Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances beten two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. Hover, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric.#R##N##R##N#  take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3.  prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as ll as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2.
['Alexandr Andoni', 'Assaf Naor', 'Ofer Neiman']
I am writing a research paper on a idea Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost
[{"from": "human", "value": "I am writing a research paper on a idea Impossibility of Sketching of the 3D Transportation Metric with Quadratic Cost"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Transportation cost metrics, also known as the Wasserstein distances W_p, are a natural choice for defining distances beten two pointsets, or distributions, and have been applied in numerous fields. From the computational perspective, there has been an intensive research effort for understanding the W_p metrics over R^k, with work on the W_1 metric (a.k.a earth mover distance) being most successful in terms of theoretical guarantees. Hover, the W_2 metric, also known as the root-mean square (RMS) bipartite matching distance, is often a more suitable choice in many application areas, e.g. in graphics. Yet, the geometry of this metric space is currently poorly understood, and efficient algorithms have been elusive. For example, there are no known non-trivial algorithms for nearest-neighbor search or sketching for this metric.#R##N##R##N#  take the first step towards explaining the lack of efficient algorithms for the W_2 metric, even over the three-dimensional Euclidean space R^3.  prove that there are no meaningful embeddings of W_2 over R^3 into a wide class of normed spaces, as ll as that there are no efficient sketching algorithms for W_2 over R^3 achieving constant approximation. For example, our results imply that: 1) any embedding into L1 must incur a distortion of Omega(sqrt(log(n))) for pointsets of size n equipped with the W_2 metric; and 2) any sketching algorithm of size s must incur Omega(sqrt(log(n))/sqrt(s)) approximation. Our results follow from a more general statement, asserting that W_2 over R^3 contains the 1/2-snowflake of all finite metric spaces with a uniformly bounded distortion. These are the first non-embeddability/non-sketchability results for W_2."}]