The Experts below are selected from a list of 160410 Experts worldwide ranked by ideXlab platform
Fanyana Ncongwane - One of the best experts on this subject based on the ideXlab platform.
-
On the bicompletion of a partial quasi-Metric Space and $$T_{0}$$T0 -quasi-Metric Spaces
Afrika Matematika, 2020Co-Authors: Seithuti P. Moshokoa, Fanyana NcongwaneAbstract:The purpose of the paper is to extend the completion theory of a partial Metric Space to the context of an asymMetric setup, namely, a partial quasi-Metric Space. We present two bicompletions of a partial quasi-Metric Space by appealing to the associated partial Metric Space on one hand and while on the other hand an associated $$T_{0}$$ -quasi-Metric Space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-Metric Spaces considered is restricted to the class of $$T_{0}$$ -quasi-Metric Spaces.
-
On the bicompletion of a partial quasi-Metric Space and $$T_{0}$$ T 0 -quasi-Metric Spaces
Afrika Matematika, 2020Co-Authors: Seithuti P. Moshokoa, Fanyana NcongwaneAbstract:The purpose of the paper is to extend the completion theory of a partial Metric Space to the context of an asymMetric setup, namely, a partial quasi-Metric Space. We present two bicompletions of a partial quasi-Metric Space by appealing to the associated partial Metric Space on one hand and while on the other hand an associated $$T_{0}$$ T 0 -quasi-Metric Space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-Metric Spaces considered is restricted to the class of $$T_{0}$$ T 0 -quasi-Metric Spaces.
Müller Hans-georg - One of the best experts on this subject based on the ideXlab platform.
-
Total Variation Regularized Fr\'echet Regression for Metric-Space Valued Data
2021Co-Authors: Lin Zhenhua, Müller Hans-georgAbstract:Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparaMetric Fr\'echet regression, which refers to a regression setting where a response residing in a Metric Space is paired with a scalar predictor and the target is a conditional Fr\'echet mean. Specifically, we seek to approximate an unknown Metric-Space valued function by an estimator that minimizes the Fr\'echet version of least squares and at the same time has small total variation, appropriately defined for Metric-Space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for Metric data objects that reside in Hadamard Spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including Metric Spaces of symMetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera--Holmes--Vogtmann Metric.Comment: 31 pages, 3 figure
-
Total Variation Regularized Fr\'echet Regression for Metric-Space Valued Data
2020Co-Authors: Lin Zhenhua, Müller Hans-georgAbstract:Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparaMetric Fr\'echet regression, which refers to a regression setting where a response residing in a generic Metric Space is paired with a scalar predictor and the target is a conditional Fr\'echet mean. Specifically, we seek to approximate an unknown Metric-Space valued function by an estimator that minimizes the Fr\'echet version of least squares and at the same time has small total variation, appropriately defined for Metric-Space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for Metric data objects that reside in Hadamard Spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including Metric Spaces of symMetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera--Holmes--Vogtmann Metric.Comment: 44 pages, 5 figure
Jingming Zhu - One of the best experts on this subject based on the ideXlab platform.
-
a Metric Space with its transfinite asymptotic dimension ω 1
Topology and its Applications, 2020Co-Authors: Jingming ZhuAbstract:Abstract We construct a Metric Space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both ω + 1 , where ω is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.
-
A Metric Space with its transfinite asymptotic dimension ω + 1
Topology and its Applications, 2020Co-Authors: Jingming ZhuAbstract:Abstract We construct a Metric Space whose transfinite asymptotic dimension and complementary-finite asymptotic dimension are both ω + 1 , where ω is the smallest infinite ordinal number. Therefore, we prove that the omega conjecture is not true.
Seithuti P. Moshokoa - One of the best experts on this subject based on the ideXlab platform.
-
On the bicompletion of a partial quasi-Metric Space and $$T_{0}$$T0 -quasi-Metric Spaces
Afrika Matematika, 2020Co-Authors: Seithuti P. Moshokoa, Fanyana NcongwaneAbstract:The purpose of the paper is to extend the completion theory of a partial Metric Space to the context of an asymMetric setup, namely, a partial quasi-Metric Space. We present two bicompletions of a partial quasi-Metric Space by appealing to the associated partial Metric Space on one hand and while on the other hand an associated $$T_{0}$$ -quasi-Metric Space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-Metric Spaces considered is restricted to the class of $$T_{0}$$ -quasi-Metric Spaces.
-
On the bicompletion of a partial quasi-Metric Space and $$T_{0}$$ T 0 -quasi-Metric Spaces
Afrika Matematika, 2020Co-Authors: Seithuti P. Moshokoa, Fanyana NcongwaneAbstract:The purpose of the paper is to extend the completion theory of a partial Metric Space to the context of an asymMetric setup, namely, a partial quasi-Metric Space. We present two bicompletions of a partial quasi-Metric Space by appealing to the associated partial Metric Space on one hand and while on the other hand an associated $$T_{0}$$ T 0 -quasi-Metric Space is utilized. The two bicompletions are not necessarily the same but they coincide once the class of partial quasi-Metric Spaces considered is restricted to the class of $$T_{0}$$ T 0 -quasi-Metric Spaces.
Lin Zhenhua - One of the best experts on this subject based on the ideXlab platform.
-
Total Variation Regularized Fr\'echet Regression for Metric-Space Valued Data
2021Co-Authors: Lin Zhenhua, Müller Hans-georgAbstract:Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparaMetric Fr\'echet regression, which refers to a regression setting where a response residing in a Metric Space is paired with a scalar predictor and the target is a conditional Fr\'echet mean. Specifically, we seek to approximate an unknown Metric-Space valued function by an estimator that minimizes the Fr\'echet version of least squares and at the same time has small total variation, appropriately defined for Metric-Space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for Metric data objects that reside in Hadamard Spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including Metric Spaces of symMetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera--Holmes--Vogtmann Metric.Comment: 31 pages, 3 figure
-
Total Variation Regularized Fr\'echet Regression for Metric-Space Valued Data
2020Co-Authors: Lin Zhenhua, Müller Hans-georgAbstract:Non-Euclidean data that are indexed with a scalar predictor such as time are increasingly encountered in data applications, while statistical methodology and theory for such random objects are not well developed yet. To address the need for new methodology in this area, we develop a total variation regularization technique for nonparaMetric Fr\'echet regression, which refers to a regression setting where a response residing in a generic Metric Space is paired with a scalar predictor and the target is a conditional Fr\'echet mean. Specifically, we seek to approximate an unknown Metric-Space valued function by an estimator that minimizes the Fr\'echet version of least squares and at the same time has small total variation, appropriately defined for Metric-Space valued objects. We show that the resulting estimator is representable by a piece-wise constant function and establish the minimax convergence rate of the proposed estimator for Metric data objects that reside in Hadamard Spaces. We illustrate the numerical performance of the proposed method for both simulated and real data, including Metric Spaces of symMetric positive-definite matrices with the affine-invariant distance, of probability distributions on the real line with the Wasserstein distance, and of phylogenetic trees with the Billera--Holmes--Vogtmann Metric.Comment: 44 pages, 5 figure
