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1. Sparse Approximation via Generating Point Sets

   https://doi.org/10.1137/1.9781611974331.ch40  

   Blum, Avrim; Har-Peled, Sariel; Raichel, Benjamin (December 2015, Proceedings of the Twenty-Seventh Annual ACM-SIAM Symposium on Discrete Algorithms (SODA2016))

   For a set P of n points in the unit ball b ⊆ R d , consider the problem of finding a small subset T ⊆ P such that its convex-hull ε-approximates the convex-hull of the original set. Specifically, the Hausdorff distance between the convex hull of T and the convex hull of P should be at most ε. We present an efficient algorithm to compute such an ε ′ -approximation of size kalg, where ε ′ is a function of ε, and kalg is a function of the minimum size kopt of such an ε-approximation. Surprisingly, there is no dependence on the dimension d in either of the bounds. Furthermore, every point of P can be ε- approximated by a convex-combination of points of T that is O(1/ε2 )-sparse. Our result can be viewed as a method for sparse, convex autoencoding: approximately representing the data in a compact way using sparse combinations of a small subset T of the original data. The new algorithm can be kernelized, and it preserves sparsity in the original input. 

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2. DeepHull: Fast Convex Hull Approximation in High Dimensions

   https://doi.org/10.1109/ICASSP43922.2022.9746031  

   Balestriero, Randall; Wang, Zichao; Baraniuk, Richard G. (May 2022, ICASSP 2022 - 2022 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP))

   Computing or approximating the convex hull of a dataset plays a role in a wide range of applications, including economics, statistics, and physics, to name just a few. However, convex hull computation and approximation is exponentially complex, in terms of both memory and computation, as the ambient space dimension increases. In this paper, we propose DeepHull, a new convex hull approximation algorithm based on convex deep networks (DNs) with continuous piecewise-affine nonlinearities and nonnegative weights. The idea is that binary classification between true data samples and adversarially generated samples with such a DN naturally induces a polytope decision boundary that approximates the true data convex hull. A range of exploratory experiments demonstrates that DeepHull efficiently produces a meaningful convex hull approximation, even in a high-dimensional ambient space. 

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3. Convexification of Bilinear Terms over Network Polytopes

   https://doi.org/10.1287/moor.2023.0001  

   Khademnia, Erfan; Davarnia, Danial (April 2024, Mathematics of Operations Research)

   It is well-known that the McCormick relaxation for the bilinear constraint z = xy gives the convex hull over the box domains for x and y. In network applications where the domain of bilinear variables is described by a network polytope, the McCormick relaxation, also referred to as linearization, fails to provide the convex hull and often leads to poor dual bounds. We study the convex hull of the set containing bilinear constraints [Formula: see text] where x_irepresents the arc-flow variable in a network polytope, and y_jis in a simplex. For the case where the simplex contains a single y variable, we introduce a systematic procedure to obtain the convex hull of the above set in the original space of variables, and show that all facet-defining inequalities of the convex hull can be obtained explicitly through identifying a special tree structure in the underlying network. For the generalization where the simplex contains multiple y variables, we design a constructive procedure to obtain an important class of facet-defining inequalities for the convex hull of the underlying bilinear set that is characterized by a special forest structure in the underlying network. Computational experiments conducted on different applications show the effectiveness of the proposed methods in improving the dual bounds obtained from alternative techniques. Funding: This work was supported by Air Force Office of Scientific Research [Grant FA9550-23-1-0183]; National Science Foundation, Division of Civil, Mechanical and Manufacturing Innovation [Grant 2338641]. Supplemental Material: The online appendix is available at https://doi.org/10.1287/moor.2023.0001 . 

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4. Probabilistic Prediction of Material Stability: Integrating Convex Hulls into Active Learning

   Novick, Andrew; Cai, Diana; Nguyen, Quan; Garnett, Roman; Adams, Ryan; Toberer Eric (February 2024, arXivorg)

   Active learning is a valuable tool for efficiently exploring complex spaces, finding a variety of uses in materials science. However, the determination of convex hulls for phase diagrams does not neatly fit into traditional active learning approaches due to their global nature. Specifically, the thermodynamic stability of a material is not simply a function of its own energy, but rather requires energetic information from all other competing compositions and phases. Here we present Convex hull-aware Active Learning (CAL), a novel Bayesian algorithm that chooses experiments to minimize the uncertainty in the convex hull. CAL prioritizes compositions that are close to or on the hull, leaving significant uncertainty in other compositions that are quickly determined to be irrelevant to the convex hull. The convex hull can thus be predicted with significantly fewer observations than approaches that focus solely on energy. Intrinsic to this Bayesian approach is uncertainty quantification in both the convex hull and all subsequent predictions (e.g., stability and chemical potential). By providing increased search efficiency and uncertainty quantification, CAL can be readily incorporated into the emerging paradigm of uncertainty-based workflows for thermodynamic prediction. 

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5. Dynamic Convex Hulls Under Window-Sliding Updates

   https://doi.org/10.1007/978-3-031-38906-1_46  

   Wang, Haitao (August 2023, Proceedings of the 18th Algorithms and Data Structures Symposium (WADS 2023))

   We consider the problem of dynamically maintaining the convex hull of a set S of points in the plane under the following special sequence of insertions and deletions (called window-sliding updates): insert a point to the right of all points of S and delete the leftmost point of S. We propose an O(|S|)-space data structure that can handle each update in O(1) amortized time, such that all standard binary-search-based queries on the convex hull of S can be answered in 𝑂(log |S|) time, and the convex hull itself can be output in time linear in its size. 

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