Algorithmic solution of higher type equations

Escardó, Martín Hötzel

doi:10.1093/logcom/exr048

Cited by 5 publications

(5 citation statements)

References 14 publications

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“…Provided that the series converges, the series can be computed in general [17] or approximated [5], but this is beyond the scope of our paper. From the point of view of regression analysis this may seem surprising, but this is a known result using searchable sets [10].…”

Section: Polynomial Regressionmentioning

confidence: 98%

“…This paper has been inspired by and relies extensively on a significant body of work by Escard ó, starting with searchable infinite sets [7]. The properties of regression established here can be equally formulated in that setting, or in related setting such as compact sets [10] and compact types [11]. What makes our approach distinct is that in the formulations above are not synthetic, in the sense of [6].…”

Section: Related Workmentioning

confidence: 99%

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A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

Ghica,

Ambridge

2020

Preprint

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We examine the connections between deterministic, complete, and general global optimisation of continuous functions and a general concept of regression from the perspective of constructive type theory via the concept of 'searchability'. We see how the property of convergence of global optimisation is a straightforward consequence of searchability. The abstract setting allows us to generalise searchability and continuity to higher-order functions, so that we can formulate novel convergence criteria for regression, derived from the convergence of global optimisation. All the theory and the motivating examples are fully formalised in the proof assistant AGDA. General:Algorithms may take into account information about the shape of the function. For example, the minimisation of functions with convex envelopes is intensively studied [27].In contrast, we will make minimal assumptions of this nature.Complete: An incomplete algorithm makes no guarantees regarding the quality of the solution it arrives at, focussing on efficiency via sophisticated heuristics, rather than correctness. The typical example of an incomplete algorithm is gradient descent, which will only find a local minimum of a function [23]. In contrast, we provide mathematical guarantees a solution is indeed optimal within some margin of error.Deterministic: A randomized algorithm can offer an asymptotic guarantee that the optimum is reached, with probability one, without actually knowing when it has been reached [26]. In contrast, we will give strong termination guarantees for the algorithm. Continuous:Many global optimisation algorithms deal with discrete problems, such as branchand-bound [14]. In contrast, we will focus on the minimisation of continuous functions.To summarise, in this paper we will concentrate on general, complete, continuous, deterministic global search, which finds one guaranteed optimal-within-epsilon global minimum of a continuous function [18]. In the sequel, by 'optimisation' this is what we mean, precisely.The first important results in the area relevant to our work appeared in the 1960s and 70s, the optimisation of rational functions using interval arithmetic by Moore and Young [16], which was 1

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Section: Polynomial Regressionmentioning

confidence: 98%

Section: Related Workmentioning

confidence: 99%

A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

Ghica,

Ambridge

2020

Preprint

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“…Indeed several notions of computability, equivalent in the Euclidean case [BrWe99], have been shown distinct for separable metric spaces [BrPr03]. In fact, compactness is well-known crucial in Pure as well as in Computable Analysis [Schr04], [Esca13], [Stei16].…”

Section: B Overview Previous and Related Workmentioning

confidence: 99%

“…the supremum norm. We are guided by the structural properties exhibited in Computational Logic [KrWe87], [Esca13] and by the well-established Euclidean case [Ko91], [BrWe99], [Weih00]:…”

Section: Introduction Motivation Backgroundmentioning

confidence: 99%

Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization

Park,

Park

et al. 2017

Preprint

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“…implicit, functional) approach to Computable Analysis developed in [18]. It follows the following general strategy [7] (also explained in [18, Section 9]) for proving computability of some object Ω living in an admissibly represented space by three steps: I) Obtain a definition of Ω as the element of a computably closed set. II) Obtain a computably compact set containing Ω. III) Find a classical proof that (I) and (II) uniquely determine Ω.…”

mentioning

confidence: 99%

Computing Haar Measures

Pauly¹,

Seon²,

Ziegler³

2019

Preprint

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According to Haar's Theorem, every compact group G admits a unique left-invariant Borel probability measure µG. Let the Haar integral (of G) denote the functional G : C(G) f → f dµG integrating any continuous function f : G → R with respect to µG. This generalizes, and recovers for the additive group G = [0; 1) mod 1, the usual Riemann integral: computable (cmp. Weihrauch 2000, Theorem 6.4.1), and of computational cost characterizing complexity class #P1 (cmp. Ko 1991, Theorem 5.32).We establish that in fact every computably compact computable metric group renders the Haar integral computable: once using an elegant synthetic argument; and once presenting and analyzing an explicit, imperative algorithm. Regarding computational complexity, for the groups SO(3) and SU(2) we reduce the Haar integral to and from Euclidean/Riemann integration. In particular both also characterize #P1. ACM Subject ClassificationTheory of computation → Constructive mathematics; Mathematics of computing → Continuous mathematics Keywords and phrases computable analysis, topological groups, exact real arithmetic, Haar measure Digital Object Identifier 10.4230/LIPIcs.CSL.2020.00 European Union's Horizon 2020 MSCA IRSES project 731143. Motivation and OverviewComplementing empirical approaches, heuristics, and recipes [23,25], Computable Analysis[30] provides a rigorous algorithmic foundation to Numerics, as well as a way of formally measuring the constructive contents of theorems in classical Calculus. Haar's Theorem is such an example, of particular beauty combining three categories: compact metric spaces, algebraic groups, and measure spaces:Fact 1. Let (G, e, •, • −1 ) denote a group and (G, d) a compact metric space such that the group operation • and inverse operation • −1 are continuous with respect to d (that is, a topological group). There exists a unique left-invariant Borel probability measure µ G , called Haar measure, on G.We refrain from expanding on generalizations to locally-compact Hausdorff spaces. Recall that a left-invariant measure satisfies µ(U ) = µ(g • U ) for every g ∈ G and every measurable

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Algorithmic solution of higher type equations

Cited by 5 publications

References 14 publications

A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

A Constructive, Type-Theoretic Approach to Regression via Global Optimisation

Computable Operations on Compact Subsets of Metric Spaces with Applications to Fréchet Distance and Shape Optimization

Computing Haar Measures

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