Max Intersection-Complete Codes Molly Hoch Wellesley College July - - PowerPoint PPT Presentation

max intersection complete codes
SMART_READER_LITE
LIVE PREVIEW

Max Intersection-Complete Codes Molly Hoch Wellesley College July - - PowerPoint PPT Presentation

Max Intersection-Complete Codes Molly Hoch Wellesley College July 17, 2017 Molly Hoch (Wellesley College) Max -Complete Codes July 17, 2017 1 / 24 Motivation The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery


slide-1
SLIDE 1

Max Intersection-Complete Codes

Molly Hoch

Wellesley College

July 17, 2017

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 1 / 24

slide-2
SLIDE 2

Motivation

The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery of place cells and grid cells

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 2 / 24

slide-3
SLIDE 3

Motivation

The 2014 Nobel Prize in Physiology or Medicine was awarded for the discovery of place cells and grid cells Place cells represent an animal’s location Multiple place cells can fire at once

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 2 / 24

slide-4
SLIDE 4

Notation

Definition

A neural code C on n neurons is a set of subsets of [n]. Given n neurons, we build neural codes from their respective receptive fields, living in Rd. The receptive field of a neuron i is denoted Ui.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 3 / 24

slide-5
SLIDE 5

Notation

Definition

A neural code C on n neurons is a set of subsets of [n]. Given n neurons, we build neural codes from their respective receptive fields, living in Rd. The receptive field of a neuron i is denoted Ui. On 5 neurons, one codeword could be {2,4}; this is where the receptive fields U2 and U4 overlap; we write this as 24.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 3 / 24

slide-6
SLIDE 6

Neural Codes and Convexity

We call a code convex if all the receptive fields from which it is built are convex.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 4 / 24

slide-7
SLIDE 7

Neural Codes and Convexity

We call a code convex if all the receptive fields from which it is built are convex. Certain types of codes are known to be convex, notably max intersection-complete codes.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 4 / 24

slide-8
SLIDE 8

Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

slide-9
SLIDE 9

Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Definition

A code C is max intersection-complete if all intersections of its facets (codewords maximal up to inclusion) are present in the code.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

slide-10
SLIDE 10

Types of Codes

Definition

A code C is intersection-complete if all intersections of its codewords are present in the code.

Definition

A code C is max intersection-complete if all intersections of its facets (codewords maximal up to inclusion) are present in the code.

Definition

The maximal code on n neurons is Cmax(n) = {σ : σ ⊆ [n]}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 5 / 24

slide-11
SLIDE 11

Max Intersection-Complete Codes

U1 U2 U3 U4

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 6 / 24

slide-12
SLIDE 12

Max Intersection-Complete Codes

1

slide-13
SLIDE 13

Max Intersection-Complete Codes

1 12 1 1

slide-14
SLIDE 14

Max Intersection-Complete Codes

1 12 1 1 13 123

slide-15
SLIDE 15

Max Intersection-Complete Codes

1 12 1 1 13 123 124 14

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 7 / 24

slide-16
SLIDE 16

13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅}

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

slide-17
SLIDE 17

13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

slide-18
SLIDE 18

13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No!

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

slide-19
SLIDE 19

13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No! Max intersection-complete?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

slide-20
SLIDE 20

13 123 124 12 14 C = {123, 124, 12, 13, 14, ∅} Intersection-complete? No! Max intersection-complete? Yes!

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 8 / 24

slide-21
SLIDE 21

Neural Ideals

From a neural code C, we obtain its neural ideal JC, defined to be JC :=

  • i∈σ

xi

  • j∈τ

(1 − xj) : σ / ∈ C, τ = [n] − σ.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 9 / 24

slide-22
SLIDE 22

Neural Ideals

From a neural code C, we obtain its neural ideal JC, defined to be JC :=

  • i∈σ

xi

  • j∈τ

(1 − xj) : σ / ∈ C, τ = [n] − σ. In our example, 24 is not a codeword of C, so x2x4(1 + x1)(1 + x3) ∈ JC

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 9 / 24

slide-23
SLIDE 23

The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

slide-24
SLIDE 24

The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal. The canonical form has three types of elements, but we focus on only two: Type 1 relations:

i xi

Type 2 relations:

i xi

  • j (1 − xj)

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

slide-25
SLIDE 25

The Canonical Form

The canonical form CF(JC) of a neural ideal consists of the minimal pseudomonomials with respect to divisibility present in the neural ideal. The canonical form has three types of elements, but we focus on only two: Type 1 relations:

i xi

Type 2 relations:

i xi

  • j (1 − xj)

If a Type 1 relation xa1 . . . xan is in the CF of JC, then the codeword c = a1 . . . an is not in C, nor is any codeword containing c. If a Type 2 relation xa1 . . . xan(1 − xb1) . . . (1 − xbm) is in the CF, then

  • i∈{a1,...,an}

Ui ⊆

  • j∈{b1,...,bm}

Uj

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 10 / 24

slide-26
SLIDE 26

Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

slide-27
SLIDE 27

Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}. Because x3x4 ∈ CF(JC), we can’t have 34 ∈ C, nor can we have 134, 234,

  • r 1234 ∈ C.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

slide-28
SLIDE 28

Canonical Form Example

Recall our code C = {123, 124, 12, 14, 13, ∅}. Here, CF(JC) = {x2(1−x1), x3(1−x1), x4(1−x1), x3x4, x1(1−x2)(1−x3)(1−x4)}. Because x3x4 ∈ CF(JC), we can’t have 34 ∈ C, nor can we have 134, 234,

  • r 1234 ∈ C.

Further, an element like x2(1 − x1) tells us that U2 ⊆ U1.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 11 / 24

slide-29
SLIDE 29

Canonical Form Example

Similarly, because x1(1 − x2)(1 − x3)(1 − x4) ∈ CF(JC), we have that U1 ⊆ U2 ∪ U3 ∪ U4.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 12 / 24

slide-30
SLIDE 30

Canonical Form Example

Similarly, because x1(1 − x2)(1 − x3)(1 − x4) ∈ CF(JC), we have that U1 ⊆ U2 ∪ U3 ∪ U4. 123 124 12 13 14

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 12 / 24

slide-31
SLIDE 31

An Existing Signature for Intersection-Complete Codes

The following theorem gives a signature in the canonical form for intersection-complete codes:

Theorem (Curto, Gross, et al. 2015)

A code C is intersection-complete if and only if CF(JC) contains only monomials and pseudomonomials of the form (1 − xj)

i xi.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 13 / 24

slide-32
SLIDE 32

Question

Research Question

Does there exist a signature in the canonical form for maximum intersection-complete codes?

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 14 / 24

slide-33
SLIDE 33

Finding the Facets

We have been able to develop an algorithm for finding the facets of a code C from CF(JC). We use the fact that if a monomial appears in CF(JC) then no codeword containing the indices of that monomial appears in C.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 15 / 24

slide-34
SLIDE 34

Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. On 4 neurons, Cmax = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 16 / 24

slide-35
SLIDE 35

Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. Removing all codewords eliminated by this monomial gives us C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 34, 123, 124, 134, 234, 1234}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 17 / 24

slide-36
SLIDE 36

Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. This leaves us with C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 123, 124}.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 18 / 24

slide-37
SLIDE 37

Example of Facet Algorithm

Recall our earlier example: C = {123, 124, 12, 13, 14, ∅}. The only monomial in CF(JC) is x3x4. This leaves us with C′

max = {∅, 1, 2, 3, 4, 12, 13, 14, 23, 24, 123, 124}.

We see that the facets of C and our reduced C′

max are the same.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 18 / 24

slide-38
SLIDE 38

Sufficient Condition for Non-maximality

Proposition (Franke-H)

Let C be a neural code, JC be its neural ideal, and CF(JC) be the corresponding canonical form. If there exist τ ⊂ [n] and σ ⊆ [n] − τ such that

i∈τ xi ∈ CF(JC) and j∈σ xj

  • i∈τ (1 − xi) ∈ CF(JC), then C is not

convex.

Corollary

For a code to be max intersection-complete, it cannot have the above condition.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 19 / 24

slide-39
SLIDE 39

Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 20 / 24

slide-40
SLIDE 40

Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅ U4 U5

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 20 / 24

slide-41
SLIDE 41

Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅, and that U1 ⊆ U4 ∪ U5.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 21 / 24

slide-42
SLIDE 42

Example

Let C = {4, 5, 1234, 1235, ∅} be a code on five neurons. The canonical form contains both x4x5 and x1(1 − x4)(1 − x5). This tells us that U4 ∩ U5 = ∅, and that U1 ⊆ U4 ∪ U5. U4 U5 U1

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 21 / 24

slide-43
SLIDE 43

An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

slide-44
SLIDE 44

An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

  • 2. Add “equivalent” neurons

Neurons which always fire together are equivalent, e.g. if xi(1 − xj) and xj(1 − xi) ∈ CF(JC), then neurons i and j are equivalent.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

slide-45
SLIDE 45

An Algorithm for Determining Missing Codewords

  • 1. Pick a complex pseudomonomial

This is a pseudomonomial with multiple (1 − xj) factors, e.g., xi(1 − xj1) . . . (1 − xjm). Write ∩k∈[m]ijk = i.

  • 2. Add “equivalent” neurons

Neurons which always fire together are equivalent, e.g. if xi(1 − xj) and xj(1 − xi) ∈ CF(JC), then neurons i and j are equivalent.

  • 3. Add all other possible neurons not prevented by monomials.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 22 / 24

slide-46
SLIDE 46

Example

Let C = {12345, 1236, 2345, ∅}

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

slide-47
SLIDE 47

Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

slide-48
SLIDE 48

Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

  • 2. x4(1 − x5), x5(1 − x4), x2(1 − x3), x3(1 − x2) ∈ CF(JC), so

U4 = U5 and U2 = U3 Potentially, we then have 2345 ∩ 236

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

slide-49
SLIDE 49

Example

Let C = {12345, 1236, 2345, ∅}

  • 1. Pick x2(1 − x4)(1 − x6) ∈ CF(JC). This gives us U2 ⊆ U4 ∪ U6.

Potentially, we have 24 ∩ 26 = 2.

  • 2. x4(1 − x5), x5(1 − x4), x2(1 − x3), x3(1 − x2) ∈ CF(JC), so

U4 = U5 and U2 = U3 Potentially, we then have 2345 ∩ 236

  • 3. x4x6, x5x6 are the only monomials in CF(JC), so we can add 1 to 2345

and 236 to get 1236 ∩ 12345 = 123

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 23 / 24

slide-50
SLIDE 50

Acknowledgments

Thank you to:

  • Dr. Anne Shiu

Megan Franke Nida Obatake Texas A&M University Department of Mathematics National Science Foundation

Molly Hoch (Wellesley College) Max ∩-Complete Codes July 17, 2017 24 / 24