shallow packing lemma and its applications in ...sri/caalm2019/slides/arijitghosh-caalm19.pdfshallow...
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Shallow Packing Lemma and its Applications inCombinatorial Geometry
Arijit Ghosh1
1Indian Statistical InstituteKolkata, India
Ghosh CAALM Workshop, 2019
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Coauthors
Kunal Dutta (INRIA, DataShape)
Esther Ezra (Georgia Tech, Math Dep.)
Bruno Jartoux (Ben-Gurion Univ., CS Dep.)
Nabil H. Mustafa (Universite Paris-Est, LIGM)
Ghosh CAALM Workshop, 2019
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Talk will be based on the following papers
Shallow packings, semialgebraic set systems, Macbeathregions and polynomial partitioning,with Bruno Jartoux, Kunal Dutta and Nabil Hassan Mustafa.Discrete & Computational Geometry, to appear.
A Simple Proof of Optimal Epsilon Nets,with Kunal Dutta and Nabil Hassan Mustafa.Combinatorica, 38(5): 1269 – 1277, 2018.
Two proofs for Shallow Packings,with Kunal Dutta and Esther Ezra.Discrete & Computational Geometry, 56(4): 910-939, 2016.
Ghosh CAALM Workshop, 2019
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Situation map: three combinatorial structures
� Classical
� Recent
� New
Geometricset systems
Shallowpackings
Upperbound
Tightlowerbound
MnetsUpperbound
Lowerbound
ε-netsAll knownresults
Ghosh CAALM Workshop, 2019
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Situation map: three combinatorial structures
� Classical
� Recent
� New
Geometricset systems
Shallowpackings
Upperbound
Tightlowerbound
MnetsUpperbound
Lowerbound
ε-netsAll knownresults
Ghosh CAALM Workshop, 2019
![Page 6: Shallow Packing Lemma and its Applications in ...sri/CAALM2019/Slides/ArijitGhosh-caalm19.pdfShallow Packing Lemma and its Applications in Combinatorial Geometry Arijit Ghosh1 1Indian](https://reader033.vdocuments.mx/reader033/viewer/2022042021/5e789ee3f921ca443c2c8a07/html5/thumbnails/6.jpg)
Geometric set systems
Point-disk incidences: an example of geometric set system
Ghosh CAALM Workshop, 2019
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Geometric set systems
Point-disk incidences: an example of geometric set system
Ghosh CAALM Workshop, 2019
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Geometric set systems
Typical applications: range searching, point set queries.
Ghosh CAALM Workshop, 2019
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Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh CAALM Workshop, 2019
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Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh CAALM Workshop, 2019
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Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh CAALM Workshop, 2019
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Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε contains one of them.
K
h
≥ ε
Ghosh CAALM Workshop, 2019
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Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for halfplanes
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any halfplane h with|h ∩ K | ≥ εn includes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh CAALM Workshop, 2019
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Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for halfplanes
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any halfplane h with|h ∩ K | ≥ εn includes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh CAALM Workshop, 2019
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Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for disks
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any disk h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh CAALM Workshop, 2019
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Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for [shapes]
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any [shape] h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh CAALM Workshop, 2019
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Mnets, or combinatorial Macbeath regions
Macbeath decomposition (Macbeath 1952)
For any convex body K with unit volume and ε > 0, there is asmall collection of convex subsets of K with volume Θ(ε) suchthat any halfplane h with vol(h ∩ K ) ≥ ε includes one of them.
Mnets – for [shapes]
For a set K of n points and ε > 0, an Mnet is a collection ofsubsets of Θ(εn) points such that any [shape] h with |h ∩ K | ≥ εnincludes one of them.
Goal: discrete analogue of Macbeath’s tool.
Ghosh CAALM Workshop, 2019
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Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh CAALM Workshop, 2019
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Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh CAALM Workshop, 2019
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Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh CAALM Workshop, 2019
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Bounds on Mnets
Question
What is the minimum size of an Mnet?
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
X Disks
X Rectangles
X Lines
X ‘Fat’ objects
× General convex sets
Theorem (D.–G.–J.–M. ’17)
This is tight for hyperplanes.
Ghosh CAALM Workshop, 2019
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Abstract set systems
X := arbitrary n-point setΣ := collection of subsets of X , i.e., Σ ⊆ 2X
The pair (X ,Σ) is called a set system
Set systems (X ,Σ) are also referred to as hypergraphs,range spaces
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Abstract set systems
Projection:For Y ⊆ X ,
ΣY := {S ∩ Y : S ∈ Σ}
andΣkY := {S ∩ Y : S ∈ Σ and |S ∩ Y | ≤ k}
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VC dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Sauer-Shelah Lemma: VC dim. d0 implies for all m ≤ n,πΣ(m) ≤ O(md0).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh CAALM Workshop, 2019
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VC dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}
Sauer-Shelah Lemma: VC dim. d0 implies for all m ≤ n,πΣ(m) ≤ O(md0).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh CAALM Workshop, 2019
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VC dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Sauer-Shelah Lemma: VC dim. d0 implies for all m ≤ n,πΣ(m) ≤ O(md0).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh CAALM Workshop, 2019
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VC dimension and shallow cell complexity
Primal Shatter function Given (X ,Σ), primal shatter function isdefined as
πΣ(m) := maxY⊆X , |Y |=m
|ΣY |
VC dimension: d0 := max {m |πΣ(m) = 2m}Sauer-Shelah Lemma: VC dim. d0 implies for all m ≤ n,πΣ(m) ≤ O(md0).
