[PPT] - Families of Mass Destruction Shagnik Das 1 aros 1 Tam as M esz 1 PowerPoint Presentation

SLIDE 1

Families of Mass Destruction

Shagnik Das1 Tam´ as M´ esz´ aros1

1Freie Universit¨

at Berlin

Symposium Diskrete Mathematik, Berlin 15th July 2016

SLIDE 2

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 4, 5} is shattered.

S. Das

FU Berlin

SLIDE 3

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 4, 5} is shattered.

S. Das

FU Berlin

SLIDE 4

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 4, 5} is shattered.

S. Das

FU Berlin

SLIDE 5

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 4, 5} is shattered.

S. Das

FU Berlin

SLIDE 6

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 2, 3} is not shattered.

S. Das

FU Berlin

SLIDE 7

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering sets

Definition (Shattering)

A family F shatters a set A if its members intersect A in every possible way: {F ∩ A : F ∈ F} = 2A. Given a family F, Sh(F) denotes the collection of sets it shatters.

Definition (Families)

Ground set: [n] = {1, 2, . . . , n}. Family F a collection of subsets: F ⊆ 2[n]. k-uniform if all members have size k: F ⊆ [n]

k

.

A (small) example

Let F be the family {{1}, {1, 2}, {2, 3}, {2, 4}, {1, 2, 5}, {1, 3, 4}, {2, 4, 5}, {3, 4, 5}, {1, 3, 4, 5}}. Then {1, 2, 3} is not shattered.

S. Das

FU Berlin

SLIDE 8

Introducing the problem Some initial bounds An exact solution Concluding remarks

VC dimension and the Sauer–Shelah Lemma

Definition (VC dimension)

The VC dimension of a family F, denoted dimVC(F), is the size of the largest set it shatters: dimVC(F) = max{|A| : A ∈ Sh(F)}.

Lemma (Sauer, Shelah (1972))

For any set family F ⊆ 2[n], |Sh(F)| ≥ |F| , and the bound is best possible.

Corollary

If dimVC(F) < k, then |F| ≤

k−1

i=0

n i

.

Many applications: computational geometry, machine learning, ...

S. Das

FU Berlin

SLIDE 9

Introducing the problem Some initial bounds An exact solution Concluding remarks

VC dimension and the Sauer–Shelah Lemma

Definition (VC dimension)

The VC dimension of a family F, denoted dimVC(F), is the size of the largest set it shatters: dimVC(F) = max{|A| : A ∈ Sh(F)}.

Lemma (Sauer, Shelah (1972))

For any set family F ⊆ 2[n], |Sh(F)| ≥ |F| , and the bound is best possible.

Corollary

If dimVC(F) < k, then |F| ≤

k−1

i=0

n i

.

Many applications: computational geometry, machine learning, ...

S. Das

FU Berlin

SLIDE 10

Introducing the problem Some initial bounds An exact solution Concluding remarks

VC dimension and the Sauer–Shelah Lemma

Definition (VC dimension)

The VC dimension of a family F, denoted dimVC(F), is the size of the largest set it shatters: dimVC(F) = max{|A| : A ∈ Sh(F)}.

Lemma (Sauer, Shelah (1972))

For any set family F ⊆ 2[n], |Sh(F)| ≥ |F| , and the bound is best possible.

Corollary

If dimVC(F) < k, then |F| ≤

k−1

i=0

n i

.

Many applications: computational geometry, machine learning, ...

S. Das

FU Berlin

SLIDE 11

Introducing the problem Some initial bounds An exact solution Concluding remarks

VC dimension and the Sauer–Shelah Lemma

Definition (VC dimension)

The VC dimension of a family F, denoted dimVC(F), is the size of the largest set it shatters: dimVC(F) = max{|A| : A ∈ Sh(F)}.

Lemma (Sauer, Shelah (1972))

For any set family F ⊆ 2[n], |Sh(F)| ≥ |F| , and the bound is best possible.

Corollary

If dimVC(F) < k, then |F| ≤

k−1

i=0

n i

.

Many applications: computational geometry, machine learning, ...

S. Das

FU Berlin

SLIDE 12

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximising the number of shattered sets

Question

Which families of m sets maximise the number of shattered sets?

Why maximise?

1. Machines are

dangerously smart

2. An old problem with

a lot of history

3. Quite a bit of fun
S. Das

FU Berlin

SLIDE 13

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximising the number of shattered sets

Question

Which families of m sets maximise the number of shattered sets?

