arXiv subjects

Jesse Geneson

Publications and source records attributed to Jesse Geneson.

At least 37 records · Page 2Linked to original sources

arXiv2024

Bounds on the price of feedback for mistake-bounded online learning

We improve several worst-case bounds for various online learning scenarios from (Auer and Long, Machine Learning, 1999). In particular, we sharpen an upper bound for delayed ambiguous reinforcement learning by a factor of 2 and an upper bound for learning compositions of families of functions by a factor of 2.41. We also improve a lower bound from the same paper for learning compositions of $k$ families of functions by a factor of $Θ(\ln{k})$, matching the upper bound up to a constant factor. In addition, we solve a problem from (Long, Theoretical Computer Science, 2020) on the price of bandit feedback with respect to standard feedback for multiclass learning, and we improve an upper bound from (Feng et al., Theoretical Computer Science, 2023) on the price of $r$-input delayed ambiguous reinforcement learning by a factor of $r$, matching a lower bound from the same paper up to the leading term.

cs.LG↗

arXiv2023

Efficient quantum loading of probability distributions through Feynman propagators

We present quantum algorithms for the loading of probability distributions using Hamiltonian simulation for one dimensional Hamiltonians of the form ${\hat H}= Δ+ V(x) \mathbb{I}$. We consider the potentials $V(x)$ for which the Feynman propagator is known to have an analytically closed form and utilize these Hamiltonians to load probability distributions including the normal, Laplace and Maxwell-Boltzmann into quantum states. We also propose a variational method for probability distribution loading based on constructing a coarse approximation to the distribution in the form of a `ladder state' and then projecting onto the ground state of a Hamiltonian chosen to have the desired probability distribution as ground state. These methods extend the suite of techniques available for the loading of probability distributions, and are more efficient than general purpose data loading methods used in quantum machine learning.

quant-ph↗

arXiv2023

Stable fixed points of combinatorial threshold-linear networks

Combinatorial threshold-linear networks (CTLNs) are a special class of recurrent neural networks whose dynamics are tightly controlled by an underlying directed graph. Recurrent networks have long been used as models for associative memory and pattern completion, with stable fixed points playing the role of stored memory patterns in the network. In prior work, we showed that target-free cliques of the graph correspond to stable fixed points of the dynamics, and we conjectured that these are the only stable fixed points possible. In this paper, we prove that the conjecture holds in a variety of special cases, including for networks with very strong inhibition and graphs of size $n \leq 4$. We also provide further evidence for the conjecture by showing that sparse graphs and graphs that are nearly cliques can never support stable fixed points. Finally, we translate some results from extremal combinatorics to obtain an upper bound on the number of stable fixed points of CTLNs in cases where the conjecture holds.

q-bio.NC↗

arXiv2023

Estimating systemic importance with missing data in input-output graphs

In the context of the Cobb-Douglas productivity model we consider the $N \times N$ input-output linkage matrix $W$ for a network of $N$ firms $f_1, f_2, \cdots, f_N$. The associated influence vector $v_w$ of $W$ is defined in terms of the Leontief inverse $L_W$ of $W$ as $v_W = \fracα{N} L_W \vec{\mathbf{1}}$ where $L_W = (I - (1-α) W')^{-1}$, $W'$ denotes the transpose of $W$ and $I$ is the identity matrix. Here $\vec{\mathbf{1}}$ is the $N \times 1$ vector whose entries are all one. The influence vector is a metric of the importance for the firms in the production network. Under the realistic assumption that the data to compute the influence vector is incomplete, we prove bounds on the worst-case error for the influence vector that are sharp up to a constant factor. We also consider the situation where the missing data is binomially distributed and contextualize the bound on the influence vector accordingly. We also investigate how far off the influence vector can be when we only have data on nodes and connections that are within distance $k$ of some source node. A comparison of our results is juxtaposed against PageRank analogues. We close with a discussion on a possible extension beyond Cobb-Douglas to the Constant Elasticity of Substitution model, as well as the possibility of considering other probability distributions for missing data.

