Technical Write-Up: A Lossless Bidirectional Tensor Matrix Embedding Framework with Hyperspherical Normalization and Complex Tensor Support.

Hi everyone,

I’ve been exploring tensor representations and recently developed a lossless, bidirectional tensor-to-matrix and matrix-to-tensor embedding framework that I wanted to share for educational discussion.

Unlike standard unfolding or reshaping, this method:

• Preserves full structural metadata so tensors of any order (3D, 4D, …) can be flattened to matrices and perfectly reconstructed.

• Works with real and complex tensors (error ~10^-16 , near machine precision).

• Supports hyperspherical normalization (projection onto a unit hypersphere) while remaining invertible.

• Explicitly defines bijective operators and such that:

This isn’t a decomposition method (like CP or Tucker), and it’s not just reshape; it’s a mathematically rigorous embedding that guarantees invertibility and precision across arbitrary orders.

Resources:

• Short technical write-up (math & proofs): Ayodele, F. (2025). A Lossless Bidirectional Tensor Matrix Embedding Framework with Hyperspherical Normalization and Complex Tensor Support. Zenodo. https://doi.org/10.5281/zenodo.16749356

• Reference implementation (open-source): fikayoAy/MatrixTransformer: MatrixTransfromer is a sophisticated mathematical utility class that enables transformation, manipulation, and analysis of matrices between different matrix types

Why I’m sharing:

I’m interested in:

• Feedback on the mathematical formulation.

• Ideas for ML or HPC use cases (e.g., working with high-order data in lower-dimensional computational forms).

• Discussion around how such embeddings could integrate into workflows (like preprocessing for deep learning or symbolic methods).

If you’re curious how it works (or skeptical), happy to clarify details (e.g., how it differs from reshape or unfolding) in comments.

Would you find this kind of approach useful for data preprocessing or tensor-based ML workflows, or is this more niche to HPC/math-heavy applications?