Why maximise?

1. Machines are

dangerously smart

2. An old problem with

a lot of history

3. Quite a bit of fun
S. Das

FU Berlin

SLIDE 14

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximising the number of shattered sets

Question

Which families of m sets maximise the number of shattered sets?

Why maximise?

1. Machines are

dangerously smart

2. An old problem with

a lot of history

3. Quite a bit of fun
S. Das

FU Berlin

SLIDE 15

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximising the number of shattered sets

Question

Which families of m sets maximise the number of shattered sets?

Why maximise?

1. Machines are

dangerously smart

2. An old problem with

a lot of history

3. Quite a bit of fun

A rose by any other name...

◮ Universal sets,

k-independent sets, covering arrays, k-faulty systems

◮ Used for software testing,

derandomisation

S. Das

FU Berlin

SLIDE 16

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximising the number of shattered sets

Question

Which families of m sets maximise the number of shattered sets?

Why maximise?

1. Machines are

dangerously smart

2. An old problem with

a lot of history

3. Quite a bit of fun

A rose by any other name...

◮ Universal sets,

k-independent sets, covering arrays, k-faulty systems

◮ Used for software testing,

derandomisation

S. Das

FU Berlin

SLIDE 17

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximally destructive animals

A bull in a china shop

S. Das

FU Berlin

SLIDE 18

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximally destructive animals

Ein Elefant im Porzellanladen

S. Das

FU Berlin

SLIDE 19

Introducing the problem Some initial bounds An exact solution Concluding remarks

Maximally destructive animals

Ein Elefant im Porzellanladen

S. Das

FU Berlin

SLIDE 20

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family

Definition (Random family)

Let Frand,m be the family {F1, F2, . . . , Fm}, where each Fi is a uniformly random set in 2[n], chosen independently.

What does Frand,m shatter?

◮ Suppose A ⊆ [n], |A| = k ◮ For each i, Fi ∩ A ∼ Unif

2A

◮ Coupon collector problem with 2k coupons ◮ P (A ∈ Sh(F)) →

1

if m = ω(k2k) if m = o(k2k)

S. Das

FU Berlin

SLIDE 21

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family

Definition (Random family)

Let Frand,m be the family {F1, F2, . . . , Fm}, where each Fi is a uniformly random set in 2[n], chosen independently.

What does Frand,m shatter?

◮ Suppose A ⊆ [n], |A| = k ◮ For each i, Fi ∩ A ∼ Unif

2A

◮ Coupon collector problem with 2k coupons ◮ P (A ∈ Sh(F)) →

1

if m = ω(k2k) if m = o(k2k)

S. Das

FU Berlin

SLIDE 22

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family

Definition (Random family)

Let Frand,m be the family {F1, F2, . . . , Fm}, where each Fi is a uniformly random set in 2[n], chosen independently.

What does Frand,m shatter?

◮ Suppose A ⊆ [n], |A| = k ◮ For each i, Fi ∩ A ∼ Unif

2A

◮ Coupon collector problem with 2k coupons ◮ P (A ∈ Sh(F)) →

1

if m = ω(k2k) if m = o(k2k)

S. Das

FU Berlin

SLIDE 23

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family

Definition (Random family)

Let Frand,m be the family {F1, F2, . . . , Fm}, where each Fi is a uniformly random set in 2[n], chosen independently.

What does Frand,m shatter?

◮ Suppose A ⊆ [n], |A| = k ◮ For each i, Fi ∩ A ∼ Unif

2A

◮ Coupon collector problem with 2k coupons ◮ P (A ∈ Sh(F)) →

1

if m = ω(k2k) if m = o(k2k)

S. Das

FU Berlin

SLIDE 24

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family — continued

Observation

◮ We need 2k sets to shatter a k-set ◮ Family of m sets shatters no set of size k > log m

What does Frand,m shatter?

Set size k Sets shattered? log m − log log m − ω(1) Almost all log m − log log m + ω(1) Very few > log m None

S. Das

FU Berlin

SLIDE 25

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family — continued

Observation

◮ We need 2k sets to shatter a k-set ◮ Family of m sets shatters no set of size k > log m

What does Frand,m shatter?

Set size k Sets shattered? log m − log log m − ω(1) Almost all log m − log log m + ω(1) Very few > log m None

S. Das

FU Berlin

SLIDE 26

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family — continued — continued

log m − log log m log m n

Could we do better?