Shallow cell complexity ϕ(·, ·) If ∀Y ⊆ X ,∣∣∣ΣkY
∣∣∣ ≤ |Y | × ϕ(|Y |, k).
Ghosh CAALM Workshop, 2019
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Shallow cell complexity of some geometric set systems
1. Points and half-spaces O(|Y |bd/2c−1kdd/2e)or orthants in Rd
2. Points and balls O(|Y |b(d+1)/2c−1kd(d+1)/2e)in Rd
3. (d − 1)-variate polynomial |Y |d−2+εk1−ε
function of constant degreeand points in Rd
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Some geometric set systems
dim = 2
dim = d+ 1
dim = d+ 1dim = d+ 2
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
Theorem (Haussler-Welzl’87)
For a set system with VC-dimen d there exists an ε-net of sizeO(dε log d
ε
)
O(
1ε
)-size ε-nets are known for special set systems
Half-spaces in R2 and R3, pseudo-disks, homothetic copies ofconvex objects, α-fat wedges etc . . .(Matousek-Seidel-Welzl, Buzaglo-Pinchasi-Rote, Pyra-Ray, . . . )
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Epsilon-nets
Epsilon-nets: For a set system (X ,Σ), Y ⊆ X is an ε-net if
∀S ∈ Σ with |S | ≥ εn, Y ∩ S 6= ∅
Theorem (Haussler-Welzl’87)
For a set system with VC-dimen d there exists an ε-net of sizeO(dε log d
ε
)O(
1ε
)-size ε-nets are known for special set systems
Half-spaces in R2 and R3, pseudo-disks, homothetic copies ofconvex objects, α-fat wedges etc . . .(Matousek-Seidel-Welzl, Buzaglo-Pinchasi-Rote, Pyra-Ray, . . . )
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Optimal size Epsilon-nets
Theorem (Varadarajan’10, Aronov et al.’10, Chan et al.’12)
Let (X ,Σ) be a set system with constant VC-dimen and shallowcell complexity ϕ(·). Then there exists an ε-net of (X ,Σ) of size
O
(1
εlogϕ
(1
ε
)).
Remark: This result gives optimal size nets for all known geometricset systems. For example the above result implies 1
ε log log 1ε size
nets for points and rectangles in plane.
Ghosh CAALM Workshop, 2019
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Optimal size Epsilon-nets
Theorem (Varadarajan’10, Aronov et al.’10, Chan et al.’12)
Let (X ,Σ) be a set system with constant VC-dimen and shallowcell complexity ϕ(·). Then there exists an ε-net of (X ,Σ) of size
O
(1
εlogϕ
(1
ε
)).
Remark: This result gives optimal size nets for all known geometricset systems.
For example the above result implies 1ε log log 1
ε sizenets for points and rectangles in plane.
Ghosh CAALM Workshop, 2019
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Optimal size Epsilon-nets
Theorem (Varadarajan’10, Aronov et al.’10, Chan et al.’12)
Let (X ,Σ) be a set system with constant VC-dimen and shallowcell complexity ϕ(·). Then there exists an ε-net of (X ,Σ) of size
O
(1
εlogϕ
(1
ε
)).
Remark: This result gives optimal size nets for all known geometricset systems. For example the above result implies 1
ε log log 1ε size
nets for points and rectangles in plane.
Ghosh CAALM Workshop, 2019
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δ-packing number
Parameter: Let δ > 0 be a integer parameter
δ-separated: A set system (X ,Σ) is δ-separated if for all S1, S2
in Σ, if the size of the symmetric difference (Hamming distance)S1∆S2 is greater than δ, i.e. |S1∆S2| > δ.
δ-packing number: The cardinality of the largest δ-separatedsubcollection of Σ is called the δ-packing number of Σ.
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Connection to Euclidean packing
This is analogous to packing maximum number of Euclidean ballsof radius δ/2 in a box with edge length n.
In Rd , this packing number is O((
nδ
)d)We have the same bound for the case of set systems with VCdimension d . (due to Haussler, Chazelle and Wernisch)
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Connection to Euclidean packing
This is analogous to packing maximum number of Euclidean ballsof radius δ/2 in a box with edge length n.
