TY - GEN
T1 - Optimality of the Johnson-Lindenstrauss Dimensionality Reduction for Practical Measures
AU - Bartal, Yair
AU - Fandina, Ora Nova
AU - Larsen, Kasper Green
N1 - Publisher Copyright: © Yair Bartal, Ora Nova Fandina, and Kasper Green Larsen; licensed under Creative Commons License CC-BY 4.0
PY - 2022/6/1
Y1 - 2022/6/1
N2 - It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [8] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1 + ? average distortion for embedding into k-dimensional Euclidean space, where k = O(1/?2), and for more general q-norms of distortion, k = O(max(1/?2, q/?)), whereas tight lower bounds were established only for large values of q via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 = q = O(log(2??2n)) and ? = v1n, where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ? then k = ?(1/?2), for any ? = v1n, matching the upper bounds of [8] and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.
AB - It is well known that the Johnson-Lindenstrauss dimensionality reduction method is optimal for worst case distortion. While in practice many other methods and heuristics are used, not much is known in terms of bounds on their performance. The question of whether the JL method is optimal for practical measures of distortion was recently raised in [8] (NeurIPS'19). They provided upper bounds on its quality for a wide range of practical measures and showed that indeed these are best possible in many cases. Yet, some of the most important cases, including the fundamental case of average distortion were left open. In particular, they show that the JL transform has 1 + ? average distortion for embedding into k-dimensional Euclidean space, where k = O(1/?2), and for more general q-norms of distortion, k = O(max(1/?2, q/?)), whereas tight lower bounds were established only for large values of q via reduction to the worst case. In this paper we prove that these bounds are best possible for any dimensionality reduction method, for any 1 = q = O(log(2??2n)) and ? = v1n, where n is the size of the subset of Euclidean space. Our results also imply that the JL method is optimal for various distortion measures commonly used in practice, such as stress, energy and relative error. We prove that if any of these measures is bounded by ? then k = ?(1/?2), for any ? = v1n, matching the upper bounds of [8] and extending their tightness results for the full range moment analysis. Our results may indicate that the JL dimensionality reduction method should be considered more often in practical applications, and the bounds we provide for its quality should be served as a measure for comparison when evaluating the performance of other methods and heuristics.
KW - JL transform
KW - average distortion
KW - practical dimensionality reduction
UR - http://www.scopus.com/inward/record.url?scp=85134314072&partnerID=8YFLogxK
U2 - https://doi.org/10.4230/LIPIcs.SoCG.2022.13
DO - https://doi.org/10.4230/LIPIcs.SoCG.2022.13
M3 - منشور من مؤتمر
T3 - Leibniz International Proceedings in Informatics, LIPIcs
SP - 13:1-13:16
BT - 38th International Symposium on Computational Geometry, SoCG 2022
A2 - Goaoc, Xavier
A2 - Kerber, Michael
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 38th International Symposium on Computational Geometry, SoCG 2022
Y2 - 7 June 2022 through 10 June 2022
ER -