◮ Almost all sets of size ≤ log m have size ⌊log m⌋ ◮ Frand,m shatters very few sets of this size

S. Das

FU Berlin

SLIDE 27

Introducing the problem Some initial bounds An exact solution Concluding remarks

A random family — continued — continued

log m − log log m log m n

Could we do better?

◮ Almost all sets of size ≤ log m have size ⌊log m⌋ ◮ Frand,m shatters very few sets of this size

S. Das

FU Berlin

SLIDE 28

Introducing the problem Some initial bounds An exact solution Concluding remarks

The uniform setting

Definition

Given 0 ≤ k ≤ n and 0 ≤ m ≤ 2n, let shmax(n, k, m) denote the maximum number of k-sets in [n] that can be shattered by a family of size m.

The smallest families

We need 2k sets to shatter a single k-set. Can we shatter many more? What is shmax(n, k, 2k)?

Universal families

How many sets are needed to shatter all k-sets? That is, what is m0(n, k) = min

m : shmax(n, k, m) =

n

k

?
S. Das

FU Berlin

SLIDE 29

Introducing the problem Some initial bounds An exact solution Concluding remarks

The uniform setting

Definition

Given 0 ≤ k ≤ n and 0 ≤ m ≤ 2n, let shmax(n, k, m) denote the maximum number of k-sets in [n] that can be shattered by a family of size m.

The smallest families

We need 2k sets to shatter a single k-set. Can we shatter many more? What is shmax(n, k, 2k)?

Universal families

How many sets are needed to shatter all k-sets? That is, what is m0(n, k) = min

m : shmax(n, k, m) =

n

k

?
S. Das

FU Berlin

SLIDE 30

Introducing the problem Some initial bounds An exact solution Concluding remarks

The uniform setting

Definition

Given 0 ≤ k ≤ n and 0 ≤ m ≤ 2n, let shmax(n, k, m) denote the maximum number of k-sets in [n] that can be shattered by a family of size m.

The smallest families

We need 2k sets to shatter a single k-set. Can we shatter many more? What is shmax(n, k, 2k)?

Universal families

How many sets are needed to shatter all k-sets? That is, what is m0(n, k) = min

m : shmax(n, k, m) =

n

k

?
S. Das

FU Berlin

SLIDE 31

Introducing the problem Some initial bounds An exact solution Concluding remarks

Universal families

Theorem (R´ enyi, 1971)

m0(n, 2) = min

m : n ≤

m−1

⌈m/2⌉

∼ log n + 1

2 log log n.

Theorem (Kleitman–Spencer, 1973)

Ω

2k log n
≤ m0(n, k) ≤ O
k2k log n
.

Theorem (Alon, 1986)

m0(n, k) ≤ 2O(k4) log n, via an explicit construction. Applications: experimental design, derandomisation

S. Das

FU Berlin

SLIDE 32

Introducing the problem Some initial bounds An exact solution Concluding remarks

Universal families

Theorem (R´ enyi, 1971)

m0(n, 2) = min

m : n ≤

m−1

⌈m/2⌉

∼ log n + 1

2 log log n.

Theorem (Kleitman–Spencer, 1973)

Ω

2k log n
≤ m0(n, k) ≤ O
k2k log n
.

Theorem (Alon, 1986)

m0(n, k) ≤ 2O(k4) log n, via an explicit construction. Applications: experimental design, derandomisation

S. Das

FU Berlin

SLIDE 33

Introducing the problem Some initial bounds An exact solution Concluding remarks

Universal families

Theorem (R´ enyi, 1971)

m0(n, 2) = min

m : n ≤

m−1

⌈m/2⌉

∼ log n + 1

2 log log n.

Theorem (Kleitman–Spencer, 1973)

Ω

2k log n
≤ m0(n, k) ≤ O
k2k log n
.

Theorem (Alon, 1986)

m0(n, k) ≤ 2O(k4) log n, via an explicit construction. Applications: experimental design, derandomisation

S. Das

FU Berlin

SLIDE 34

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 35

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 36

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 37

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 38

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 39

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 40

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 41

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 42

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 43

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 44

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 45

Introducing the problem Some initial bounds An exact solution Concluding remarks

Shattering with 2k sets

A random fact

◮ If A ∈

[n]

k

, P
A ∈ Sh(Frand,2k)
=

(2k)! (2k)2k ≈ e−2k ◮ ⇒ shmax

n, k, 2k

≥ e−2kn

k

.

