Tensor-based modulation (TBM) schemes are a promising approach for unsourced random access (URA), where user separation relies on decomposing the received signal tensor via the canonical polyadic decomposition (CPD), typically computed with alternating least squares (ALS). Standard ALS, however, treats the factor matrices as unstructured and fails to exploit the discrete structure of the tensor sub-constellations. We propose DVB-ALS, a discrete variational Bayesian CPD framework with specific priors, tailored to a tensor structure with the corresponding encoding strategy, combined with iterative computation of an approximate posterior distribution. A discrete Gaussian mixture prior on one Grassmannian factor softly aligns the estimates toward the constellation points. The remaining factors are jointly modeled with a structured Gaussian prior whose posterior mean is constrained to the Khatri-Rao product manifold and posterior variance upper-bounded to prevent norm divergence during inference. The resulting closed-form coordinate ascent algorithm jointly estimates all latent factors and their uncertainties. We integrate DVB-ALS into DVB-TBM to design a complete URA receiver with single-user demapping, polar decoding with cyclic redundancy check (CRC) verification, and successive interference cancellation (SIC). Simulation results show significant gains over standard ALS-based decomposition and robust detection performance in URA settings, outperforming state-of-the-art schemes under high system loads.

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