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This paper proposes and evaluates a sketching language to author crowd motion. It focuses on the path, speed, thickness, and density parameters of crowd motion. A sketch-based vocabulary is proposed for each parameter and evaluated in a user study against complex crowd scenes. A sketch recognition pipeline converts the sketches into a crowd simulation. The user study results show that 1) participants at various skill levels and can draw accurate crowd motion through sketching, 2) certain sketch styles lead to a more accurate representation of crowd parameters, and 3) sketching allows to produce complex crowd motions in a few seconds. The results show that some styles although accurate actually are less preferred over less accurate ones.
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Stochastic Gradient Langevin Dynamics with Variance Reduction
Sketching is a stochastic dimension reduction method that preserves geometric structures of data and has applications in high-dimensional regression, low rank approximation and graph sparsification. In this work, we show that sketching can be used to compress simulation data and still accurately estimate time autocorrelation and power spectral density. For a given compression ratio, the accuracy is much higher than using previously known methods. In addition to providing theoretical guarantees, we apply sketching to a molecular dynamics simulation of methanol and find that the estimate of spectral density is 90% accurate using only 10% of the data.
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- Award ID(s):
- 1819251
- PAR ID:
- 10284268
- Date Published:
- Journal Name:
- Proceedings of International Joint Conference on Neural Networks
- ISSN:
- 2161-4407
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Solving Rank Constrained Least Squares via Recursive Importance Sketching In statistics and machine learning, we sometimes run into the rank-constrained least squares problems, for which we need to find the best low-rank fit between sets of data, such as trying to figure out what factors are affecting the data, filling in missing information, or finding connections between different sets of data. This paper introduces a new method for solving this problem called the recursive importance sketching algorithm (RISRO), in which the central idea is to break the problem down into smaller, easier parts using a unique technique called “recursive importance sketching.” This new method is not only easy to use, but it is also very efficient and gives accurate results. We prove that RISRO converges in a local quadratic-linear and quadratic rate under some mild conditions. Simulation studies also demonstrate the superior performance of RISRO.more » « less
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Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.more » « less
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Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.more » « less
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Recently, a number of variants of the notion of cut-preserving hypergraph sparsification have been studied in the literature. These variants include directed hypergraph sparsification, submodular hypergraph sparsification, general notions of approximation including spectral approximations, and more general notions like sketching that can answer cut queries using more general data structures than just sparsifiers. In this work, we provide reductions between these different variants of hypergraph sparsification and establish new upper and lower bounds on the space complexity of preserving their cuts. Specifically, we show that: 1) (1 ± ε) directed hypergraph spectral (respectively cut) sparsification on n vertices efficiently reduces to (1 ± ε) undirected hypergraph spectral (respectively cut) sparsification on n² + 1 vertices. Using the work of Lee and Jambulapati, Liu, and Sidford (STOC 2023) this gives us directed hypergraph spectral sparsifiers with O(n² log²(n) / ε²) hyperedges and directed hypergraph cut sparsifiers with O(n² log(n)/ ε²) hyperedges by using the work of Chen, Khanna, and Nagda (FOCS 2020), both of which improve upon the work of Oko, Sakaue, and Tanigawa (ICALP 2023). 2) Any cut sketching scheme which preserves all cuts in any directed hypergraph on n vertices to a (1 ± ε) factor (for ε = 1/(2^{O(√{log(n)})})) must have worst-case bit complexity n^{3 - o(1)}. Because directed hypergraphs are a subclass of submodular hypergraphs, this also shows a worst-case sketching lower bound of n^{3 - o(1)} bits for sketching cuts in general submodular hypergraphs. 3) (1 ± ε) monotone submodular hypergraph cut sparsification on n vertices efficiently reduces to (1 ± ε) symmetric submodular hypergraph sparsification on n+1 vertices. Using the work of Jambulapati et. al. (FOCS 2023) this gives us monotone submodular hypergraph sparsifiers with Õ(n / ε²) hyperedges, improving on the O(n³ / ε²) hyperedge bound of Kenneth and Krauthgamer (arxiv 2023). At a high level, our results use the same general principle, namely, by showing that cuts in one class of hypergraphs can be simulated by cuts in a simpler class of hypergraphs, we can leverage sparsification results for the simpler class of hypergraphs.more » « less
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