cs.DM↗

arXiv2023

Random processes for generating task-dependency graphs

We investigate random processes for generating task-dependency graphs of order $n$ with $m$ edges and a specified number of initial vertices and terminal vertices. In order to do so, we consider two random processes for generating task-dependency graphs that can be combined to accomplish this task. In the $(x, y)$ edge-removal process, we start with a maximally connected task-dependency graph and remove edges uniformly at random as long as they do not cause the number of initial vertices to exceed $x$ or the number of terminal vertices to exceed $y$. In the $(x, y)$ edge-addition process, we start with an empty task-dependency graph and add edges uniformly at random as long as they do not cause the number of initial vertices to be less than $x$ or the number of terminal vertices to be less than $y$. In the $(x, y)$ edge-addition process, we halt if there are exactly $x$ initial vertices and $y$ terminal vertices. For both processes, we determine the values of $x$ and $y$ for which the resulting task-dependency graph is guaranteed to have exactly $x$ initial vertices and $y$ terminal vertices, and we also find the extremal values for the number of edges in the resulting task-dependency graphs as a function of $x$, $y$, and the number of vertices. Furthermore, we asymptotically bound the expected number of edges in the resulting task-dependency graphs. Finally, we define a random process using only edge-addition and edge-removal, and we show that with high probability this random process generates an $(x, y)$ task-dependency graph of order $n$ with $m$ edges.

cs.DM↗

arXiv2023

Online Learning of Smooth Functions

In this paper, we study the online learning of real-valued functions where the hidden function is known to have certain smoothness properties. Specifically, for $q \ge 1$, let $\mathcal F_q$ be the class of absolutely continuous functions $f: [0,1] \to \mathbb R$ such that $\|f'\|_q \le 1$. For $q \ge 1$ and $d \in \mathbb Z^+$, let $\mathcal F_{q,d}$ be the class of functions $f: [0,1]^d \to \mathbb R$ such that any function $g: [0,1] \to \mathbb R$ formed by fixing all but one parameter of $f$ is in $\mathcal F_q$. For any class of real-valued functions $\mathcal F$ and $p>0$, let $\text{opt}_p(\mathcal F)$ be the best upper bound on the sum of $p^{\text{th}}$ powers of absolute prediction errors that a learner can guarantee in the worst case. In the single-variable setup, we find new bounds for $\text{opt}_p(\mathcal F_q)$ that are sharp up to a constant factor. We show for all $\varepsilon \in (0, 1)$ that $\text{opt}_{1+\varepsilon}(\mathcal{F}_{\infty}) = Θ(\varepsilon^{-\frac{1}{2}})$ and $\text{opt}_{1+\varepsilon}(\mathcal{F}_q) = Θ(\varepsilon^{-\frac{1}{2}})$ for all $q \ge 2$. We also show for $\varepsilon \in (0,1)$ that $\text{opt}_2(\mathcal F_{1+\varepsilon})=Θ(\varepsilon^{-1})$. In addition, we obtain new exact results by proving that $\text{opt}_p(\mathcal F_q)=1$ for $q \in (1,2)$ and $p \ge 2+\frac{1}{q-1}$. In the multi-variable setup, we establish inequalities relating $\text{opt}_p(\mathcal F_{q,d})$ to $\text{opt}_p(\mathcal F_q)$ and show that $\text{opt}_p(\mathcal F_{\infty,d})$ is infinite when $p d$. We also obtain sharp bounds on learning $\mathcal F_{\infty,d}$ for $p < d$ when the number of trials is bounded.

cs.LG↗

arXiv2022

Sharp bounds on the price of bandit feedback for several models of mistake-bounded online learning