In Rd , this packing number is O((
nδ
)d)
We have the same bound for the case of set systems with VCdimension d . (due to Haussler, Chazelle and Wernisch)
Ghosh CAALM Workshop, 2019
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Connection to Euclidean packing
This is analogous to packing maximum number of Euclidean ballsof radius δ/2 in a box with edge length n.
In Rd , this packing number is O((
nδ
)d)We have the same bound for the case of set systems with VCdimension d . (due to Haussler, Chazelle and Wernisch)
Ghosh CAALM Workshop, 2019
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Shallow packing result
Theorem (Dutta-Ezra-G.’15 and Mustafa’16)
Let (X ,Σ) be a set system with VC-dim d and shallow cellcomplexity ϕ(·) on a n-point set X . Let δ ≥ 1 and k ≤ n be twointeger parameters such that:
1. ∀S ∈∑
, |S | ≤ k , and
2.∑
is δ-packed.
Then
|Σ| ≤ dn
δϕ
(d n
δ,d k
δ
)We can show that the above bound is tight.
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Applications: Optimal nets
Theorem (Dutta-G.-Mustafa’17)
Let (X ,Σ) be a set system with VC-dim d and shallow cellcomplexity ϕ(·) on a n-point set X . Then there exists an ε-net ofsize
1
εlogϕ
(d
ε, d
)+
d
ε
Remark: The proof just uses the Shallow Packing Lemma and theAlteration Technique from The Probabilistic Method.
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Applications: Optimal nets
Theorem (Dutta-G.-Mustafa’17)
Let (X ,Σ) be a set system with VC-dim d and shallow cellcomplexity ϕ(·) on a n-point set X . Then there exists an ε-net ofsize
1
εlogϕ
(d
ε, d
)+
d
ε
Remark: The proof just uses the Shallow Packing Lemma and theAlteration Technique from The Probabilistic Method.
Ghosh CAALM Workshop, 2019
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Application: Mnets bound
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
Remark: The proof just uses Guth-Katz’s Polynomial PartitioningTheorem together with Shallow Packing Lemma.
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Application: Mnets bound
Theorem (D.–G.–J.–M. ’17)
Semialgebraic set systems with VC-dim. d <∞ and shallowcell complexity ϕ have an ε-Mnet of size
O
(d
ε· ϕ(d
ε, d
)).
Remark: The proof just uses Guth-Katz’s Polynomial PartitioningTheorem together with Shallow Packing Lemma.
Ghosh CAALM Workshop, 2019
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From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
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From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
Ghosh CAALM Workshop, 2019
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From Mnets to ε-nets
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
This gives ε-nets of size dε logϕ
(dε , d)for semialgebraic set
systems.
Yields best known bounds on ε-nets for geometric set systemswith bounded VC-dim.
Table: Upper bounds on Mnets and ε-nets
Mnet ε-net
Disks ε−1 ε−1
Rectangles 1ε log 1
ε1ε log log 1
ε
Halfspaces (Rd) O(ε−bd/2c) d
ε log 1ε
Ghosh CAALM Workshop, 2019
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Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
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Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh CAALM Workshop, 2019
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Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh CAALM Workshop, 2019
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Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh CAALM Workshop, 2019
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Probabilistic proof
Theorem (D.–G.–J.–M. ’17)
(ε-Mnet of size M
with sets of size ≥ τεn
)=⇒ ε-net of size
log(εM)/τ + 1
ε
Proof.
1 M is such an Mnet. Let p = 1τεn log(εM).
2 Pick every point into a sample S with probability p.
3 ∀m ∈M, Pr[S ∩m = ∅] = (1− p)|m| ≤ e−p|m| ≤ 1εM
4 In expectation, |S |+ |m ∈M : S ∩m = ∅| ≤ np + 1ε .
5 so there is an ε-net of size ≤ np + 1ε (why?).
Ghosh CAALM Workshop, 2019
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Proof of Shallow Packing
Theorem follows from the following result:
Theorem
Let (X ,R) be a δ-separated set system with VC dimension at mostd . Then
|R| ≤ 2E [|RA′ |]
where A′ is an uniformaly random subset of X of size 4dnδ − 1.
For the proof of the main result consider:
R′ :=
{σ ∈ R : |σ ∩ A′| > 3× 4dk
δ
}(1)
R′′ := R \R′ (2)
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Proof of Shallow Packing
Theorem follows from the following result:
Theorem
Let (X ,R) be a δ-separated set system with VC dimension at mostd . Then
|R| ≤ 2E [|RA′ |]
where A′ is an uniformaly random subset of X of size 4dnδ − 1.
For the proof of the main result consider:
R′ :=
{σ ∈ R : |σ ∩ A′| > 3× 4dk
δ
}(1)
R′′ := R \R′ (2)
Ghosh CAALM Workshop, 2019
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Conclusion
Ideally we want a combinatorial proof of the Mnets bound forset systems.
Improve the current lower bound.
Find more applications/connections of Mnets in combinatorialgeometry.
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Thank you.
Ghosh CAALM Workshop, 2019