The total loss:
Resulting sketch (\tildeX) ∈ ℝ^N × S is , can be computed on‑the‑fly, and fits comfortably in GPU memory for S ≈ 10³–10⁴.
a.sanchez@uv.es Abstract The exponential growth of data in scientific, industrial, and social domains has made the analysis of ultra‑high‑dimensional (UHD) datasets a pressing challenge. Conventional dimensionality‑reduction techniques (e.g., PCA, t‑SNE, UMAP) either suffer from prohibitive computational costs or fail to preserve intricate feature‑level relationships when the dimensionality exceeds 10⁶. We introduce XFREDHD (e X treme F eature‑Rich E mbedding for D ata in H igh D imensions), a scalable, end‑to‑end framework that couples a feature‑wise random projection with an adaptive hierarchical auto‑encoder and a graph‑preserving regularizer . XFREDHD reduces dimensionality by up to three orders of magnitude while maintaining > 95 % of pairwise cosine similarity and enabling downstream tasks (classification, clustering, anomaly detection) to achieve state‑of‑the‑art performance. Extensive experiments on synthetic benchmarks, genomics, hyperspectral imaging, and large‑scale recommender‑system logs demonstrate that XFREDHD outperforms existing baselines in both accuracy and runtime (up to 12× speed‑up on a 64‑GPU cluster). We release the open‑source implementation (Apache 2.0) and a curated suite of UHD datasets to foster reproducibility. 1. Introduction High‑dimensional data arise in numerous domains: xfredhd
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 The total loss: Resulting sketch (\tildeX) ∈ ℝ^N
[ \mathcalL \textGPR = \frac1E\sum (i,j)\in E\bigl(\textsim Z(z_i, z_j) - \textsim \tildeX(\tildex_i, \tildex_j)\bigr)^2 ]
[ \textsim_X (x_i, x_j) \approx \textsim_Z (f(x_i), f(x_j)) ] We introduce XFREDHD (e X treme F eature‑Rich
[ \big|\langle \tildex_i, \tildex_j\rangle - \langle x_i, x_j\rangle\big| \le \epsilon |x_i|,|x_j| ]