scholarly journals Shattering-Extremal Set Systems of VC Dimension at most 2

10.37236/4548 ◽  
2014 ◽  
Vol 21 (4) ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai
Keyword(s):  
Vc Dimension ◽  
Set Systems ◽  
Dimension 2 ◽  
Extremal Set Systems ◽  
Extremal Set ◽  
Open Question

We say that a set system $\mathcal{F}\subseteq 2^{[n]}$ shatters a given set $S\subseteq [n]$ if $2^S=\{F~\cap~S ~:~F~\in~\mathcal{F}\}$. The Sauer inequality states that in general, a set system $\mathcal{F}$ shatters at least $|\mathcal{F}|$ sets. Here we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly $|\mathcal{F}|$ sets. In this paper we characterize shattering-extremal set systems of Vapnik-Chervonenkis dimension $2$ in terms of their inclusion graphs, and as a corollary we answer an open question about leaving out elements from shattering-extremal set systems in the case of families of Vapnik-Chervonenkis dimension $2$.

scholarly journals Shattering-Extremal Set Systems of Small VC-Dimension

2013 ◽  
Vol 2013 ◽  
pp. 1-8 ◽  
Author(s):  
Tamás Mészáros ◽  
Lajos Rónyai
Keyword(s):  
Vc Dimension ◽  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

We say that a set system ℱ⊆2[n] shatters a given set S⊆[n] if 2S={F∩S:F∈ℱ}. The Sauer inequality states that in general, a set system ℱ shatters at least |ℱ| sets. Here, we concentrate on the case of equality. A set system is called shattering-extremal if it shatters exactly |ℱ| sets. We characterize shattering extremal set systems of Vapnik-Chervonenkis dimension 1 in terms of their inclusion graphs. Also, from the perspective of extremality, we relate set systems of bounded Vapnik-Chervonenkis dimension to their projections.

scholarly journals Shattering-extremal set systems from Sperner families

2020 ◽  
Vol 276 ◽  
pp. 92-101
Author(s):  
Christopher Kusch ◽  
Tamás Mészáros
Keyword(s):  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

scholarly journals Two-Dimensional Partial Cubes

10.37236/8934 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Victor Chepoi ◽  
Kolja Knauer ◽  
Manon Philibert
Keyword(s):  
Cell Structure ◽  
Low Rank ◽  
Complete Graphs ◽  
Two Dimensional ◽  
Vc Dimension ◽  
Set Systems ◽  
Sample Compression ◽  
Vertex Set ◽  
Dimension 2 ◽  
Pseudoline Arrangements

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

scholarly journals Extremal set systems with restricted k-wise intersections

2004 ◽  
Vol 105 (1) ◽  
pp. 143-159 ◽  
Author(s):  
Zoltán Füredi ◽  
Benny Sudakov
Keyword(s):  
Set Systems ◽  
Extremal Set Systems ◽  
Extremal Set

scholarly journals VC Dimension and a Union Theorem for Set Systems

10.37236/8288 ◽  
2019 ◽  
Vol 26 (3) ◽  
Author(s):  
Stijn Cambie ◽  
António Girão ◽  
Ross J. Kang
Keyword(s):  
Difference Operator ◽  
Negative Answer ◽  
Symmetric Difference ◽  
Vc Dimension ◽  
Compression Method ◽  
Set Systems ◽  
Theoretic Problem ◽  
Extremal Set ◽  
Positive Integers

Fix positive integers $k$ and $d$. We show that, as $n\to\infty$, any set system $\mathcal{A} \subset 2^{[n]}$ for which the VC dimension of $\{ \triangle_{i=1}^k S_i \mid S_i \in \mathcal{A}\}$ is at most $d$ has size at most $(2^{d\bmod{k}}+o(1))\binom{n}{\lfloor d/k\rfloor}$. Here $\triangle$ denotes the symmetric difference operator. This is a $k$-fold generalisation of a result of Dvir and Moran, and it settles one of their questions.  A key insight is that, by a compression method, the problem is equivalent to an extremal set theoretic problem on $k$-wise intersection or union that was originally due to Erdős and Frankl. We also give an example of a family $\mathcal{A} \subset 2^{[n]}$ such that the VC dimension of $\mathcal{A}\cap \mathcal{A}$ and of $\mathcal{A}\cup \mathcal{A}$ are both at most $d$, while $\lvert \mathcal{A} \rvert = \Omega(n^d)$. This provides a negative answer to another question of Dvir and Moran.

scholarly journals The VC Dimension of Metric Balls under Fréchet and Hausdorff Distances

