $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$4, each block $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$5 or $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$6 has a conditionally Gaussian full conditional. For example,

$$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$7

with

$$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$8

and $$\mathbb Y \;=\;\bigl\langle\,\mathbb X\,,\,\mathbb B\,\bigr\rangle_{\!L}\;+\;\mathbb E,$$9 updated from an inverse-gamma conditional (Lock, 2017). The same paper also describes alternating least squares on the penalized objective, with warm starts at large $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$0 followed by decreasing $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$1, described there as “tempered regularization” (Lock, 2017).

In BayTensor, posterior inference alternates between Gibbs updates conditional on fixed multilinear rank $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$2 and a rank–Metropolis–Hastings step on neighboring rank configurations satisfying $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$3 (Wang et al., 2022). Rather than full Reversible Jump MCMC, the method uses a fractional-Bayes-factor Metropolis–Hastings move to jointly update dimension and associated parameters (Wang et al., 2022). The same work also provides an optimization-based alternative: closed-form MAP updates for $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$4 at fixed $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$5, combined with simulated annealing over $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$6 using a BIC objective (Wang et al., 2022).

The M–DGDP model likewise uses a blocked Gibbs sampler. The main blocks are the rank weights $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$7, the margin-specific parameters $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$8, and $$\mathbb X\in\mathbb R^{N\times P_1\times\cdots\times P_L}$$9, with generalized inverse-Gaussian, Gamma, multivariate normal, and inverse-gamma updates arising from conjugacy and scale-mixture representations (Guhaniyogi et al., 2015).

CoMET uses a two-block collapsed Gibbs sampler. One block samples compressed random effects $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$0 and covariance parameters $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$1; the other marginalizes over $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$2, whitens the resulting regression blocks, and samples the mode-specific factor margins $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$3, local shrinkages, global shrinkages, and $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$4 from closed-form conditionals (Sarkar et al., 22 Feb 2026). The paper states that all full-conditionals are in closed form and that each sweep costs only $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$5, roughly linear in the original tensor dimensions (Sarkar et al., 22 Feb 2026).

For mixed-type outcomes, BSKPD combines TPBN shrinkage with Polya-Gamma augmentation. Binary responses are converted to a Gaussian-form conditional likelihood using $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$6 and $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$7, after which $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$8, $$\mathbb Y\in\mathbb R^{N\times Q_1\times\cdots\times Q_M}$$9, $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$0, $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$1, and $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$2 admit closed-form Gibbs updates (Wang et al., 20 May 2025).

CBTR also uses a custom Gibbs sampler with Gaussian updates for the margin vectors, generalized inverse-Gaussian and inverse-gamma updates for hyperparameters, and Gaussian/inverse-gamma updates for $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$3 and $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$4 (Casarin et al., 2 Oct 2025). To reduce sensitivity to a single projection, it further employs Bayesian model averaging over multiple independent random projections, with model evidences estimated by reverse logistic regression (Casarin et al., 2 Oct 2025).

5. Theory: concentration, consistency, and dimension recovery

Several strands of theory support compressed Bayesian tensor regression, though the precise guarantees depend on the compression mechanism. In the scalar-on-tensor CP model with M–DGDP prior, posterior consistency holds under mild regularity, including a true coefficient tensor of rank-$$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$5 CP form, bounded true margins, and growth condition $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$6 (Guhaniyogi et al., 2015). The paper states that the posterior concentrates on any Kullback–Leibler neighborhood of the true $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$7 at the usual root-$$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$8 rate up to logarithmic factors (Guhaniyogi et al., 2015).

CBTR adds theory for random projection itself. Using projection entries drawn i.i.d. from the “database-friendly” distribution of Achlioptas, the paper proves Johnson–Lindenstrauss-type distance preservation: provided the projected dimension satisfies $$\mathbb B\in\mathbb R^{P_1\times\cdots\times P_L\times Q_1\times\cdots\times Q_M}$$9, pairwise Frobenius distances among tensors are preserved within factors $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$00 with high probability (Casarin et al., 2 Oct 2025). Under additional regularity conditions, Theorem 4.1 and Theorem 4.2 establish posterior contraction for Gaussian and PARAFAC priors after compression (Casarin et al., 2 Oct 2025).

For Tucker tensor-on-tensor regression, the principal theoretical contribution is inferential rather than asymptotic: the MCMC sampler jointly estimates both model dimension and parameters, addressing a limitation of earlier Tucker methods that either assumed the core dimension to be known or selected it through cross-validation or model selection criteria (Wang et al., 2022). The empirical section further reports Bayesian dimension recovery rates of 80–100% for MCMC and 40–80% for the simulated-annealing alternative in the simulated settings described in the paper (Wang et al., 2022).

CoMET provides a high-dimensional predictive guarantee. Under assumptions labeled (A1)–(A3), including bounded operator norms, sub-Gaussian covariates, and compression-rank conditions such as $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$01 and $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$02, the posterior predictive mean $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$03 satisfies

$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$04

so the posterior predictive risk tends to zero (Sarkar et al., 22 Feb 2026).

BSKPD proves both identifiability and posterior consistency. Theorem 1 establishes asymptotic identifiability of the sparse Kronecker product decomposition up to the usual orthogonal non-identifiability in $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$05, and Theorem 2 shows posterior concentration on Frobenius-norm neighborhoods of $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$06 in both low-dimensional and high-dimensional regimes under stated sparsity and growth conditions (Wang et al., 20 May 2025).

