Canonical Variates in Wasserstein Metric Space
Summary
Paper digest
What problem does the paper attempt to solve? Is this a new problem?
The paper aims to address the problem of developing a method called Canonical Variates in Wasserstein space (CVW) for classification based on pairwise distances, specifically focusing on clustering subjects with individual variability to ensure unique characteristics are not overlooked . This problem is not entirely new, as the paper builds upon existing methodologies and algorithms in the field of classification and clustering, such as Support Vector Machine (SVM), Random Forest (RF), and Logistic Regression (LR) . The novelty lies in the proposed CVW method and its variants, CVW-C and CVW-S, which utilize Wasserstein space and Gaussian Mixture Models (GMMs) to address the challenges associated with data pooling and individual subject clustering .
What scientific hypothesis does this paper seek to validate?
This paper aims to validate the scientific hypothesis related to the development and application of Canonical Variates in Wasserstein Metric Space for various scientific purposes, such as single-cell data integration, immune profile prediction, distance-based mixture modeling, and immunotherapy outcome prediction . The research explores topics like optimal transportation, Wasserstein distances, and statistical and machine learning methods for immunoprofiling based on single-cell data . Additionally, the paper delves into the analysis of pulmonary fibrosis data using single-nucleus sequencing and focuses on specific cell types to understand disease mechanisms and identify potential biomarkers . The experiments conducted in the paper involve classification based on pairwise distances, utilizing the kernel version of the pseudo-mixture model, to provide theoretical guarantees and investigate convergence characteristics .
What new ideas, methods, or models does the paper propose? What are the characteristics and advantages compared to previous methods?
The paper "Canonical Variates in Wasserstein Metric Space" introduces several innovative ideas, methods, and models related to data analysis and classification using Wasserstein distances and Gaussian Mixture Models (GMMs) . One key contribution is the concept of Minimized Aggregated Wasserstein (MAW) distance, which involves modeling continuous distributions with GMMs to compute Wasserstein distances efficiently . This approach allows for the calculation of Wasserstein distances between Gaussian distributions in a closed-form manner, eliminating the need for generating large samples from high-dimensional spaces . The MAW distance is particularly useful in scenarios with high-dimensional data, as it requires a low number of components to accurately model the data, ensuring computational efficiency .
Furthermore, the paper discusses the application of the MAW distance in clustering and classification tasks, specifically in the context of single-cell data analysis . By transforming data clouds into feature vectors via clustering and assigning coherent labels to clusters, attributes such as cluster proportions and mean values can be aggregated into feature vectors for analysis . This transformation from data clouds to vectors is beneficial for classification and clustering methods tailored for distributional data, with broader applications in model integration .
Moreover, the paper explores the use of Gaussian Mixture Models (GMMs) in defining the Wasserstein distance between distributions in a dimension-reduced space . This method, known as the Pseudo-Mixture Model (PMM), involves constructing feature vectors for each subject by concatenating cluster proportions and average values of variables within clusters . The study evaluates the performance of the PMM method in the original space and the dimension-reduced space, demonstrating its effectiveness in analyzing datasets through clustering and feature vector construction .
Overall, the paper presents novel approaches for data analysis, classification, and clustering using Wasserstein distances, Gaussian Mixture Models, and innovative distance metrics like the Minimized Aggregated Wasserstein (MAW) distance, offering valuable insights into the field of statistical and machine learning methods for immunoprofiling based on single-cell data . The paper "Canonical Variates in Wasserstein Metric Space" introduces novel approaches for data analysis and classification, offering distinct characteristics and advantages compared to previous methods .
Characteristics:
- The paper focuses on classifying instances characterized by distributions in a vector space using the Wasserstein metric .
- It employs distance-based classification algorithms like k-nearest neighbors, k-means, and pseudo-mixture modeling .
- The study emphasizes dimension reduction within the Wasserstein metric space to enhance classification accuracy, introducing discriminant coordinates or canonical variates axes .
- The approach optimizes Fisher's ratio, maximizing between-class variation to within-class variation, to identify discriminant directions .
- The paper utilizes Gaussian Mixture Models (GMMs) and the Minimized Aggregated Wasserstein (MAW) distance for efficient optimization .
Advantages:
- The proposed method significantly enhances classification performance compared to established algorithms operating on vector representations derived from distributional data .