2021 ◽  
Author(s):  
Anne Driemel ◽  
André Nusser ◽  
Jeff M. Phillips ◽  
Ioannis Psarros
Keyword(s):  
Density Estimation ◽  
Lower Bounds ◽  
Upper And Lower Bounds ◽  
Similarity Metrics ◽  
Vc Dimension ◽  
Set Systems ◽  
Polygonal Curves ◽  
The Individual ◽  
Curve Similarity ◽  
Metric Balls

AbstractThe Vapnik–Chervonenkis dimension provides a notion of complexity for systems of sets. If the VC dimension is small, then knowing this can drastically simplify fundamental computational tasks such as classification, range counting, and density estimation through the use of sampling bounds. We analyze set systems where the ground set X is a set of polygonal curves in $$\mathbb {R}^d$$ R d and the sets $$\mathcal {R}$$ R are metric balls defined by curve similarity metrics, such as the Fréchet distance and the Hausdorff distance, as well as their discrete counterparts. We derive upper and lower bounds on the VC dimension that imply useful sampling bounds in the setting that the number of curves is large, but the complexity of the individual curves is small. Our upper and lower bounds are either near-quadratic or near-linear in the complexity of the curves that define the ranges and they are logarithmic in the complexity of the curves that define the ground set.

scholarly journals Shattering, Graph Orientations, and Connectivity

10.37236/3326 ◽  
2013 ◽  
Vol 20 (3) ◽  
Author(s):  
László Kozma ◽  
Shay Moran
Keyword(s):  
Graph Theory ◽  
Combinatorial Optimization ◽  
Main Tool ◽  
The Other ◽  
Vc Dimension ◽  
Forbidden Subgraphs ◽  
Set Systems ◽  
Flows In Networks ◽  
Extremal Systems

We present a connection between two seemingly disparate fields: VC-theory and graph theory. This connection yields natural correspondences between fundamental concepts in VC-theory, such as shattering and VC-dimension, and well-studied concepts of graph theory related to connectivity, combinatorial optimization, forbidden subgraphs, and others.In one direction, we use this connection to derive results in graph theory. Our main tool is a generalization of the Sauer-Shelah Lemma (Pajor, 1985; Bollobás and Radcliffe, 1995; Dress, 1997; Holzman and Aharoni). Using this tool we obtain a series of inequalities and equalities related to properties of orientations of a graph. Some of these results appear to be new, for others we give new and simple proofs.In the other direction, we present new illustrative examples of shattering-extremal systems - a class of set-systems in VC-theory whose understanding is considered by some authors to be incomplete Bollobás and Radcliffe, 1995; Greco, 1998; Rónyai and Mészáros, 2011). These examples are derived from properties of orientations related to distances and flows in networks.

scholarly journals The VC-dimension of set systems defined by graphs

1997 ◽  
Vol 77 (3) ◽  
pp. 237-257 ◽  
Author(s):  
Evangelos Kranakis ◽  
Danny Krizanc ◽  
Berthold Ruf ◽  
Jorge Urrutia ◽  
Gerhard Woeginger
Keyword(s):  
Vc Dimension ◽  
Set Systems

Mars: was There an Ancient Eden?

1997 ◽  
Vol 161 ◽  
pp. 203-218 ◽  
Author(s):  
Tobias C. Owen
Keyword(s):  
Positive Answer ◽  
Biogenic Elements ◽  
Habitable Zones ◽  
The Face ◽  
The Origin Of Life ◽  
Early Mars ◽  
Favorable Conditions ◽  
Open Question ◽  
Rocky Planets ◽  
Impact Erosion

AbstractThe clear evidence of water erosion on the surface of Mars suggests an early climate much more clement than the present one. Using a model for the origin of inner planet atmospheres by icy planetesimal impact, it is possible to reconstruct the original volatile inventory on Mars, starting from the thin atmosphere we observe today. Evidence for cometary impact can be found in the present abundances and isotope ratios of gases in the atmosphere and in SNC meteorites. If we invoke impact erosion to account for the present excess of129Xe, we predict an early inventory equivalent to at least 7.5 bars of CO2. This reservoir of volatiles is adequate to produce a substantial greenhouse effect, provided there is some small addition of SO2(volcanoes) or reduced gases (cometary impact). Thus it seems likely that conditions on early Mars were suitable for the origin of life – biogenic elements and liquid water were present at favorable conditions of pressure and temperature. Whether life began on Mars remains an open question, receiving hints of a positive answer from recent work on one of the Martian meteorites. The implications for habitable zones around other stars include the need to have rocky planets with sufficient mass to preserve atmospheres in the face of intensive early bombardment.

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