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No-dimensional results of combinatorial convexity. Dimension strikes back

Grigory Ivanov

Abstract

We discuss no-dimensional (approximate) versions of Carathéodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several questions. We also point out a simple way to ``bring the dimension back into the picture'': by combining no-dimensional statements with dimension-dependent norm comparisons, one can transfer problems in $\ell_1^d$, $\ell_\infty^d$, and Schatten classes $S_1, S_\infty$ to nearby $\ell_p^d$ or $S_p$ spaces with better geometry. As elementary applications, we obtain a weak additive analogue of the Johnson--Lindenstrauss flattening lemma, local-to-global estimates for Chebyshev regression over the $\ell_1$ ball, and a local-to-global guarantee for quantum feasibility from locally consistent linear measurements.

No-dimensional results of combinatorial convexity. Dimension strikes back

Abstract

We discuss no-dimensional (approximate) versions of Carathéodory's and Helly's theorems. Our goal is to draw attention to open problems and potential applications related to these results. We survey recent progress and pose several questions. We also point out a simple way to ``bring the dimension back into the picture'': by combining no-dimensional statements with dimension-dependent norm comparisons, one can transfer problems in , , and Schatten classes to nearby or spaces with better geometry. As elementary applications, we obtain a weak additive analogue of the Johnson--Lindenstrauss flattening lemma, local-to-global estimates for Chebyshev regression over the ball, and a local-to-global guarantee for quantum feasibility from locally consistent linear measurements.
Paper Structure (14 sections, 10 theorems, 54 equations)

This paper contains 14 sections, 10 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Applications of no-dimensional results
  3. Definitions and results from Banach space theory
  4. No-dimensional Carathéodory theorem
  5. No-dimensional Helly theorem
  6. A trick for bringing dimension back
  7. Dimension-dependent norm comparisons
  8. How the trick is used
  9. Dimension strikes back. Carathéodory-type results and problems
  10. Johnson--Lindenstrauss-type sketching
  11. Positive semidefinite matrix sparsification
  12. Dimension strikes back. Helly-type results and problems
  13. Chebyshev regression
  14. Quantum feasibility from local consistency

Key Result

Theorem 1.1

Fix $d\ge 1$ and $n \ge 8,$ and let $f_1,\dots,f_n\colon [0,1]^d\to[-1,1]$ be continuous. A sketching protocol consists of an encoding map and a decoding rule where $\mathop{\mathrm{Dec}}\nolimits(\mathop{\mathrm{Enc}}\nolimits(f_i),\mathop{\mathrm{Enc}}\nolimits(f_j))$ is interpreted as an estimate of the squared $L_2$-distance Then for every $\varepsilon\in(0,1)$ there exist $k\le C\,\frac{\l

Theorems & Definitions (20)

  • Theorem 1.1: Communication-efficient similarity search for signals
  • Theorem 1.2: Local-to-global Chebyshev regression
  • Theorem 1.3: Quantum feasibility from local consistency
  • Proposition 2.1
  • Proposition 2.2
  • Definition 3.1
  • Definition 3.2
  • Proposition 3.1
  • Lemma 3.1
  • Remark 3.3
  • ...and 10 more