arXiv subjects
Publications and source records attributed to David Jensen.
We provide a new perspective on the divisor theory of graphs, using additive combinatorics. As a test case for this perspective, we compute the gonality of certain families of outerplanar graphs, specifically the strip graphs. The Jacobians of such graphs are always cyclic of Fibonacci order. As a consequence, we obtain several results on the additive properties of Fibonacci numbers.
A Ferrers rook graph is a graph whose vertices correspond to the dots in a Ferrers diagram, and where two vertices are adjacent if they are in the same row or the same column. We propose a conjectural formula for the gonality of Ferrers rook graphs, and prove this conjecture for a few infinite families of Ferrers diagrams. We also prove the conjecture for all Ferrers diagrams $F$ with $|F| \leq 8$.
Reinforcement learning (RL) algorithms often struggle to learn policies that generalize to novel situations due to issues such as causal confusion, overfitting to irrelevant factors, and failure to isolate control of state factors. These issues stem from a common source: a failure to accurately identify and exploit state-specific causal relationships in the environment. While some prior works in RL aim to identify these relationships explicitly, they rely on informal domain-specific heuristics such as spatial and temporal proximity. Actual causality offers a principled and general framework for determining the causes of particular events. However, existing definitions of actual cause often attribute causality to a large number of events, even if many of them rarely influence the outcome. Prior work on actual causality proposes normality as a solution to this problem, but its existing implementations are challenging to scale to complex and continuous-valued RL environments. This paper introduces functional actual cause (FAC), a framework that uses context-specific independencies in the environment to restrict the set of actual causes. We additionally introduce Joint Optimization for Actual Cause Inference (JACI), an algorithm that learns from observational data to infer functional actual causes. We demonstrate empirically that FAC agrees with known results on a suite of examples from the actual causality literature, and JACI identifies actual causes with significantly higher accuracy than existing heuristic methods in a set of complex, continuous-valued environments.
About 40 transneptunian binaries (TNBs) have fully determined orbits with about 10 others being solved except for breaking the mirror ambiguity. Despite decades of study almost all TNBs have only ever been analyzed with a model that assumes perfect Keplerian motion (e.g., two point masses). In reality, all TNB systems are non-Keplerian due to non-spherical shapes, possible presence of undetected system components, and/or solar perturbations. In this work, we focus on identifying candidates for detectable non-Keplerian motion based on sample of 45 well-characterized binaries. We use MultiMoon, a non-Keplerian Bayesian inference tool, to analyze published relative astrometry allowing for non-spherical shapes of each TNB system's primary. We first reproduce the results of previous Keplerian fitting efforts with MultiMoon, which serves as a comparison for the non-Keplerian fits and confirms that these fits are not biased by the assumption of a Keplerian orbit. We unambiguously detect non-Keplerian motion in 8 TNB systems across a range of primary radii, mutual orbit separations, and system masses. As a proof of concept for non-Keplerian fitting, we perform detailed fits for (66652) Borasisi-Pabu, possibly revealing a $J_2 \approx 0.44$, implying Borasisi (and/or Pabu) may be a contact binary or an unresolved compact binary. However, full confirmation of this result will require new observations. This work begins the next generation of TNB analyses that go beyond the point mass assumption to provide unique and valuable information on the physical properties of TNBs with implications for their formation and evolution.
Algorithmic robustness refers to the sustained performance of a computational system in the face of change in the nature of the environment in which that system operates or in the task that the system is meant to perform. Below, we motivate the importance of algorithmic robustness, present a conceptual framework, and highlight the relevant areas of research for which algorithmic robustness is relevant. Why robustness? Robustness is an important enabler of other goals that are frequently cited in the context of public policy decisions about computational systems, including trustworthiness, accountability, fairness, and safety. Despite this dependence, it tends to be under-recognized compared to these other concepts. This is unfortunate, because robustness is often more immediately achievable than these other ultimate goals, which can be more subjective and exacting. Thus, we highlight robustness as an important goal for researchers, engineers, regulators, and policymakers when considering the design, implementation, and deployment of computational systems. We urge researchers and practitioners to elevate the attention paid to robustness when designing and evaluating computational systems. For many key systems, the immediate question after any demonstration of high performance should be: "How robust is that performance to realistic changes in the task or environment?" Greater robustness will set the stage for systems that are more trustworthy, accountable, fair, and safe. Toward that end, this document provides a brief roadmap to some of the concepts and existing research around the idea of algorithmic robustness.
The $r$th gonality of a graph is the smallest degree of a divisor on the graph with rank $r$. The gonality sequence of a graph is a tropical analogue of the gonality sequence of an algebraic curve. We show that the set of truncated gonality sequences of graphs forms a semigroup under addition. Using this, we study which triples $(x,y,z)$ can be the first 3 terms of a graph gonality sequence. We show that nearly every such triple with $z \geq \frac{3}{2}x+2$ is the first three terms of a graph gonality sequence, and also exhibit triples where the ratio $\frac{z}{x}$ is an arbitrary rational number between 1 and 3. In the final section, we study algebraic curves whose $r$th and $(r+1)$st gonality differ by 1, and posit several questions about graphs with this property.
We generalize the Embedding Theorem of Eisenbud-Harris from classical Brill-Noether theory to the setting of Hurwitz-Brill-Noether theory. More precisely, in classical Brill-Noether theory, the embedding theorem states that a general linear series of degree d and rank r on a general curve of genus g is an embedding if r is at least 3. If \(f \colon C \to \mathbb{P}^1\) is a general cover of degree k, and L is a line bundle on C, recent work of the authors shows that the splitting type of \(f_* L\) provides the appropriate generalization of the pair (r, d) in classical Brill--Noether theory. In the context of Hurwitz-Brill-Noether theory, the condition that r is at least 3 is no longer sufficient to guarantee that a general such linear series is an embedding. We show that the additional condition needed to guarantee that a general linear series |L| is an embedding is that the splitting type of \(f_* L\) has at least three nonnegative parts. This new extra condition reflects the unique geometry of k-gonal curves, which lie on scrolls in \(\mathbb{P}^r\).
Learning causal structure is useful in many areas of artificial intelligence, including planning, robotics, and explanation. Constraint-based structure learning algorithms such as PC use conditional independence (CI) tests to infer causal structure. Traditionally, constraint-based algorithms perform CI tests with a preference for smaller-sized conditioning sets, partially because the statistical power of conventional CI tests declines rapidly as the size of the conditioning set increases. However, many modern conditional independence tests are model-based, and these tests use well-regularized models that maintain statistical power even with very large conditioning sets. This suggests an intriguing new strategy for constraint-based algorithms which may result in a reduction of the total number of CI tests performed: Test variable pairs with large conditioning sets first, as a pre-processing step that finds some conditional independencies quickly, before moving on to the more conventional strategy that favors small conditioning sets. We propose such a pre-processing step for the PC algorithm which relies on performing CI tests on a few randomly selected large conditioning sets. We perform an empirical analysis on directed acyclic graphs (DAGs) that correspond to real-world systems and both empirical and theoretical analyses for Erdős-Renyi DAGs. Our results show that Pre-Processing Plus PC (P3PC) performs far fewer CI tests than the original PC algorithm, between 0.5% to 36%, and often less than 10%, of the CI tests that the PC algorithm alone performs. The efficiency gains are particularly significant for the DAGs corresponding to real-world systems.
A growing family of approaches to causal inference rely on Bayesian formulations of assumptions that go beyond causal graph structure. For example, Bayesian approaches have been developed for analyzing instrumental variable designs, regression discontinuity designs, and within-subjects designs. This paper introduces simulation-based identifiability (SBI), a procedure for testing the identifiability of queries in Bayesian causal inference approaches that are implemented as probabilistic programs. SBI complements analytical approaches to identifiability, leveraging a particle-based optimization scheme on simulated data to determine identifiability for analytically intractable models. We analyze SBI's soundness for a broad class of differentiable, finite-dimensional probabilistic programs with bounded effects. Finally, we provide an implementation of SBI using stochastic gradient descent, and show empirically that it agrees with known identification results on a suite of graph-based and quasi-experimental design benchmarks, including those using Gaussian processes.
Segerman's 15+4 puzzle is a hinged version of the classic 15-puzzle, in which the tiles rotate as they slide around. In 1974, Wilson classified the groups of solutions to sliding block puzzles. We generalize Wilson's result to puzzles like the 15+4 puzzle, where the tiles can rotate, and the sets of solutions are subgroups of the generalized symmetric groups. Aside from two exceptional cases, we see that the group of solutions to such a puzzle is always either the entire generalized symmetric group or one of two special subgroups of index two.
We propose a definition of tropical linear series that isolates some of the essential combinatorial properties of tropicalizations of not-necessarily-complete linear series on algebraic curves. The definition combines the Baker-Norine notion of rank with the notion of tropical independence and has the property that the restriction of a tropical linear series of rank r to a connected subgraph is a tropical linear series of rank r. We show that tropical linear series of rank 1 are finitely generated as tropical modules and state a number of open problems related to algebraic, combinatorial, and topological properties of higher rank tropical linear series
Recent work in reinforcement learning has focused on several characteristics of learned policies that go beyond maximizing reward. These properties include fairness, explainability, generalization, and robustness. In this paper, we define interventional robustness (IR), a measure of how much variability is introduced into learned policies by incidental aspects of the training procedure, such as the order of training data or the particular exploratory actions taken by agents. A training procedure has high IR when the agents it produces take very similar actions under intervention, despite variation in these incidental aspects of the training procedure. We develop an intuitive, quantitative measure of IR and calculate it for eight algorithms in three Atari environments across dozens of interventions and states. From these experiments, we find that IR varies with the amount of training and type of algorithm and that high performance does not imply high IR, as one might expect.
Planetary systems with mean-motion resonances (MMRs) hold special value in terms of their dynamical complexity and their capacity to constrain planet formation and migration histories. The key towards making these connections, however, is to have a reliable characterization of the resonant dynamics, especially the so-called "libration amplitude", which qualitatively measures how deep the system is into the resonance. In this work, we identify an important complication with the interpretation of libration amplitude estimates from observational data of resonant systems. Specifically, we show that measurement noise causes inferences of the libration amplitude to be systematically biased to larger values, with noisier data yielding a larger bias. We demonstrated this through multiple approaches, including using dynamical fits of synthetic radial velocity data to explore how the the libration amplitude distribution inferred from the posterior parameter distribution varies with the degree of measurement noise. We find that even modest levels of noise still result in a slight bias. The origin of the bias stems from the topology of the resonant phase space and the fact that the available phase space volume increases non-uniformly with increasing libration amplitude. We highlight strategies for mitigating the bias through the usage of particular priors. Our results imply that many known resonant systems are likely deeper in resonance than previously appreciated.
We define a new graph invariant called the scramble number. We show that the scramble number of a graph is a lower bound for the gonality and an upper bound for the treewidth. Unlike the treewidth, the scramble number is not minor monotone, but it is subgraph monotone and invariant under refinement. We compute the scramble number and gonality of several families of graphs for which these invariants are strictly greater than the treewidth.
We briefly survey recent results related to linear series on curves that are general in various moduli spaces, highlighting the interplay between algebraic geometry on a general curve and the combinatorics of its degenerations. Breakthroughs include the proof of the Maximal Rank Theorem, which determines the Hilbert function of the general linear series of given degree and rank on the general curve in M_g, and complete analogs of the standard Brill-Noether theorems for curves that are general in Hurwitz spaces. Other advances include partial results in a similar direction for linear series in the Prym locus of a general unramified double cover of a general k-gonal curve and instances of the Strong Maximal Rank Conjecture.
Methods that infer causal dependence from observational data are central to many areas of science, including medicine, economics, and the social sciences. A variety of theoretical properties of these methods have been proven, but empirical evaluation remains a challenge, largely due to the lack of observational data sets for which treatment effect is known. We describe and analyze observational sampling from randomized controlled trials (OSRCT), a method for evaluating causal inference methods using data from randomized controlled trials (RCTs). This method can be used to create constructed observational data sets with corresponding unbiased estimates of treatment effect, substantially increasing the number of data sets available for empirical evaluation of causal inference methods. We show that, in expectation, OSRCT creates data sets that are equivalent to those produced by randomly sampling from empirical data sets in which all potential outcomes are available. We then perform a large-scale evaluation of seven causal inference methods over 37 data sets, drawn from RCTs, as well as simulators, real-world computational systems, and observational data sets augmented with a synthetic response variable. We find notable performance differences when comparing across data from different sources, demonstrating the importance of using data from a variety of sources when evaluating any causal inference method.
The advances in artificial intelligence enabled by deep learning architectures are undeniable. In several cases, deep neural network driven models have surpassed human level performance in benchmark autonomy tasks. The underlying policies for these agents, however, are not easily interpretable. In fact, given their underlying deep models, it is impossible to directly understand the mapping from observations to actions for any reasonably complex agent. Producing this supporting technology to "open the black box" of these AI systems, while not sacrificing performance, was the fundamental goal of the DARPA XAI program. In our journey through this program, we have several "big picture" takeaways: 1) Explanations need to be highly tailored to their scenario; 2) many seemingly high performing RL agents are extremely brittle and are not amendable to explanation; 3) causal models allow for rich explanations, but how to present them isn't always straightforward; and 4) human subjects conjure fantastically wrong mental models for AIs, and these models are often hard to break. This paper discusses the origins of these takeaways, provides amplifying information, and suggestions for future work.
The ubiquity of mobile devices has led to the proliferation of mobile services that provide personalized and context-aware content to their users. Modern mobile services are distributed between end-devices, such as smartphones, and remote servers that reside in the cloud. Such services thrive on their ability to predict future contexts to pre-fetch content or make context-specific recommendations. An increasingly common method to predict future contexts, such as location, is via machine learning (ML) models. Recent work in context prediction has focused on ML model personalization where a personalized model is learned for each individual user in order to tailor predictions or recommendations to a user's mobile behavior. While the use of personalized models increases efficacy of the mobile service, we argue that it increases privacy risk since a personalized model encodes contextual behavior unique to each user. To demonstrate these privacy risks, we present several attribute inference-based privacy attacks and show that such attacks can leak privacy with up to 78% efficacy for top-3 predictions. We present Pelican, a privacy-preserving personalization system for context-aware mobile services that leverages both device and cloud resources to personalize ML models while minimizing the risk of privacy leakage for users. We evaluate Pelican using real world traces for location-aware mobile services and show that Pelican can substantially reduce privacy leakage by up to 75%.