We determine sharp bounds on the price of bandit feedback for several variants of the mistake-bound model. The first part of the paper presents bounds on the $r$-input weak reinforcement model and the $r$-input delayed, ambiguous reinforcement model. In both models, the adversary gives $r$ inputs in each round and only indicates a correct answer if all $r$ guesses are correct. The only difference between the two models is that in the delayed, ambiguous model, the learner must answer each input before receiving the next input of the round, while the learner receives all $r$ inputs at once in the weak reinforcement model. In the second part of the paper, we introduce models for online learning with permutation patterns, in which a learner attempts to learn a permutation from a set of permutations by guessing statistics related to sub-permutations. For these permutation models, we prove sharp bounds on the price of bandit feedback.

cs.LG↗

arXiv2021

Peripherality in networks: theory and applications

We investigate several related measures of peripherality and centrality for vertices and edges in networks, including the Mostar index which was recently introduced as a measure of peripherality for both edges and networks. We refute a conjecture on the maximum possible Mostar index of bipartite graphs from (Došlić et al, Journal of Mathematical Chemistry, 2018) and (Ali and Došlić, Applied Mathematics and Computation, 2021). We also correct a result from the latter paper, where they claimed that the maximum possible value of the terminal Mostar index among all trees of order $n$ is $(n-1)(n-2)$. We show that this maximum is $(n-1)(n-3)$ for $n \ge 3$, and that it is only attained by the star. We asymptotically answer another problem on the maximum difference between the Mostar index and the irregularity of trees from (F. Gao et al, On the difference of Mostar index and irregularity of graphs, Bulletin of the Malaysian Mathematical Sciences Society, 2021). We also prove a number of extremal bounds and computational complexity results about the Mostar index, irregularity, and measures of peripherality and centrality. We discuss graphs where the Mostar index is not an accurate measure of peripherality. We construct a general family of graphs with the property that the Mostar index is strictly greater for edges that are closer to the center. We also investigate centrality and peripherality in two graphs which represent the SuperFast and MOZART-4 systems of atmospheric chemical reactions by computing various measures of peripherality and centrality for the vertices and edges in these graphs. For both of these graphs, we find that the Mostar index is closer to a measure of centrality than peripherality of the edges. We also introduce some new indices which perform well as measures of peripherality on the SuperFast and MOZART-4 graphs.

math.CO↗

arXiv2021

Formations and generalized Davenport-Schinzel sequences

Let $up(r, t) = (a_1 a_2 \dots a_r)^t$. We investigate the problem of determining the maximum possible integer $n(r, t)$ for which there exist $2t-1$ permutations $π_1, π_2, \dots, π_{2t-1}$ of $1, 2, \dots, n(r, t)$ such that the concatenated sequence $π_1 π_2 \dots π_{2t-1}$ has no subsequence isomorphic to $up(r,t)$. This quantity has been used to obtain an upper bound on the maximum number of edges in $k$-quasiplanar graphs. It was proved by (Geneson, Prasad, and Tidor, Electronic Journal of Combinatorics, 2014) that $n(r, t) \le (r-1)^{2^{2t-2}}$. We prove that $n(r,t) = Θ(r^{2t-1 \choose t})$, where the constant in the bound depends only on $t$. Using our upper bound in the case $t = 2$, we also sharpen an upper bound of (Klazar, Integers, 2002), who proved that $Ex(up(r,2),n) < (2n+1)L$ where $L = Ex(up(r,2),K-1)+1$, $K = (r-1)^4 + 1$, and $Ex(u, n)$ denotes the extremal function for forbidden generalized Davenport-Schinzel sequences. We prove that $K = (r-1)^4 + 1$ in Klazar's bound can be replaced with $K = (r-1) \binom{r}{2}+1$. We also prove a conjecture from (Geneson, Prasad, and Tidor, Electronic Journal of Combinatorics, 2014) by showing for $t \geq 1$ that $Ex(a b c (a c b)^{t} a b c, n) = n 2^{\frac{1}{t!}α(n)^{t} \pm O(α(n)^{t-1})}$. In addition, we prove that $Ex(a b c a c b (a b c)^{t} a c b, n) = n 2^{\frac{1}{(t+1)!}α(n)^{t+1} \pm O(α(n)^{t})}$ for all $t \geq 1$.

