arXiv subjects

Kaitlyn Phillipson

Publications and source records attributed to Kaitlyn Phillipson.

4 recordsLinked to original sources

arXiv2019

Classification of Open and Closed Convex Codes on Five Neurons

Neural codes, represented as collections of binary strings, encode neural activity and show relationships among stimuli. Certain neurons, called place cells, have been shown experimentally to fire in convex regions in space. A natural question to ask is: Which neural codes can arise as intersection patterns of convex sets? While past research has established several criteria, complete conditions for convexity are not yet known for codes with more than four neurons. We classify all neural codes with five neurons as convex/non-convex codes. Furthermore, we investigate which of these codes can be represented by open versus closed convex sets. Interestingly, we find a code which is an open but not closed convex code and demonstrate a minimal example for this phenomenon.

q-bio.NC↗

arXiv2019

A Faster Solution to Smale's 17th Problem I: Real Binomial Systems

Suppose $F:=(f_1,\ldots,f_n)$ is a system of random $n$-variate polynomials with $f_i$ having degree $\leq\!d_i$ and the coefficient of $x^{a_1}_1\cdots x^{a_n}_n$ in $f_i$ being an independent complex Gaussian of mean $0$ and variance $\frac{d_i!}{a_1!\cdots a_n!\left(d_i-\sum^n_{j=1}a_j \right)!}$. Recent progress on Smale's 17th Problem by Lairez --- building upon seminal work of Shub, Beltran, Pardo, Bürgisser, and Cucker --- has resulted in a deterministic algorithm that finds a single (complex) approximate root of $F$ using just $N^{O(1)}$ arithmetic operations on average, where $N\!:=\!\sum^n_{i=1}\frac{(n+d_i)!}{n!d_i!}$ ($=n(n+\max_i d_i)^{O(\min\{n,\max_i d_i)\}}$) is the maximum possible total number of monomial terms for such an $F$. However, can one go faster when the number of terms is smaller, and we restrict to real coefficient and real roots? And can one still maintain average-case polynomial-time with more general probability measures? We show the answer is yes when $F$ is instead a binomial system --- a case whose numerical solution is a key step in polyhedral homotopy algorithms for solving arbitrary polynomial systems. We give a deterministic algorithm that finds a real approximate root (or correctly decides there are none) using just $O(n^2(\log(n)+\log\max_i d_i))$ arithmetic operations on average. Furthermore, our approach allows Gaussians with arbitrary variance. We also discuss briefly the obstructions to maintaining average-case time polynomial in $n\log \max_i d_i$ when $F$ has more terms.

math.AG↗

arXiv2018

Gröbner Bases of Neural Ideals

The brain processes information about the environment via neural codes. The neural ideal was introduced recently as an algebraic object that can be used to better understand the combinatorial structure of neural codes. Every neural ideal has a particular generating set, called the canonical form, that directly encodes a minimal description of the receptive field structure intrinsic to the neural code. On the other hand, for a given monomial order, any polynomial ideal is also generated by its unique (reduced) Gröbner basis with respect to that monomial order. How are these two types of generating sets -- canonical forms and Gröbner bases -- related? Our main result states that if the canonical form of a neural ideal is a Gröbner basis, then it is the universal Gröbner basis (that is, the union of all reduced Gröbner bases). Furthermore, we prove that this situation -- when the canonical form is a Gröbner basis -- occurs precisely when the universal Gröbner basis contains only pseudo-monomials (certain generalizations of monomials). Our results motivate two questions: (1)~When is the canonical form a Gröbner basis? (2)~When the universal Gröbner basis of a neural ideal is {\em not} a canonical form, what can the non-pseudo-monomial elements in the basis tell us about the receptive fields of the code? We give partial answers to both questions. Along the way, we develop a representation of pseudo-monomials as hypercubes in a Boolean lattice.

q-bio.NC↗

arXiv2012

Fewnomial Systems with Many Roots, and an Adelic Tau Conjecture

Consider a system F of n polynomials in n variables, with a total of n+k distinct exponent vectors, over any local field L. We discuss conjecturally tight bounds on the maximal number of non-degenerate roots F can have over L, with all coordinates having fixed phase, as a function of n, k, and L only. In particular, we give new explicit systems with number of roots approaching the best known upper bounds. We also briefly review the background behind such bounds, and their application, including connections to computational number theory and variants of the Shub-Smale tau-Conjecture and the P vs. NP Problem. One of our key tools is the construction of combinatorially constrained tropical varieties with maximally many intersections.

math.AG↗