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PreprintCC BY 4.0arXiv

Fitting Gaussian vectors in normed spaces

  • Guillaume Aubrun
  • Emmanouil Stavrakakis
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Abstract

Given a family of N i.i.d. Gaussian points in a d-dimensional normed space, what is the probability that some linear transformation maps it to the unit sphere? The case of a Euclidean space is known as the ellipsoid fitting problem and has recently attracted major attention. We initiate a study of this fitting problem in a general normed space, both asymptotically and in a fixed dimension. We specifically focus on the space ℓ∞d, where a combinatorial reformulation allows us to show that cd2/log d points can be fitted with high probability. We illustrate that the geometry of the normed space is reflected in the fitting problem by showing that in dimension $2$, the spaces ℓ22 and ℓ∞2 are characterized by having extremal fittability properties.

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Cite

BibTeXRIS
BibTeX
@misc{aubrun2026fitting,
  title = {{Fitting Gaussian vectors in normed spaces}},
  author = {Guillaume Aubrun and Emmanouil Stavrakakis},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.20299},
  eprint = {2610.20299},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.20299},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - Fitting Gaussian vectors in normed spaces
AU  - Guillaume Aubrun
AU  - Emmanouil Stavrakakis
PY  - 2026
DA  - 2026/10/14
PB  - arXiv
DO  - 10.48550/arxiv.2610.20299
UR  - https://arxiv.org/abs/2610.20299
AB  - Given a family of N i.i.d. Gaussian points in a d-dimensional normed space, what is the probability that some linear transformation maps it to the unit sphere? The case of a Euclidean space is known as the ellipsoid fitting problem and has recently attracted major attention. We initiate a study of this fitting problem in a general normed space, both asymptotically and in a fixed dimension. We specifically focus on the space ℓ_∞^d, where a combinatorial reformulation allows us to show that cd^2/log d points can be fitted with high probability. We illustrate that the geometry of the normed space is reflected in the fitting problem by showing that in dimension $2$, the spaces ℓ_2^2 and ℓ_∞^2 are characterized by having extremal fittability properties.
ER  -