math.CO↗

arXiv2021

The distance-k dimension of graphs

The metric dimension, $\dim(G)$, of a graph $G$ is a graph parameter motivated by robot navigation that has been studied extensively. Let $G$ be a graph with vertex set $V(G)$, and let $d(x,y)$ denote the length of a shortest $x-y$ path in $G$. For a positive integer $k$ and for distinct $x,y \in V(G)$, let $d_k(x,y)=\min\{d(x,y), k+1\}$ and let $R_k\{x,y\}=\{z\in V(G): d_k(x,z) \neq d_k(y,z)\}$. A subset $S\subseteq V(G)$ is a distance-$k$ resolving set of $G$ if $|S \cap R_k\{x,y\}| \ge 1$ for any pair of distinct $x,y \in V(G)$, and the distance-$k$ dimension, $\dim_k(G)$, of $G$ is the minimum cardinality over all distance-$k$ resolving sets of $G$. In this paper, we study the distance-$k$ dimension of graphs. We obtain some general bounds for distance-$k$ dimension. For all $k \ge 1$, we characterize connected graphs $G$ of order $n$ with $\dim_k(G) \ge n-2$. We determine $\dim_k(G)$ when $G$ is a cycle or a path. We also examine the effect of vertex or edge deletion on the distance-$k$ dimension of graphs.

math.CO↗

arXiv2021

Sharper bounds for online learning of smooth functions of a single variable

We investigate the generalization of the mistake-bound model to continuous real-valued single variable functions. Let $\mathcal{F}_q$ be the class of absolutely continuous functions $f: [0, 1] \rightarrow \mathbb{R}$ with $||f'||_q \le 1$, and define $opt_p(\mathcal{F}_q)$ as the best possible bound on the worst-case sum of the $p^{th}$ powers of the absolute prediction errors over any number of trials. Kimber and Long (Theoretical Computer Science, 1995) proved for $q \ge 2$ that $opt_p(\mathcal{F}_q) = 1$ when $p \ge 2$ and $opt_p(\mathcal{F}_q) = \infty$ when $p = 1$. For $1 < p < 2$ with $p = 1+ε$, the only known bound was $opt_p(\mathcal{F}_{q}) = O(ε^{-1})$ from the same paper. We show for all $ε\in (0, 1)$ and $q \ge 2$ that $opt_{1+ε}(\mathcal{F}_q) = Θ(ε^{-\frac{1}{2}})$, where the constants in the bound do not depend on $q$. We also show that $opt_{1+ε}(\mathcal{F}_{\infty}) = Θ(ε^{-\frac{1}{2}})$.

cs.LG↗

arXiv2021

Continuous Turán numbers

In this paper, we define a notion of containment and avoidance for subsets of $\mathbb{R}^2$. Then we introduce a new, continuous and super-additive extremal function for subsets $P \subseteq \mathbb{R}^2$ called $px(n, P)$, which is the supremum of $μ_2(S)$ over all open $P$-free subsets $S \subseteq [0, n]^2$, where $μ_2(S)$ denotes the Lebesgue measure of $S$ in $\mathbb{R}^2$. We show that $px(n, P)$ fully encompasses the Zarankiewicz problem and more generally the 0-1 matrix extremal function $ex(n, M)$ up to a constant factor. More specifically, we define a natural correspondence between finite subsets $P \subseteq \mathbb{R}^2$ and 0-1 matrices $M_P$, and we prove that $px(n, P) = Θ(ex(n, M_P))$ for all finite subsets $P \subseteq \mathbb{R}^2$, where the constants in the bounds depend only on the distances between the points in $P$. We also discuss bounded infinite subsets $P$ for which $px(n, P)$ grows faster than $ex(n, M)$ for all fixed 0-1 matrices $M$. In particular, we show that $px(n, P) = Θ(n^{2})$ for any open subset $P \subseteq \mathbb{R}^2$. We prove an even stronger result, that if $Q_P$ is the set of points with rational coordinates in any open subset $P \subseteq \mathbb{R}^2$, then $px(n, Q_P) = Θ(n^2)$. Finally, we obtain a strengthening of the Kővari-Sós-Turán theorem that applies to infinite subsets of $\mathbb{R}^2$. Specifically, for subsets $P_{s, t, c} \subseteq \mathbb{R}^2$ consisting of $t$ horizontal line segments of length $s$ with left endpoints on the same vertical line with consecutive segments a distance of $c$ apart, we prove that $px(n, P_{s, t,c}) = O(s^{\frac{1}{t}}n^{2-\frac{1}{t}})$, where the constant in the bound depends on $t$ and $c$. When $t = 2$, we show that this bound is sharp up to a constant factor that depends on $c$.

