Skip to content
DraconiticResearch
Search Sources API
197,761 studies31 sourcesUpdated 36 min ago
PreprintarXiv licencearXiv

Functional Projection and Complexity Reduction

  • Vera Fischer
  • Yurii Khomskii
  • Yipu Li
  • Julia Millhouse
Open at sourcePDFDOICite

Abstract

Törnquist (arXiv:1310.3250) proved that if there exists a Σ12 maximal almost disjoint (MAD) family, then there exists a Π11 MAD family. Since then, several other "complexity reduction" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on Π11-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call "Functional Projection", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in L[U], the canonical inner model for a measurable cardinal, Functional Projection holds for Π12 relations. As an immediate corollary, we obtain that in L[U] there are Π12 witnesses for many combinatorial families.

Related

  1. Variable-Projection Sparse Functional Principal Component Analysis: Interpretable Functional Dimensionality Reduction with Applications to Raman Spectral Data
    Han Ying Lim, Dharini Pathmanathan, Philipp Otto +1
    arXivPreprintarXiv licenceFull textStatisticsarXiv
  2. Oblique Projection Networks (OPN) Reference guide - variants, code, tests, complexity, and the application scope of oblique projections
    Olivier Caspary
    HALPreprintCC BY 4.0Full textGeneral mathematicsHAL
  3. Curvature, Scaling, and Complexity in Riemannian Projection-Free Optimization
    Binh T. Nguyen
    arXivPreprintCC BY-NC-ND 4.0Full textGeometry & topologyOptimisation & controlarXiv
  4. Risk Reduction Through Reverse Engineering of a Comparison Functional
    Michael Todinov
    MathematicsPeer-reviewedCC BY 4.0General mathematicsCrossref
  5. Massive parallelization of projection-based depths
    Leonardo Leone, Pavlo Mozharovskyi, David Bounie
    Computational Statistics and Data AnalysisPeer-reviewedLicence not statedFull textStatisticsHAL

Cite

BibTeXRIS
BibTeX
@misc{fischer2026functional,
  title = {{Functional Projection and Complexity Reduction}},
  author = {Vera Fischer and Yurii Khomskii and Yipu Li and Julia Millhouse},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.17455},
  eprint = {2610.17455},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.17455},
}
RIS
TY  - UNPB
TI  - Functional Projection and Complexity Reduction
AU  - Vera Fischer
AU  - Yurii Khomskii
AU  - Yipu Li
AU  - Julia Millhouse
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.17455
UR  - https://arxiv.org/abs/2610.17455
AB  - Törnquist (arXiv:1310.3250) proved that if there exists a Σ^1_2 maximal almost disjoint (MAD) family, then there exists a Π^1_1 MAD family. Since then, several other "complexity reduction" proofs have been found for other combinatorial objects, such as maximal independent families or ultrafilters. All of these proofs have relied on Π^1_1-uniformization, so they did not easily lift to higher projective levels. In this paper, we introduce a principle weaker than Uniformization which we call "Functional Projection", and show that this principle is sufficient for most complexity reduction proofs. On the other hand, we prove that in L[U], the canonical inner model for a measurable cardinal, Functional Projection holds for Π^1_2 relations. As an immediate corollary, we obtain that in L[U] there are Π^1_2 witnesses for many combinatorial families.
ER  -