- It demonstrates robustness against variations in the distributional representations of data clouds, ensuring reliable classification outcomes .
- By focusing on data clouds as distributions, the approach addresses the complexities of handling discrete or continuous distributions, offering a more tailored and effective classification strategy .
- The use of GMMs and the MAW distance facilitates efficient optimization algorithms, enhancing computational efficiency in data analysis and classification tasks .
- The paper's methodology provides a unique perspective on dimension reduction for data instances represented as distributions, offering a specialized approach to classification tailored to the inherent characteristics of distributional data .
In summary, the paper's innovative methods leverage the Wasserstein metric, GMMs, and novel distance metrics to enhance classification accuracy, computational efficiency, and robustness in handling distributional data, marking a significant advancement in data analysis and classification techniques .
Do any related researches exist? Who are the noteworthy researchers on this topic in this field?What is the key to the solution mentioned in the paper?
Several related research papers and notable researchers in the field of Wasserstein metric space and optimal transport have been identified:
- Noteworthy researchers in this field include Flamary, Cuturi, Courty, Rakotomamonjy, Furman, Jojic, Kidd, Shen-Orr, Price, Jarrell, Tse, Huang, Lund, Maecker, Utz, Dekker, Koller, Davis, Habermann, Gutierrez, Bui, Yahn, Winters, Calvi, Chung, Taylor, Jetter, Raju, Roberson, Ding, Wood, Sucre, Richmond, Serezani, McDonnell, Mallal, Bacchetta, Loyd, Shaver, Ware, Bremner, Walia, Blackwell, Banovich, Kropski, Benamou, Carlier, Nenna, Peyr´e, Bonneel, Rabin, Pfister, Brummelman, Haftmann, N´u˜nez, Alvisi, Mazza, Becher, Lugli, Chen, Ye, Li, Kolouri, Nadjahi, Simsekli, Badeau, Rohde, Korthauer, Chu, Newton, Deshpande, Hu, Sun, Pyrros, Siddiqui, Koyejo, Zhao, Forsyth, Schwing, Delon, Desolneux, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kleiner, Lyttle, Mendez, Mulder, Mendu, Oostenink, Rooker, Russo, Salimbangon, Saksena, Shin, Sominsky, Van Oekelen, Barcessat, Bhardwaj, Kim-Schulze, Gnjatic, Brown, Cordon-Cardo, Krammer, Merad, Jagannath, Wajnberg, Simon, Parekh, Yang, Toh, Ye, Lim, Zhang, Lin, Finak, Ushey, Seshadri, Hawn, Frahm, Scriba, Mahomed, Hanekom, Bart, Pantaleo, Tomaras, Rerks-Ngarm, Kaewkungwal, Nitayaphan, Pitisuttithum, Michael, Kim, Robb, O’Connell, Karasavvas, Gilbert, De Rosa, McElrath, Gottardo, Rubio, Everaert, Damme, Preter, Vermaelen, Stefanski, Rincon-Arevalo, Schrezenmeier, Karberg, Szelinski, Ritter, Jahrsd¨orfer, Ludwig, Lino, D¨orner, Villani, Wang, Shafto, Wu, Al-Eryani, Roden, Junankar, Harvey, Andersson, Thenna- van, Wang, Torpy, Bartonicek, Larsson, Kaczorowski, Weisenfeld, Uytingco, Chew, Bent, Chan, Gnanasam- bandapillai, Dutertre, Gluch, Hui, Beith, Parker, Robbins, Durante, Rodriguez, Kurtenbach, Kuznetsov, Sanchez, Decatur, Snyder, Feun, Livingstone, Harbour, Aleman, Upadhyaya, Tuballes, Kappes, Gleason, Beach, Agte, Srivastava, Andre, Azad, Banu, Berm´udez-Gonz´alez, Cai, Cognigni, David, Floda, Firpo, Kle
How were the experiments in the paper designed?
The experiments in the paper were designed to investigate the convergence characteristics of the algorithm through classification based on pairwise distances alone . The experiments involved employing the kernel version of the pseudo-mixture model developed by Qiao and Li (2016) for classification purposes . Different types of feature vectors, including cluster proportions and cluster means, were explored, and the best results from these three types were reported . Additionally, the experiments included analyzing single-nucleus sequencing data related to pulmonary fibrosis, focusing on SCG3A2+ cell types . The experiments aimed to assess the performance of the algorithm in various scenarios and datasets, providing a fair and effective comparison against widely-used classification techniques such as SVM, RF, and LR .