math.CO↗

arXiv2021

A note on the price of bandit feedback for mistake-bounded online learning

The standard model and the bandit model are two generalizations of the mistake-bound model to online multiclass classification. In both models the learner guesses a classification in each round, but in the standard model the learner recieves the correct classification after each guess, while in the bandit model the learner is only told whether or not their guess is correct in each round. For any set $F$ of multiclass classifiers, define $opt_{std}(F)$ and $opt_{bandit}(F)$ to be the optimal worst-case number of prediction mistakes in the standard and bandit models respectively. Long (Theoretical Computer Science, 2020) claimed that for all $M > 2$ and infinitely many $k$, there exists a set $F$ of functions from a set $X$ to a set $Y$ of size $k$ such that $opt_{std}(F) = M$ and $opt_{bandit}(F) \ge (1 - o(1))(|Y|\ln{|Y|})opt_{std}(F)$. The proof of this result depended on the following lemma, which is false e.g. for all prime $p \ge 5$, $s = \mathbf{1}$ (the all $1$ vector), $t = \mathbf{2}$ (the all $2$ vector), and all $z$. Lemma: Fix $n \ge 2$ and prime $p$, and let $u$ be chosen uniformly at random from $\left\{0, \dots, p-1\right\}^n$. For any $s, t \in \left\{1, \dots, p-1\right\}^n$ with $s \neq t$ and for any $z \in \left\{0, \dots, p-1\right\}$, we have $\Pr(t \cdot u = z \mod p \text{ } | \text{ } s \cdot u = z \mod p) = \frac{1}{p}$. We show that this lemma is false precisely when $s$ and $t$ are multiples of each other mod $p$. Then using a new lemma, we fix Long's proof.

cs.DM↗

arXiv2020

Almost all permutation matrices have bounded saturation functions

Saturation problems for forbidden graphs have been a popular area of research for many decades, and recently Brualdi and Cao initiated the study of a saturation problem for 0-1 matrices. We say that 0-1 matrix $A$ is saturating for the forbidden 0-1 matrix $P$ if $A$ avoids $P$ but changing any zero to a one in $A$ creates a copy of $P$. Define $sat(n, P)$ to be the minimum possible number of ones in an $n \times n$ 0-1 matrix that is saturating for $P$. Fulek and Keszegh proved that for every 0-1 matrix $P$, either $sat(n, P) = O(1)$ or $sat(n, P) = Θ(n)$. They found two 0-1 matrices $P$ for which $sat(n, P) = O(1)$, as well as infinite families of 0-1 matrices $P$ for which $sat(n, P) = Θ(n)$. Their results imply that $sat(n, P) = Θ(n)$ for almost all $k \times k$ 0-1 matrices $P$. Fulek and Keszegh conjectured that there are many more 0-1 matrices $P$ such that $sat(n, P) = O(1)$ besides the ones they found, and they asked for a characterization of all permutation matrices $P$ such that $sat(n, P) = O(1)$. We affirm their conjecture by proving that almost all $k \times k$ permutation matrices $P$ have $sat(n, P) = O(1)$. We also make progress on the characterization problem, since our proof of the main result exhibits a family of permutation matrices with bounded saturation functions.

