((hot)) | Xfredhd

[ \mathcalL \textGPR = \frac1\sum (i,j)\in E\bigl(\textsim Z(z_i, z_j) - \textsim \tildeX(\tildex_i, \tildex_j)\bigr)^2 ]

[ \big|\langle \tildex_i, \tildex_j\rangle - \langle x_i, x_j\rangle\big| \le \epsilon |x_i|,|x_j| ]

Resulting sketch (\tildeX) ∈ ℝ^N × S is , can be computed on‑the‑fly, and fits comfortably in GPU memory for S ≈ 10³–10⁴.

| Domain | Typical Dimensionality | Example | |----------------------------|------------------------|-----------------------------------------| | Genomics & Transcriptomics | 10⁶ – 10⁸ | Single‑cell RNA‑seq expression matrices | | Remote Sensing | 10⁴ – 10⁶ | Hyperspectral cubes (hundreds of bands) | | Recommender Systems | 10⁶ – 10⁹ | User–item interaction tensors | | Natural Language Processing| 10⁵ – 10⁷ | Contextualized token embeddings |

¹ Department of Computer Science, University of Valencia, Spain ² Department of Electrical Engineering, Indian Institute of Technology Delhi, India ³ Data Science Lab, Stanford University, USA

Theoretical guarantee: With high probability, for any two samples i , j :

XFREDHD: A Novel Framework for Extreme‑Scale Feature‑Rich Embedding and Dimensionality Reduction in High‑Dimensional Data Authors: Dr. A. M. Sanchez¹, Prof. L. K. Rao², Dr. J. H. Miller³