P1 . . . P2 . . . Pk . . .

Beating random

⇒ shmax

n, k, 2k
n

k

k ≈ e−kn

k

.
S. Das

FU Berlin

SLIDE 46

Introducing the problem Some initial bounds An exact solution Concluding remarks

Our main result

Theorem (D.–M´ esz´ aros, 2016+)

If 2k − 1|n, we have shmax

n, k, 2k

=

n

2k − 1 k · 2k2 k!

k

i=1
1 − 2−i

.

◮ As soon as we can shatter a single k-set, we can shatter a

constant proportion of all k-sets

◮ Optimal family is explicitly constructed

S. Das

FU Berlin

SLIDE 47

Introducing the problem Some initial bounds An exact solution Concluding remarks

Our main result

Theorem (D.–M´ esz´ aros, 2016+)

If n ≫ k2k, we have shmax

n, k, 2k

= (1 + o(1))ck n k

,

where ck = k

i=1

1 − 2−i

→ 0.288788 . . . as k → ∞.

◮ As soon as we can shatter a single k-set, we can shatter a

constant proportion of all k-sets

◮ Optimal family is explicitly constructed

S. Das

FU Berlin

SLIDE 48

Introducing the problem Some initial bounds An exact solution Concluding remarks

Our main result

Theorem (D.–M´ esz´ aros, 2016+)

If n ≫ k2k, we have shmax

n, k, 2k

= (1 + o(1))ck n k

,

where ck = k

i=1

1 − 2−i

→ 0.288788 . . . as k → ∞.

◮ As soon as we can shatter a single k-set, we can shatter a

constant proportion of all k-sets

◮ Optimal family is explicitly constructed

S. Das

FU Berlin

SLIDE 49

Introducing the problem Some initial bounds An exact solution Concluding remarks

A dual perspective

Definition (Incidence vector)

Given i ∈ [n] and a family F = {F1, F2, . . . , Fm} over [n], the incidence vector wi ∈ {±1}m is defined by wi,j =

+1

if i ∈ Fj −1 if i / ∈ Fj .

Observation

The set A = {a1, a2, . . . , ak} is shattered if and only if the m × k matrix

wa1
wa2

. . .

wak
has all 2k vectors in {±1}k

appearing as rows.

S. Das

FU Berlin

SLIDE 50

Introducing the problem Some initial bounds An exact solution Concluding remarks

A dual perspective

Definition (Incidence vector)

Given i ∈ [n] and a family F = {F1, F2, . . . , Fm} over [n], the incidence vector wi ∈ {±1}m is defined by wi,j =

+1

if i ∈ Fj −1 if i / ∈ Fj .

Observation

The set A = {a1, a2, . . . , ak} is shattered if and only if the m × k matrix

wa1
wa2

. . .

wak
has all 2k vectors in {±1}k

appearing as rows.

S. Das

FU Berlin

SLIDE 51

Introducing the problem Some initial bounds An exact solution Concluding remarks

An auxiliary hypergraph

Definition (Shatter hypergraph)

The k-uniform shatter hypergraph Hk = (Vk, Ek) has

◮ Vk = {±1}2k ◮ Ek = {{

v1, . . . , vk} :

v1

. . .

vk
has all 2k distinct rows}.

Key properties

◮ Positive degree ⇔ exactly 2k−1 coordinates equal to +1 ◮ Link hypergraph of such a vertex: Hk−1 ⊗ Hk−1 ◮ Positive codegree ⇒ vectors orthogonal

S. Das

FU Berlin

SLIDE 52

Introducing the problem Some initial bounds An exact solution Concluding remarks

An auxiliary hypergraph

Definition (Shatter hypergraph)

The k-uniform shatter hypergraph Hk = (Vk, Ek) has

◮ Vk = {±1}2k ◮ Ek = {{

v1, . . . , vk} :

v1

. . .

vk
has all 2k distinct rows}.

Key properties

◮ Positive degree ⇔ exactly 2k−1 coordinates equal to +1 ◮ Link hypergraph of such a vertex: Hk−1 ⊗ Hk−1 ◮ Positive codegree ⇒ vectors orthogonal

S. Das

FU Berlin

SLIDE 53

Introducing the problem Some initial bounds An exact solution Concluding remarks

Hadamard matrices

Definition (Sylvester’s Hadamard matrices)

The matrix Mk is an orthogonal 2k × 2k matrix defined recursively by M0 =

1
and, for k ≥ 0,

Mk+1 = Mk Mk Mk −Mk

.