math.CO↗

arXiv2020

Sharper bounds and structural results for minimally nonlinear 0-1 matrices

The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally non-linear if $ex(n, P) = ω(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the maximum number of ones and the maximum number of columns in a minimally non-linear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the bound on the maximum number of ones in a minimally non-linear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally non-linear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally non-linear matrix and that there are not more than six ones in any other row of a minimally non-linear matrix. Furthermore, we prove that if a minimally non-linear 0-1 matrix has ones in the same row with exactly $d$ columns between them, then within these columns there are at most $2d-1$ rows above and $2d-1$ rows below with ones.

math.CO↗

arXiv2020

Reconfiguration graphs of zero forcing sets

This paper begins the study of reconfiguration of zero forcing sets, and more specifically, the zero forcing graph. Given a base graph $G$, its zero forcing graph, $\mathscr{Z}(G)$, is the graph whose vertices are the minimum zero forcing sets of $G$ with an edge between vertices $B$ and $B'$ of $\mathscr{Z}(G)$ if and only if $B$ can be obtained from $B'$ by changing a single vertex of $G$. It is shown that the zero forcing graph of a forest is connected, but that many zero forcing graphs are disconnected. We characterize the base graphs whose zero forcing graphs are either a path or the complete graph, and show that the star cannot be a zero forcing graph. We show that computing $\mathscr{Z}(G)$ takes $2^{Θ(n)}$ operations in the worst case for a graph $G$ of order $n$.

math.CO↗

arXiv2020

Extremal results for graphs of bounded metric dimension

Metric dimension is a graph parameter motivated by problems in robot navigation, drug design, and image processing. In this paper, we answer several open extremal problems on metric dimension and pattern avoidance in graphs from (Geneson, Metric dimension and pattern avoidance, Discrete Appl. Math. 284, 2020, 1-7). Specifically, we construct a new family of graphs that allows us to determine the maximum possible degree of a graph of metric dimension at most $k$, the maximum possible degeneracy of a graph of metric dimension at most $k$, the maximum possible chromatic number of a graph of metric dimension at most $k$, and the maximum $n$ for which there exists a graph of metric dimension at most $k$ that contains $K_{n, n}$. We also investigate a variant of metric dimension called edge metric dimension and solve another problem from the same paper for $n$ sufficiently large by showing that the edge metric dimension of $P_n^{d}$ is $d$ for $n \geq d^{d-1}$. In addition, we use a probabilistic argument to make progress on another open problem from the same paper by showing that the maximum possible clique number of a graph of edge metric dimension at most $k$ is $2^{Θ(k)}$. We also make progress on a problem from (N. Zubrilina, On the edge dimension of a graph, Discrete Math. 341, 2018, 2083-2088) by finding a family of new triples $(x, y, n)$ for which there exists a graph of metric dimension $x$, edge metric dimension $y$, and order $n$. In particular, we show that for each integer $k > 0$, there exist graphs $G$ with metric dimension $k$, edge metric dimension $3^k(1-o(1))$, and order $3^k(1+o(1))$.

math.CO↗

arXiv2020

The damage throttling number of a graph

The cop throttling number of a graph, introduced in 2018 by Breen et al., optimizes the balance between the number of cops used and the number of rounds required to catch the robber in a game of Cops and Robbers. In 2019, Cox and Sanaei studied a variant of Cops and Robbers in which the robber tries to occupy (or damage) as many vertices as possible and the cop tries to minimize this damage. In their paper, they study the minimum number of vertices damaged by the robber over all games played on a given graph $G$, called the damage number of $G$. We introduce the natural parameter called the damage throttling number of a graph, denoted $\operatorname{th}_d(G)$, which optimizes the balance between the number of cops used and the number of vertices damaged in the graph. To this end, we formalize the definition of $k$-damage number, which extends the damage number to games played with $k$ cops. We show that damage throttling and cop throttling share many properties, yet they exhibit interesting differences. We prove that the damage throttling number is tightly bounded above by one less than the cop throttling number. Infinite families of examples and non-examples of tightness in this bound are given. We also find an infinite family of connected graphs $G$ of order $n$ for which $\operatorname{th}_d(G) = Ω(n^{2/3})$.

math.CO↗