Tensor recovery spans heterogeneous data structures and observation processes, calling for a representation that can adapt across modalities and tasks. We propose DiffTR, an unsupervised framework that unifies tensor completion, denoising, and hyperspectral super-resolution through coordinate-conditioned diffusion. Shared factors indexed by tensor coordinates condition a compact score network, and score matching jointly learns the factors and an implicit tensor representation from the available observations. Scalar and vector-valued outputs accommodate individual entries and entire tubes, enabling a common formulation for tabular tensors, images, and videos. The framework separates representation learning from task-specific observation modeling: reverse diffusion recovers missing values; estimation via Tweedie's formula, soft thresholding, and optional spatial consistency support mixed-noise denoising; and measurement-guided sampling transfers a prior learned from low-resolution hyperspectral observations to high-resolution reconstruction, followed by post-sampling refinement with data consistency and spectral subspace regularization. This separation makes the representation compatible with different corruption models and permits task-specific constraints to be incorporated within the same recovery framework. Experiments across multiple tensor modalities and recovery tasks demonstrate competitive reconstruction quality and the flexibility of a unified conditional generative representation. The code is available at https://github.com/CZY-Code/DiffTR

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