What is the dataset used for quantitative evaluation? Is the code open source?
The dataset used for quantitative evaluation in the study is comprised of GMMs created under different setups, with 8 datasets each containing GMMs generated under specific configurations . The code used in the study is not explicitly mentioned to be open source in the provided context.
Do the experiments and results in the paper provide good support for the scientific hypotheses that need to be verified? Please analyze.
The experiments and results presented in the paper provide substantial support for the scientific hypotheses that need to be verified. The research conducted includes various experiments and analyses that contribute to validating the hypotheses . The paper explores classification based on pairwise distances using a kernel version of the pseudo-mixture model, demonstrating a thorough investigation of the convergence characteristics of the algorithm through experiments . Additionally, the study delves into single-cell RNA sequencing data related to pulmonary fibrosis, focusing on specific cell types and analyzing gene expression patterns to gain insights into the disease .
Moreover, the paper references a wide range of related studies and methodologies, such as Wasserstein barycenters, optimal transport, and Gaussian mixture models, which collectively strengthen the scientific foundation of the research . The utilization of advanced statistical approaches and machine learning techniques in the analyses further enhances the credibility and robustness of the findings .
Overall, the comprehensive nature of the experiments, the incorporation of diverse datasets, and the application of sophisticated analytical methods collectively provide strong empirical support for the scientific hypotheses under investigation in the paper.
What are the contributions of this paper?
The paper "Canonical Variates in Wasserstein Metric Space" makes several contributions:
- It introduces the concept of multisource single-cell data integration using the MAW barycenter for Gaussian mixture models .
- The paper discusses the baseline immune profile by CyTOF for predicting the response to an investigational adjuvanted vaccine in elderly adults .
- It presents subspace robust Wasserstein distances for robust transportation problems .
- The paper explores distance-based mixture modeling for classification through hypothetical local mapping .
- It delves into the use of aggregated Wasserstein distance and state registration for hidden Markov models .
- The research introduces statistical and machine learning methods for immunoprofiling based on single-cell data .
- It discusses the efficient discretization of optimal transport in the context of Wasserstein metric space .
What work can be continued in depth?
To delve deeper into the research, several avenues can be explored further:
- Efficient discretization of optimal transport: Further investigation into the efficient discretization of optimal transport can provide insights into enhancing computational efficiency and accuracy in transportation problems .
- Max-sliced Wasserstein distance for GANs: Exploring the application and implications of the Max-sliced Wasserstein distance for Generative Adversarial Networks (GANs) can lead to advancements in GAN models and their performance .
- Multisource single-cell data integration: Delving into the MAW barycenter method for Gaussian mixture models can offer valuable insights into integrating data from multiple sources for enhanced analysis and interpretation .
- Subspace robust Wasserstein distances: Further research on subspace robust Wasserstein distances can contribute to developing more robust distance metrics for various applications, particularly in machine learning and data analysis .
- Distance-based mixture modeling: Exploring distance-based mixture modeling for classification via hypothetical local mapping can provide a deeper understanding of classification methods and their applications in statistical analysis and data mining .
- Single-cell analysis in uveal melanoma: Investigating the new evolutionary complexity revealed in uveal melanoma through single-cell analysis can lead to advancements in understanding the disease and potential treatment strategies .
- Circulating immune cell dynamics in immunotherapy: Further exploration of circulating immune cell dynamics as outcome predictors for immunotherapy in non-small cell lung cancer can contribute to improving immunotherapy outcomes and patient care .
- B cell characteristics predicting vaccination response: Researching how B cell characteristics at baseline can predict vaccination response in patients treated with RTX can provide valuable insights into personalized vaccination strategies and treatment outcomes .
- Topics in Optimal Transportation: Delving into the various topics in Optimal Transportation can offer a comprehensive understanding of this field and its applications in different domains .
- Fast and robust Earth Mover’s Distances: Further exploration of fast and robust Earth Mover’s Distances can lead to advancements in distance metrics and their efficient computation for various applications .