M0 =

1
M1 =

1 1 1 −1

M2 =

    1 1 1 1 1 − 1 1 −1 1 1 −1 −1 1 −1 −1 1    

S. Das

FU Berlin

SLIDE 54

Introducing the problem Some initial bounds An exact solution Concluding remarks

Hadamard matrices

Definition (Sylvester’s Hadamard matrices)

The matrix Mk is an orthogonal 2k × 2k matrix defined recursively by M0 =

1
and, for k ≥ 0,

Mk+1 = Mk Mk Mk −Mk

.

M0 =

1
M1 =

1 1 1 −1

M2 =

    1 1 1 1 1 − 1 1 −1 1 1 −1 −1 1 −1 −1 1    

S. Das

FU Berlin

SLIDE 55

Introducing the problem Some initial bounds An exact solution Concluding remarks

Hadamard matrices

Definition (Sylvester’s Hadamard matrices)

The matrix Mk is an orthogonal 2k × 2k matrix defined recursively by M0 =

1
and, for k ≥ 0,

Mk+1 = Mk Mk Mk −Mk

.

M0 =

1
M1 =

1 1 1 −1

M2 =

    1 1 1 1 1 − 1 1 −1 1 1 −1 −1 1 −1 −1 1    

S. Das

FU Berlin

SLIDE 56

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1     The construction for k = 2

S. Das

FU Berlin

SLIDE 57

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 1 1 −1 1 −1 1 1 −1 −1 1 −1 −1 1     The construction for k = 2

S. Das

FU Berlin

SLIDE 58

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     The construction for k = 2

S. Das

FU Berlin

SLIDE 59

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     S1 S2 S3 The construction for k = 2

S. Das

FU Berlin

SLIDE 60

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     S1 S2 S3 The construction for k = 2

S. Das

FU Berlin

SLIDE 61

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     S1 S2 S3 The construction for k = 2

S. Das

FU Berlin

SLIDE 62

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     S1 S2 S3 The construction for k = 2

S. Das

FU Berlin

SLIDE 63

Introducing the problem Some initial bounds An exact solution Concluding remarks

The construction

Construction

1. Start with the matrix Mk
2. Delete the first (all ‘+1’) column
3. Let each of the 2k − 1 remaining columns be the incidence

vector for a set Si of

n 2k−1 elements

    1 1 1 − 1 1 − 1 1 − 1 − 1 − 1 − 1 1     S1 S2 S3 The construction for k = 2

S. Das

FU Berlin

SLIDE 64

Introducing the problem Some initial bounds An exact solution Concluding remarks

Counting shattered sets

The construction

◮ Hadamard recursion + induction + symmetry ⇒ vectors in

the construction induce ek edges in Hk, where ek = 2k2 k!

k

i=1

(1 − 2−i).

◮ Blow-up ⇒

n

2k−1

k · 2k2

k!

k

i=1(1 − 2−i) sets shattered

The upper bound

◮ Upper bound: determine Lagrangian of Hk ◮ Link graphs obey similar recursion as Hadamard matrices

S. Das

FU Berlin

SLIDE 65

Introducing the problem Some initial bounds An exact solution Concluding remarks

Counting shattered sets

The construction

◮ Hadamard recursion + induction + symmetry ⇒ vectors in

the construction induce ek edges in Hk, where ek = 2k2 k!

k

i=1

(1 − 2−i).

◮ Blow-up ⇒

n

2k−1

k · 2k2

k!

k

i=1(1 − 2−i) sets shattered

The upper bound

◮ Upper bound: determine Lagrangian of Hk ◮ Link graphs obey similar recursion as Hadamard matrices

S. Das

FU Berlin

SLIDE 66

Introducing the problem Some initial bounds An exact solution Concluding remarks

Some final thoughts

Universal constructions

◮ Random copies of Hadamard construction:

m0(n, k) = O

k2k log n
◮ Constant worse than for Frand,m

◮ But uses less randomness

◮ Can we use the Hadamard construction more efficiently?

Larger families

◮ What is shmax(n, k, m) for m ≥ 2k + 1?

◮ Lose linear algebraic characterisation of shattering

S. Das

FU Berlin

SLIDE 67

Introducing the problem Some initial bounds An exact solution Concluding remarks

S. Das

FU Berlin