TY - GEN
T1 - Moderate Dimension Reduction for k-Center Clustering
AU - Jiang, Shaofeng H.C.
AU - Krauthgamer, Robert
AU - Sapir, Shay
N1 - Publisher Copyright: © Shaofeng H.-C. Jiang, Robert Krauthgamer, and Shay Sapir.
PY - 2024/6
Y1 - 2024/6
N2 - The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of n points in Rd and any fixed ϵ > 0, it reduces the dimension d to O(log n) while preserving, with high probability, all the pairwise Euclidean distances within factor 1 + ϵ. Perhaps surprisingly, the target dimension can be lower if one only wishes to preserve the optimal value of a certain problem on the pointset, e.g., Euclidean max-cut or k-means. However, for some notorious problems, like diameter (aka furthest pair), dimension reduction via the JL map to below O(log n) does not preserve the optimal value within factor 1 + ϵ. We propose to focus on another regime, of moderate dimension reduction, where a problem’s value is preserved within factor α > 1 using target dimension log n/poly(α). We establish the viability of this approach and show that the famous k-center problem is α-approximated when reducing to dimension O(logα2n + log k). Along the way, we address the diameter problem via the special case k = 1. Our result extends to several important variants of k-center (with outliers, capacities, or fairness constraints), and the bound improves further with the input’s doubling dimension. While our poly(α)-factor improvement in the dimension may seem small, it actually has significant implications for streaming algorithms, and easily yields an algorithm for k-center in dynamic geometric streams, that achieves O(α)-approximation using space poly(kdn1/α2 ). This is the first algorithm to beat O(n) space in high dimension d, as all previous algorithms require space at least exp(d). Furthermore, it extends to the k-center variants mentioned above.
AB - The Johnson-Lindenstrauss (JL) Lemma introduced the concept of dimension reduction via a random linear map, which has become a fundamental technique in many computational settings. For a set of n points in Rd and any fixed ϵ > 0, it reduces the dimension d to O(log n) while preserving, with high probability, all the pairwise Euclidean distances within factor 1 + ϵ. Perhaps surprisingly, the target dimension can be lower if one only wishes to preserve the optimal value of a certain problem on the pointset, e.g., Euclidean max-cut or k-means. However, for some notorious problems, like diameter (aka furthest pair), dimension reduction via the JL map to below O(log n) does not preserve the optimal value within factor 1 + ϵ. We propose to focus on another regime, of moderate dimension reduction, where a problem’s value is preserved within factor α > 1 using target dimension log n/poly(α). We establish the viability of this approach and show that the famous k-center problem is α-approximated when reducing to dimension O(logα2n + log k). Along the way, we address the diameter problem via the special case k = 1. Our result extends to several important variants of k-center (with outliers, capacities, or fairness constraints), and the bound improves further with the input’s doubling dimension. While our poly(α)-factor improvement in the dimension may seem small, it actually has significant implications for streaming algorithms, and easily yields an algorithm for k-center in dynamic geometric streams, that achieves O(α)-approximation using space poly(kdn1/α2 ). This is the first algorithm to beat O(n) space in high dimension d, as all previous algorithms require space at least exp(d). Furthermore, it extends to the k-center variants mentioned above.
UR - http://www.scopus.com/inward/record.url?scp=85195473820&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.SoCG.2024.64
DO - https://doi.org/10.4230/LIPIcs.SoCG.2024.64
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 40th International Symposium on Computational Geometry, SoCG 2024
A2 - Mulzer, Wolfgang
A2 - Phillips, Jeff M.
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 40th International Symposium on Computational Geometry, SoCG 2024
Y2 - 11 June 2024 through 14 June 2024
ER -