arXiv subjects
Publications and source records attributed to David Jensen.
We produce new combinatorial methods for approaching the tropical maximal rank conjecture, including inductive procedures for deducing new cases of the conjecture on graphs of increasing genus from any given case. Using explicit calculations in a range of base cases, we prove this conjecture for the canonical divisor, and in a wide range of cases for m=3, extending previous results for m=2.
We provide lower bounds on the gonality of a graph in terms of its spectral and edge expansion. As a consequence, we see that the gonality of a random 3-regular graph is asymptotically almost surely greater than one seventh its genus.
The predominant method for evaluating the quality of causal models is to measure the graphical accuracy of the learned model structure. We present an alternative method for evaluating causal models that directly measures the accuracy of estimated interventional distributions. We contrast such distributional measures with structural measures, such as structural Hamming distance and structural intervention distance, showing that structural measures often correspond poorly to the accuracy of estimated interventional distributions. We use a number of real and synthetic datasets to illustrate various scenarios in which structural measures provide misleading results with respect to algorithm selection and parameter tuning, and we recommend that distributional measures become the new standard for evaluating causal models.
Yield and quality improvement is of paramount importance to any manufacturing company. One of the ways of improving yield is through discovery of the root causal factors affecting yield. We propose the use of data-driven interpretable causal models to identify key factors affecting yield. We focus on factors that are measured in different stages of production and testing in the manufacturing cycle of a product. We apply causal structure learning techniques on real data collected from this line. Specifically, the goal of this work is to learn interpretable causal models from observational data produced by manufacturing lines. Emphasis has been given to the interpretability of the models to make them actionable in the field of manufacturing. We highlight the challenges presented by assembly line data and propose ways to alleviate them.We also identify unique characteristics of data originating from assembly lines and how to leverage them in order to improve causal discovery. Standard evaluation techniques for causal structure learning shows that the learned causal models seem to closely represent the underlying latent causal relationship between different factors in the production process. These results were also validated by manufacturing domain experts who found them promising. This work demonstrates how data mining and knowledge discovery can be used for root cause analysis in the domain of manufacturing and connected industry.
We study the Cox ring and monoid of effective divisor classes of $\overline{M}_{0,n} = Bl\mathbb{P}^{n-3}$, over a ring R. We provide a bijection between elements of the Cox ring, not divisible by any exceptional divisor section, and pure-dimensional singular simplicial complexes on {1,...,n-1} with nonzero weights in R satisfying a zero-tension condition. This leads to a combinatorial criterion, satisfied by many triangulations of closed manifolds, for a divisor class to be among the minimal generators for the effective monoid. For classes obtained as the strict transform of quadrics, we present a complete classification of minimal generators, generalizing to all n the well-known Keel-Vermeire classes for n=6. We use this classification to construct new divisors with interesting properties for all n > 6.
We provide an example of a trivalent, 3-connected graph G such that, for any choice of metric on G, the resulting metric graph is Brill-Noether special.
We use geometric invariant theory (GIT) to construct a large class of compactifications of the moduli space M_{0,n}. These compactifications include many previously known examples, as well as many new ones. As a consequence of our GIT approach, we exhibit explicit flips and divisorial contractions between these spaces.
In this survey, we discuss linear series on tropical curves and their relation to classical algebraic geometry, describe the main techniques of the subject, and survey some of the recent major developments in the field, with an emphasis on applications to problems in Brill-Noether theory and arithmetic geometry.
To any graph $G$ one can associate a toric variety $X(\mathcal{P}G)$, obtained as a blowup of projective space along coordinate subspaces corresponding to connected subgraphs of $G$. The polytope of this toric variety is the graph associahedron of $G$, a class of polytopes that includes the permutohedron, associahedron, and stellahedron. We show that the space $X(\mathcal{P}{G})$ is isomorphic to a Hassett compactification of $M_{0,n}$ precisely when $G$ is an iterated cone over a discrete set. This may be viewed as a generalization of the well-known fact that the Losev--Manin moduli space is isomorphic to the toric variety associated to the permutohedron.
Building on our earlier results on tropical independence and shapes of divisors in tropical linear series, we give a tropical proof of the maximal rank conjecture for quadrics. We also prove a tropical analogue of Max Noether's theorem on quadrics containing a canonically embedded curve, and state a combinatorial conjecture about tropical independence on chains of loops that implies the maximal rank conjecture for algebraic curves.
With the proliferation of network data, researchers are increasingly focusing on questions investigating phenomena occurring on networks. This often includes analysis of peer-effects, i.e., how the connections of an individual affect that individual's behavior. This type of influence is not limited to direct connections of an individual (such as friends), but also to individuals that are connected through longer paths (for example, friends of friends, or friends of friends of friends). In this work, we identify an ambiguity in the definition of what constitutes the extended neighborhood of an individual. This ambiguity gives rise to different semantics and supports different types of underlying phenomena. We present experimental results, both on synthetic and real networks, that quantify differences among the sets of extended neighbors under different semantics. Finally, we provide experimental evidence that demonstrates how the use of different semantics affects model selection.
We investigate the problem of learning to generate complex networks from data. Specifically, we consider whether deep belief networks, dependency networks, and members of the exponential random graph family can learn to generate networks whose complex behavior is consistent with a set of input examples. We find that the deep model is able to capture the complex behavior of small networks, but that no model is able capture this behavior for networks with more than a handful of nodes.
We develop a framework to apply tropical and nonarchimedean analytic techniques to multiplication maps on linear series and study degenerations of these multiplications maps when the special fiber is not of compact type. As an application, we give a tropical criterion for a curve over a valued field to be Gieseker-Petri general.
Let C be a curve over a complete valued field with infinite residue field whose skeleton is a chain of loops with generic edge lengths. We prove that any divisor on the chain of loops that is rational over the value group lifts to a divisor of the same rank on C, confirming a conjecture of Cools, Draisma, Robeva, and the third author.
We show that the expected gonality of a random graph is asymptotic to the number of vertices.
Recommendation systems for online dating have recently attracted much attention from the research community. In this paper we proposed a two-side matching framework for online dating recommendations and design an LDA model to learn the user preferences from the observed user messaging behavior and user profile features. Experimental results using data from a large online dating website shows that two-sided matching improves significantly the rate of successful matches by as much as 45%. Finally, using simulated matchings we show that the the LDA model can correctly capture user preferences.
We extend the theory of d-separation to cases in which data instances are not independent and identically distributed. We show that applying the rules of d-separation directly to the structure of probabilistic models of relational data inaccurately infers conditional independence. We introduce relational d-separation, a theory for deriving conditional independence facts from relational models. We provide a new representation, the abstract ground graph, that enables a sound, complete, and computationally efficient method for answering d-separation queries about relational models, and we present empirical results that demonstrate effectiveness.
The PC algorithm learns maximally oriented causal Bayesian networks. However, there is no equivalent complete algorithm for learning the structure of relational models, a more expressive generalization of Bayesian networks. Recent developments in the theory and representation of relational models support lifted reasoning about conditional independence. This enables a powerful constraint for orienting bivariate dependencies and forms the basis of a new algorithm for learning structure. We present the relational causal discovery (RCD) algorithm that learns causal relational models. We prove that RCD is sound and complete, and we present empirical results that demonstrate effectiveness.