Mixed-Integer Convex Nonlinear Optimization with Gradient-Boosted Trees Embedded

Decision trees usefully represent sparse, high dimensional and noisy data. Having learned a function from this data, we may want to thereafter integrate the function into a larger decision-making problem, e.g., for picking the best chemical process catalyst. We study a large-scale, industrially-relevant mixed-integer nonlinear nonconvex optimization problem involving both gradient-boosted trees and penalty functions mitigating risk. This mixed-integer optimization problem with convex penalty terms broadly applies to optimizing pre-trained regression tree models. Decision makers may wish to optimize discrete models to repurpose legacy predictive models, or they may wish to optimize a discrete model that particularly well-represents a data set. We develop several heuristic methods to find feasible solutions, and an exact, branch-and-bound algorithm leveraging structural properties of the gradient-boosted trees and penalty functions. We computationally test our methods on concrete mixture design instance and a chemical catalysis industrial instance

Manual measurement of retinal bifurcation features

This paper introduces a new computerized tool for accurate manual measurement of features of retinal bifurcation geometry, designed for use in investigating correlations between measurement features and clinical conditions. The tool uses user-placed rectangles to measure the vessel width, and lines placed along vessel center lines to measure the angles. An analysis is presented of measurements taken from 435 bifurcations. These are compared with theoretical predictions based on optimality principles presented in the literature. The new tool shows better agreement with the theoretical predictions than a simpler manual method published in the literature, but there remains a significant discrepancy between current theory and measured geometry

Clear Visual Separation of Temporal Event Sequences

Extracting and visualizing informative insights from temporal event sequences becomes increasingly difficult when data volume and variety increase. Besides dealing with high event type cardinality and many distinct sequences, it can be difficult to tell whether it is appropriate to combine multiple events into one or utilize additional information about event attributes. Existing approaches often make use of frequent sequential patterns extracted from the dataset, however, these patterns are limited in terms of interpretability and utility. In addition, it is difficult to assess the role of absolute and relative time when using pattern mining techniques. In this paper, we present methods that addresses these challenges by automatically learning composite events which enables better aggregation of multiple event sequences. By leveraging event sequence outcomes, we present appropriate linked visualizations that allow domain experts to identify critical flows, to assess validity and to understand the role of time. Furthermore, we explore information gain and visual complexity metrics to identify the most relevant visual patterns. We compare composite event learning with two approaches for extracting event patterns using real world company event data from an ongoing project with the Danish Business Authority.Comment: In Proceedings of the 3rd IEEE Symposium on Visualization in Data Science (VDS), 201

Data complexity measured by principal graphs

How to measure the complexity of a finite set of vectors embedded in a multidimensional space? This is a non-trivial question which can be approached in many different ways. Here we suggest a set of data complexity measures using universal approximators, principal cubic complexes. Principal cubic complexes generalise the notion of principal manifolds for datasets with non-trivial topologies. The type of the principal cubic complex is determined by its dimension and a grammar of elementary graph transformations. The simplest grammar produces principal trees. We introduce three natural types of data complexity: 1) geometric (deviation of the data's approximator from some "idealized" configuration, such as deviation from harmonicity); 2) structural (how many elements of a principal graph are needed to approximate the data), and 3) construction complexity (how many applications of elementary graph transformations are needed to construct the principal object starting from the simplest one). We compute these measures for several simulated and real-life data distributions and show them in the "accuracy-complexity" plots, helping to optimize the accuracy/complexity ratio. We discuss various issues connected with measuring data complexity. Software for computing data complexity measures from principal cubic complexes is provided as well.Comment: Computers and Mathematics with Applications, in pres

Element Distinctness, Frequency Moments, and Sliding Windows

We derive new time-space tradeoff lower bounds and algorithms for exactly computing statistics of input data, including frequency moments, element distinctness, and order statistics, that are simple to calculate for sorted data. We develop a randomized algorithm for the element distinctness problem whose time T and space S satisfy T in O (n^{3/2}/S^{1/2}), smaller than previous lower bounds for comparison-based algorithms, showing that element distinctness is strictly easier than sorting for randomized branching programs. This algorithm is based on a new time and space efficient algorithm for finding all collisions of a function f from a finite set to itself that are reachable by iterating f from a given set of starting points. We further show that our element distinctness algorithm can be extended at only a polylogarithmic factor cost to solve the element distinctness problem over sliding windows, where the task is to take an input of length 2n-1 and produce an output for each window of length n, giving n outputs in total. In contrast, we show a time-space tradeoff lower bound of T in Omega(n^2/S) for randomized branching programs to compute the number of distinct elements over sliding windows. The same lower bound holds for computing the low-order bit of F_0 and computing any frequency moment F_k, k neq 1. This shows that those frequency moments and the decision problem F_0 mod 2 are strictly harder than element distinctness. We complement this lower bound with a T in O(n^2/S) comparison-based deterministic RAM algorithm for exactly computing F_k over sliding windows, nearly matching both our lower bound for the sliding-window version and the comparison-based lower bounds for the single-window version. We further exhibit a quantum algorithm for F_0 over sliding windows with T in O(n^{3/2}/S^{1/2}). Finally, we consider the computations of order statistics over sliding windows.Comment: arXiv admin note: substantial text overlap with arXiv:1212.437