On Vapnik‐Chervonenkis density over indiscernible sequences

Guingona, Vincent; Hill, Cameron Donnay

doi:10.1002/malq.201300040

Cited by 8 publications

(10 citation statements)

References 16 publications

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“…Suppose we have picked I * j for j < k, and define I * k be a sequence indexed by ω as in Proposition 3. 3 (Sub-additivity of dp-rank for dp-minimal types) Let tp(a i /A) be dp-minimal for 1 ≤ i ≤ k. Then the dp-rank of tp(a 1 . .…”

Section: Additivity Of the Dp-rankmentioning

confidence: 99%

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Additivity of the dp-rank

Kaplan¹,

Onshuus

Usvyatsov

2013

Trans. Amer. Math. Soc.

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The main result of this article is sub-additivity of the dp-rank. We also show that the study of theories of finite dp-rank can not be reduced to the study of its dp-minimal types, and discuss the possible relations between dp-rank and VC-density. introductionThis paper grew out of discussions that the authors had during a meeting in Oberwolfach in January 2010, following a talk of Deirdre Haskell, and conversations with Sergei Starchenko, on their recent joint work with Aschenbrenner, Dolich and Macpherson [2]. Haskell's talk made it apparent to us that the notion of VC-density (Vapnik-Chervonenkis density), investigated in [2], is closely related to "dependence rank" (dp-rank) introduced by the third author in [14]. Discussions with Starchenko helped us realize that certain questions, such as additivity, which were (and still are, to our knowledge) open for VC-density, may be approached more easily in the context of dp-rank. This paper is the first step in the program of investigating basic properties of dp-rank and its connections with VC-density.Whereas dp-rank is a relatively new notion, VC-density and related concepts have been studied for quite some time in the frameworks of machine learning, computational geometry, and other branches of theoretical computer science. Recent developments point to a connection between VC-density and dp-rank, strengthening the bridge between model theory and these subjects. We believe that investigating properties of dp-rank is important for discovering the nature of this connection. Furthermore, once this relation is better understood, theorems about dp-rank are likely to prove useful in the study of finite and infinite combinatorics related to VC-classes.Dp-rank was originally defined in [14] as an attempt to capture how far a certain type (or a theory) is from having the independence property. It also helped us to isolate a minimality notion of dependence for types and theories (that is, having rank 1). We called this notion dp-minimality and investigated it in [7]. Both dprank and dp-minimality were simplifications of Shelah's various ranks from [12],

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Section: Additivity Of the Dp-rankmentioning

confidence: 99%

“…We have recently learned that Vincent Guingona and Cameron Hill have investigated V C * -density (and other properties) over indiscernible sequences in much greater detail in [3].…”

Section: Now |S P(y)mentioning

confidence: 99%

Additivity of the dp-rank

Kaplan¹,

Onshuus

Usvyatsov

2013

Trans. Amer. Math. Soc.

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“…In the recent past there have been a number of papers relating various measures of the combinatorial structure of NIP theories to one another [1,5,7]. One fact which emerged from this is the close relation of dp-rank and VCdensity restricted to indiscernible sequences.…”

Section: Introductionmentioning

confidence: 99%

“…Guingona and Hill have used the term VC ind -density to describe VC-density restricted to indiscernibles. At the end of their paper [5], Guingona and Hill ask if there is a useful characterization of when a formula has VC ind -density equal to VC-density. We offer such a characterization below (Corollary 3.4), and use it to compute the exact VC ind -density of certain formulas.…”

Section: Introductionmentioning

confidence: 99%

Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property

Johnson¹

2015

Notre Dame J. Formal Logic

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This paper presents some finite combinatorics of set systems with applications to model theory, particularly the study of dependent theories. There are two main results. First, we give a way of producing lower bounds on VC ind -density, and use it to compute the exact VC inddensity of polynomial inequalities, and a variety of geometric set families. The main technical tool used is the notion of a maximum set system, which we juxtapose to indiscernibles. In the second part of the paper we give a maximum set system analogue to Shelah's characterization of stability using indiscernible sequences.1991 Mathematics Subject Classification. 03C45.

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“…Their desirable features include a certain kind of recursive structure which allows for, among other things, the existence of so-called sample compression schemes, and as such they are central to most approaches to proving the long outstanding sample compression conjecture [1,6,7,15,12]. Some further uses of maximum set systems exist in machine learning and model theory [13,8,9].…”

Section: Introductionmentioning

confidence: 99%

Some new maximum VC classes

Johnson

2014

Information Processing Letters

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Set systems of finite VC dimension are frequently used in applications relating to machine learning theory and statistics. Two simple types of VC classes which have been widely studied are the maximum classes (those which are extremal with respect to Sauer's lemma) and so-called Dudley classes, which arise as sets of positivity for linearly parameterized functions. These two types of VC class were related by Floyd, who gave sufficient conditions for when a Dudley class is maximum. It is widely known that Floyd's condition applies to positive Euclidean halfspaces and certain other classes, such as sets of positivity for univariate polynomials.In this paper we show that Floyd's lemma applies to a wider class of linearly parameterized functions than has been formally recognized to date. In particular we show that, modulo some minor technicalities, the sets of positivity for any linear combination of real analytic functions is maximum on points in general position. This includes sets of positivity for multivariate polynomials as a special case.

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On Vapnik‐Chervonenkis density over indiscernible sequences

Abstract: In this paper, we study Vapnik-Chervonenkis density (VC-density) over indiscernible sequences (denoted VC inddensity).We answer an open question in [1], showing that VC ind -density is always integer valued. We also show that VC ind -density and dp-rank coincide in the natural way.

Cited by 8 publications

References 16 publications

Additivity of the dp-rank

Additivity of the dp-rank

Vapnik–Chervonenkis Density on Indiscernible Sequences, Stability, and the Maximum Property

Some new maximum VC classes

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