Xfredhd [99% BEST]
[ \textsim_X (x_i, x_j) \approx \textsim_Z (f(x_i), f(x_j)) ]
[ \big|\langle \tildex_i, \tildex_j\rangle - \langle x_i, x_j\rangle\big| \le \epsilon |x_i|,|x_j| ]
Theoretical guarantee: With high probability, for any two samples i , j : xfredhd
Resulting sketch (\tildeX) ∈ ℝ^N × S is , can be computed on‑the‑fly, and fits comfortably in GPU memory for S ≈ 10³–10⁴.
[ \mathcalL = \sum_k=1^3\lambda_k,\mathcalL \textrec^(k) + \lambda_g ,\mathcalL \textGPR ] [ \textsim_X (x_i, x_j) \approx \textsim_Z (f(x_i), f(x_j))
| 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 |
[ \mathcalL \textGPR = \frac1\sum (i,j)\in E\bigl(\textsim Z(z_i, z_j) - \textsim \tildeX(\tildex_i, \tildex_j)\bigr)^2 ] [ \textsim_X (x_i
The total loss:
