Algebraic Transformations of Convex Codes

Transcription

1 Algebraic Transformations of Convex Codes Amzi Jeffs Advisor: Mohamed Omar Feb 2, 2016

2 Outline Neuroscience: Place cells and neural codes.

3 Outline Neuroscience: Place cells and neural codes. Algebra Background: Neural ideals and the canonical form.

4 Outline Neuroscience: Place cells and neural codes. Algebra Background: Neural ideals and the canonical form. My Thesis Work: Understanding how codes relate to one another algebraically.

5 Biological Motivation Place cells: Neurons which are active in a particular region of an animal s environment.

6 Biological Motivation How is data on place cells collected? Time 1 2 3

7 Biological Motivation How is data on place cells collected? Time 1 2 3

8 Biological Motivation How is data on place cells collected? Time

9 Biological Motivation How is data on place cells collected? Time 1 2 3

10 Mathematical Formulation Neural codes capture an animal s response to a stimulus. We assume that the receptive fields for place cells are open convex sets in Euclidean space.

11 Mathematical Formulation We associate collections of convex sets to binary codes, and attempt to classify these codes. Definition Let U U 1,..., U n be a collection of convex open sets. The code of U is CˆU > 0, 1 nrrrrrrrrrr v R v i 1 U i v j 0 U j x g

12 Mathematical Formulation Definition Let C b 0, 1 n be a code. If there exists a collection of convex open sets U so that C CˆU we say that C is convex. We call U a convex realization of C. From last time: Not all codes are convex!

13 Mathematical Formulation Definition Let C b 0, 1 n be a code. If there exists a collection of convex open sets U so that C CˆU we say that C is convex. We call U a convex realization of C. From last time: Not all codes are convex! Question How can we detect whether a code C is convex?

14 Classifying Convex Codes Question Can we find meaningful criteria that guarantee a code is convex? Answer: Yes! Simplicial complex codes, intersection complete codes, codes with 11 1 in them, and many more!

15 Classifying Convex Codes Question Can we find meaningful criteria that guarantee a code is convex? Answer: Yes! Simplicial complex codes, intersection complete codes, codes with 11 1 in them, and many more! A Constructive Approach: Take a realization U, modify it, and see how that affects C ˆU.

16 Restricting a Convex Realization

17 An Algebraic Approach We will work in the polynomial ring F 2 x 1,..., x n.

18 An Algebraic Approach We will work in the polynomial ring F 2 x 1,..., x n. Definition (CIVCY2013) A pseudomonomial is a polynomial of the form f M x i Mˆ1 x j i >σ j >τ where σ, τ b 1, 2,..., n are disjoint.

19 An Algebraic Approach We will work in the polynomial ring F 2 x 1,..., x n. Definition (CIVCY2013) A pseudomonomial is a polynomial of the form f M x i Mˆ1 x j i >σ j >τ where σ, τ b 1, 2,..., n are disjoint. Example: x 1 x 2ˆ1 x 3 and ˆ1 x 1 ˆ1 x 5 are both pseudmonomials. x 1ˆ1 x 1 x 2 and x 3 2 are NOT pseudomonomials.

20 An Algebraic Approach For any f > F 2 x 1,..., x n and v > 0, 1 n we define f ˆv to be the result of replacing x i by v i, the i-th bit of v. Example: Let f x 1 x 2ˆ1 x 3 and v 110. Then f ˆv 1 1 ˆ1 0 1.

21 An Algebraic Approach Definition (CIVCY2013) Let v > 0, 1 n. Then indicator pseudomonomial for v is ρ v M x i M ˆ1 x j. v i 1 v j 0 Example: ρ 110 x 1 x 2ˆ1 x 3

22 An Algebraic Approach Definition (CIVCY2013) Let v > 0, 1 n. Then indicator pseudomonomial for v is ρ v M x i M ˆ1 x j. v i 1 v j 0 Example: ρ 110 x 1 x 2ˆ1 x 3 Note that ρ v is always a pseudomonomial of degree n.

23 An Algebraic Approach Definition (CIVCY2013) Let v > 0, 1 n. Then indicator pseudomonomial for v is ρ v M x i M ˆ1 x j. v i 1 v j 0 Example: ρ 110 x 1 x 2ˆ1 x 3 Note that ρ v is always a pseudomonomial of degree n. Proposition Let u, v > 0, 1 n. Then ρ v ˆu 1 if and only if u v. That is, ρ v vanishes everywhere in 0, 1 n except for at v.

24 The Neural Ideal of a Code Definition (CIVCY2013) Let C be a code. The neural ideal of C is J C `ρ v S v C e.

25 The Neural Ideal of a Code Definition (CIVCY2013) Let C be a code. The neural ideal of C is J C `ρ v S v C e. Proposition Let f > J C and c > C. Then f ˆc 0. Proof Idea: J C is generated by polynomials which vanish on all of C.