Taken together, these results suggest that compression in Bayesian tensor regression is not treated solely as a computational heuristic. In these papers it is paired with explicit guarantees on posterior concentration, dimension recovery, distance preservation, or predictive risk.

6. Empirical behavior, applications, and limitations

The empirical literature emphasizes prediction, uncertainty quantification, and parameter savings. In Lock’s facial-attribute experiment, images were represented as $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$07 RGB tensors and outcomes as 72 standardized attributes, with $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$08 training faces and $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$09 held-out faces (Lock, 2017). A grid search over $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$10 and $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$11 up to $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$12 selected $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$13, $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$14, yielding relative prediction error approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$15, compared with greater than $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$16 for unregularized or full-rank ridge, and overall correlation approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$17 between observed and predicted attributes (Lock, 2017). Posterior sampling with $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$18 draws gave out-of-sample 95% credible-interval coverage approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$19 and 90% coverage approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$20 (Lock, 2017).

The Tucker-based BayTensor method reports stronger performance on related tasks. On facial-image-to-attribute data with $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$21 images and $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$22 labels, BayTensor MCMC uses approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$23 parameters and attains RPE $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$24, whereas the CP comparator uses $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$25 parameters with RPE $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$26 (Wang et al., 2022). On MoCap data with $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$27 frames and $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$28, BayTensor MCMC has RPE approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$29 versus $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$30 for CP (Wang et al., 2022). Posterior predictive credible intervals achieve approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$31–$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$32 empirical coverage for nominal $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$33 intervals in simulated and real-data settings (Wang et al., 2022).

In the mixed-effects tensor-response fMRI model of Spencer et al., rank $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$34 minimized DIC, achieved root-MSE around $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$35 versus $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$36 for the vectorized model, yielded activation detection AUC approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$37 versus approximately $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$38, and used $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$39 parameters versus $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$40 in the vectorized model (Spencer et al., 2019). In the Balloon-Analog Risk-Taking application, the rank-$$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$41 model also had healthy MCMC ESS $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$42 and produced posterior partial-correlation patterns consistent with known risk-processing circuits (Spencer et al., 2019).

CBTR’s simulation study varies tensor size, sample size, projection scheme, sparsity of the projection, and PARAFAC rank. The reported findings are that mode-wise GTRP outperforms tensor-wise in most settings, moderate sparsity $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$43 is often optimal, and model averaging over $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$44 projections stabilizes predictions (Casarin et al., 2 Oct 2025). In the oil-volatility and S&P 500 application, CBTR-MW and CBTR-MW(1,2) outperform uncompressed BTR out of sample with RMSE $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$45 versus $$\mathcal X_i\in\mathbb R^{p_1\times\cdots\times p_D}$$46, while maintaining similar in-sample fits (Casarin et al., 2 Oct 2025).

CoMET is reported to outperform penalized competitors across simulation studies and benchmark applications involving facial-expression prediction and music emotion modeling (Sarkar et al., 22 Feb 2026). The paper attributes tractability to structured, mode-wise random projection of the random-effects covariance and a collapsed Gibbs sampler whose computational complexity grows approximately linearly with the tensor covariate dimensions (Sarkar et al., 22 Feb 2026).

The literature also identifies limitations. BayTensor notes that very large tensors may require more advanced samplers such as variational or pseudo-marginal methods (Wang et al., 2022). CBTR observes sensitivity to the choice of random projection and addresses this by Bayesian model averaging; it also notes that loss of interpretability may occur if projection is too aggressive (Casarin et al., 2 Oct 2025). These points counter a common misconception that compression is uniformly benign: the cited work instead treats compression as a tunable statistical approximation whose benefits depend on preserving relevant tensor structure.

7. Relationship to adjacent methods

Compressed Bayesian tensor regression is closely related to reduced-rank regression, tensor decomposition, structured shrinkage, and high-dimensional Bayesian regression, but it is distinguished by its simultaneous use of tensor algebra and posterior uncertainty quantification. Lock’s tensor-on-tensor regression explicitly unifies tensor-on-tensor, reduced-rank, and ridge-penalized models in a single framework (Lock, 2017). Guhaniyogi et al. connect tensor regression to parafac decomposition, multiway shrinkage, and posterior consistency in growing dimensions (Guhaniyogi et al., 2015). Wang and Xu extend the tensor-on-tensor setting by replacing CP with Tucker decomposition and by estimating model dimension inside the Bayesian algorithm rather than by external cross-validation (Wang et al., 2022).

Random-projection approaches such as CBTR differ from low-rank coefficient models in that they compress the covariates before or during inference, with explicit Johnson–Lindenstrauss-type guarantees and Bayesian model averaging across sketches (Casarin et al., 2 Oct 2025). Mixed-effects formulations such as CoMET show that compression can be applied not only to coefficient tensors but also to random-effects covariance structure (Sarkar et al., 22 Feb 2026). BSKPD indicates that a low-rank Kronecker product decomposition can support unified Gaussian, Poisson, and Bernoulli responses while maintaining closed-form Gibbs updates through Polya-Gamma augmentation (Wang et al., 20 May 2025).

A plausible implication is that the field has moved from a relatively narrow focus on scalar-on-tensor CP models toward a broader class of compressed Bayesian tensor regressions in which the point of compression is chosen to match the inferential bottleneck: coefficient dimensionality, unknown multilinear rank, predictor size, random-effects covariance, or mixed-response coupling. The common design principle remains the same: preserve multiway structure, reduce effective dimension, and maintain a fully probabilistic inferential pipeline.