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Homogeneous algorithms and solvable problems on cones

David Krieg, Peter Kritzer

TL;DR

The paper proves that homogeneous algorithms are essentially optimal for linear problems in the worst-case setting, up to a constant factor, by relating the recovery error to the diameter of information and constructing homogeneous approximate splines. This foundational result yields solvability guarantees for problems where inputs lie in cones, providing explicit cost bounds that combine the complexities of auxiliary problems and adaptive strategies. The authors illustrate the theory with concrete examples including bounded kurtosis, inverse Poincaré, and diagonal operators between Banach spaces, and develop practical cone constructions based on pilot samples, including L2-approximation in weighted Korobov spaces using standard information. The impact lies in unifying the treatment of cone-based problems under a robust framework that informs algorithm design and complexity analysis in continuous, high-dimensional settings.

Abstract

We consider linear problems in the worst case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the values of the operator uniformly on a convex and balanced set by means of algorithms that use at most $n$ such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show in this paper, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest to us for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario that is far less understood than the classical situation of balls. We use the optimality of homogeneous algorithms to prove solvability for a family of problems defined on cones. We illustrate our results by several examples.

Homogeneous algorithms and solvable problems on cones

TL;DR

The paper proves that homogeneous algorithms are essentially optimal for linear problems in the worst-case setting, up to a constant factor, by relating the recovery error to the diameter of information and constructing homogeneous approximate splines. This foundational result yields solvability guarantees for problems where inputs lie in cones, providing explicit cost bounds that combine the complexities of auxiliary problems and adaptive strategies. The authors illustrate the theory with concrete examples including bounded kurtosis, inverse Poincaré, and diagonal operators between Banach spaces, and develop practical cone constructions based on pilot samples, including L2-approximation in weighted Korobov spaces using standard information. The impact lies in unifying the treatment of cone-based problems under a robust framework that informs algorithm design and complexity analysis in continuous, high-dimensional settings.

Abstract

We consider linear problems in the worst case setting. That is, given a linear operator and a pool of admissible linear measurements, we want to approximate the values of the operator uniformly on a convex and balanced set by means of algorithms that use at most such measurements. It is known that, in general, linear algorithms do not yield an optimal approximation. However, as we show in this paper, an optimal approximation can always be obtained with a homogeneous algorithm. This is of interest to us for two reasons. First, the homogeneity allows us to extend any error bound on the unit ball to the full input space. Second, homogeneous algorithms are better suited to tackle problems on cones, a scenario that is far less understood than the classical situation of balls. We use the optimality of homogeneous algorithms to prove solvability for a family of problems defined on cones. We illustrate our results by several examples.
Paper Structure (14 sections, 9 theorems, 96 equations, 1 figure)

This paper contains 14 sections, 9 theorems, 96 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Linear Problems
  3. Homogeneous algorithms for linear problems
  4. Solvability of problems with inputs lying in a cone
  5. Structure of the paper and notation
  6. Homogeneous algorithms and linear problems
  7. Homogeneous algorithms in other settings
  8. Solvable problems and problems on cones
  9. Bounded kurtosis
  10. Inverse Poincaré
  11. Diagonal operators between Banach spaces
  12. Simple cones based on a pilot sample
  13. Simple cones based on a pilot sample: $L_2$-approximation in weighted Korobov spaces using standard information
  14. Outlook on future work: Multi-layer cones

Key Result

Theorem 1

Let $(S\colon{\mathcal{F}}\to\mathcal{G}, F,\Lambda)$ be a linear problem and let $\mathcal{G}$ be complete. For $n\in\mathbb{N}$, we let $\mathcal{A}_n^*$ denote the class of all homogeneous and non-adaptive algorithms of the form eq:alg_general, and, as above, let $\mathcal{A}_n$ denote the more g

Figures (1)

  • Figure 1: The situation described in Example \ref{['ex_bisection']}.

Theorems & Definitions (28)

  • Theorem 1
  • Remark 1
  • Remark 2
  • Remark 3
  • Remark 4
  • Theorem 2: Short version
  • Proposition 1
  • Remark 5
  • proof : Proof of Proposition \ref{['prop:main']}
  • Lemma 1: see GM80TW80
  • ...and 18 more