26 Presenting the Neural Ideal Theorem Neural ideals are precisely the ideals generated by pseudomonomials.

27 Presenting the Neural Ideal Theorem Neural ideals are precisely the ideals generated by pseudomonomials. Definition (CIVCY2013) Let J C be a neural ideal. The canonical form of J C is the set of minimal pseudomonomials in J C with respect to division. Equilvalently: CF ˆJ C f > J C S f is a PM and no proper divisor of f is in J C.

28 A Concrete Example x 1 x 2 x 3 > CF ˆJ C

29 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C

30 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C

31 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty!

32 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty! Second Piece of Information: No divisor of x 1 x 2 x 3 is in J C.

33 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty! Second Piece of Information: No divisor of x 1 x 2 x 3 is in J C. So x 1 x 2 is NOT in J C.

34 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty! Second Piece of Information: No divisor of x 1 x 2 x 3 is in J C. So x 1 x 2 is NOT in J C. Must have 110 or 111 in C.

35 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty! Second Piece of Information: No divisor of x 1 x 2 x 3 is in J C. So x 1 x 2 is NOT in J C. Must have 110 or 111 in C.

36 A Concrete Example x 1 x 2 x 3 > CF ˆJ C First Piece of Information: This polynomial vanishes on all of C So 111 is NOT in C So U 1 9 U 2 9 U 3 is empty! Second Piece of Information: No divisor of x 1 x 2 x 3 is in J C. So x 1 x 2 is NOT in J C. Must have 110 or 111 in C. So U 1 9 U 2 is nonempty! (Likewise for U 1 9 U 3 and U 2 9 U 3 )

37 The Neural Ideal in Summary C Ð J C Ð CF ˆJ C We associate codes to neural ideals, and use the canonical form to compactly present the neural ideal and encode information about the code and its realizations.

38 The Neural Ideal in Summary C Ð J C Ð CF ˆJ C We associate codes to neural ideals, and use the canonical form to compactly present the neural ideal and encode information about the code and its realizations. We hope to understand convex codes by examining neural ideals and their canonical forms.

39 Operations on Convex Codes and the Canonical Form

40 An Interesting Homomorphism The map φ F 2 n F 2 n 1 given by f ˆx 1,..., x n1, x n ( f ˆx 1,..., x n1, 1 is a homomorphism. Furthermore, it maps neural ideals to neural ideals! Most Importantly: The action of φ is exactly that restricting to the set U n in any realization of C. We have described a geometric operation purely algebraically! The map φ also sends convex neural ideals to convex neural ideals!

41 Homomorphisms Respecting Neural Ideals Definition We say a homomorphism φ F 2 n F 2 m respects neural ideals if for every C b 0, 1 n there exists D b 0, 1 n so that φˆjc J D. That is, if φ maps neural ideals to neural ideals. Can we classify all such homomorphisms? Do they have geometric meaning?

42 Homomorphisms Respecting Neural Ideals Restriction: Mapping x i ( 1 or x i ( 0 for some i. - x i ( 1 corresponds with replacing each U j by U j 9 U i. - x i ( 0 corresponds with replacing each U j by U j U i.

43 Homomorphisms Respecting Neural Ideals Restriction: Mapping x i ( 1 or x i ( 0 for some i. - x i ( 1 corresponds with replacing each U j by U j 9 U i. - x i ( 0 corresponds with replacing each U j by U j U i. Bit Flipping: Mapping x i ( 1 x i for some i. - Corresponds to taking the complement of U i.

44 Homomorphisms Respecting Neural Ideals Restriction: Mapping x i ( 1 or x i ( 0 for some i. - x i ( 1 corresponds with replacing each U j by U j 9 U i. - x i ( 0 corresponds with replacing each U j by U j U i. Bit Flipping: Mapping x i ( 1 x i for some i. - Corresponds to taking the complement of U i. Permutation: Permuting labels on the variables in F 2 n. - Corresponds to permuting labels on the sets in a realization.

45 Classifying Homomorphisms Respecting Neural Ideals Theorem Let φ F 2 n F 2 m be a homomorphism respecting neural ideals. Then φ is the composition of the three types of maps previously described: Permutation Restriction Bit flipping

46 Mapping the Work

47 Conclusion In This Talk: We associated polynomial ideals to codes. We used these ideals to understand codes and their realizations We described a class of homomorphisms which play nicely with these ideals. These homomorphisms can be used to understand convex codes, and also computationally.

48 Conclusion In This Talk: We associated polynomial ideals to codes. We used these ideals to understand codes and their realizations We described a class of homomorphisms which play nicely with these ideals. These homomorphisms can be used to understand convex codes, and also computationally. What s Next? How do maps respecting neural ideals affect canonical forms? What other algebraic techniques can be leveraged? What can we do to understand convex codes without the algebraic approach?