core-set: summary and open problems
TRANSCRIPT
![Page 1: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/1.jpg)
Core-set: Summary and Open Problems
Hu Ding
Computer Science and Engineering, Michigan State University
![Page 2: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/2.jpg)
Clustering in High Dimension
• K-means/median clustering: use core-set and JL-transform,reduce the number of points from n to roughlyO(k2(log n)2/ε4). “On Coresets for k-Median and k-MeansClustering in Metric and Euclidean Spaces and Their Applications”by Chen.
• Streaming model: a very similar idea to dynamically updatinghash table:
• construct a sequence of buckets with doubling capacities,where only the first one stores the new arriving data, andothers store core-sets.
• The basic idea in Chen’s paper is partition+uniform sampling.It was improved by a more sophisticated adaptive samplingstrategy. “A unified framework for approximating andclustering data” by Feldman and Langberg.
![Page 3: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/3.jpg)
Clustering in High Dimension
• K-means/median clustering: use core-set and JL-transform,reduce the number of points from n to roughlyO(k2(log n)2/ε4). “On Coresets for k-Median and k-MeansClustering in Metric and Euclidean Spaces and Their Applications”by Chen.
• Streaming model: a very similar idea to dynamically updatinghash table:
• construct a sequence of buckets with doubling capacities,where only the first one stores the new arriving data, andothers store core-sets.
• The basic idea in Chen’s paper is partition+uniform sampling.It was improved by a more sophisticated adaptive samplingstrategy. “A unified framework for approximating andclustering data” by Feldman and Langberg.
![Page 4: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/4.jpg)
Clustering in High Dimension
• K-means/median clustering: use core-set and JL-transform,reduce the number of points from n to roughlyO(k2(log n)2/ε4). “On Coresets for k-Median and k-MeansClustering in Metric and Euclidean Spaces and Their Applications”by Chen.
• Streaming model: a very similar idea to dynamically updatinghash table:
• construct a sequence of buckets with doubling capacities,where only the first one stores the new arriving data, andothers store core-sets.
• The basic idea in Chen’s paper is partition+uniform sampling.It was improved by a more sophisticated adaptive samplingstrategy. “A unified framework for approximating andclustering data” by Feldman and Langberg.
![Page 5: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/5.jpg)
Clustering in High Dimension
• K-means/median clustering: use core-set and JL-transform,reduce the number of points from n to roughlyO(k2(log n)2/ε4). “On Coresets for k-Median and k-MeansClustering in Metric and Euclidean Spaces and Their Applications”by Chen.
• Streaming model: a very similar idea to dynamically updatinghash table:
• construct a sequence of buckets with doubling capacities,where only the first one stores the new arriving data, andothers store core-sets.
• The basic idea in Chen’s paper is partition+uniform sampling.It was improved by a more sophisticated adaptive samplingstrategy. “A unified framework for approximating andclustering data” by Feldman and Langberg.
![Page 6: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/6.jpg)
Clustering in High Dimension (cont.)
• Projective clustering: instead of k cluster centers, it seeks kj-dimensional flats (shifted subspaces) to minimize theclustering cost.
• This problem is extremely hard, imagine the generalization forboth k-means/median and PCA in high dimension.
• (1 + ε, log n)-bicriteria approximation: resulting in at most(1 + ε) times the optimal clustering cost and k log n flats.“Bi-criteria linear-time approximations for generalizedk-mean/median/center” by Feldman et al.
• The basic idea is about uniform sampling + peeling.• If k is constant, there exists (1 + ε)-approximation. “A unified
framework for approximating and clustering data” by Feldmanand Langberg.
![Page 7: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/7.jpg)
Clustering in High Dimension (cont.)
• Projective clustering: instead of k cluster centers, it seeks kj-dimensional flats (shifted subspaces) to minimize theclustering cost.
• This problem is extremely hard, imagine the generalization forboth k-means/median and PCA in high dimension.
• (1 + ε, log n)-bicriteria approximation: resulting in at most(1 + ε) times the optimal clustering cost and k log n flats.“Bi-criteria linear-time approximations for generalizedk-mean/median/center” by Feldman et al.
• The basic idea is about uniform sampling + peeling.• If k is constant, there exists (1 + ε)-approximation. “A unified
framework for approximating and clustering data” by Feldmanand Langberg.
![Page 8: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/8.jpg)
Clustering in High Dimension (cont.)
• Projective clustering: instead of k cluster centers, it seeks kj-dimensional flats (shifted subspaces) to minimize theclustering cost.
• This problem is extremely hard, imagine the generalization forboth k-means/median and PCA in high dimension.
• (1 + ε, log n)-bicriteria approximation: resulting in at most(1 + ε) times the optimal clustering cost and k log n flats.“Bi-criteria linear-time approximations for generalizedk-mean/median/center” by Feldman et al.
• The basic idea is about uniform sampling + peeling.• If k is constant, there exists (1 + ε)-approximation. “A unified
framework for approximating and clustering data” by Feldmanand Langberg.
![Page 9: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/9.jpg)
Clustering in High Dimension (cont.)
• Projective clustering: instead of k cluster centers, it seeks kj-dimensional flats (shifted subspaces) to minimize theclustering cost.
• This problem is extremely hard, imagine the generalization forboth k-means/median and PCA in high dimension.
• (1 + ε, log n)-bicriteria approximation: resulting in at most(1 + ε) times the optimal clustering cost and k log n flats.“Bi-criteria linear-time approximations for generalizedk-mean/median/center” by Feldman et al.
• The basic idea is about uniform sampling + peeling.
• If k is constant, there exists (1 + ε)-approximation. “A unifiedframework for approximating and clustering data” by Feldmanand Langberg.
![Page 10: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/10.jpg)
Clustering in High Dimension (cont.)
• Projective clustering: instead of k cluster centers, it seeks kj-dimensional flats (shifted subspaces) to minimize theclustering cost.
• This problem is extremely hard, imagine the generalization forboth k-means/median and PCA in high dimension.
• (1 + ε, log n)-bicriteria approximation: resulting in at most(1 + ε) times the optimal clustering cost and k log n flats.“Bi-criteria linear-time approximations for generalizedk-mean/median/center” by Feldman et al.
• The basic idea is about uniform sampling + peeling.• If k is constant, there exists (1 + ε)-approximation. “A unified
framework for approximating and clustering data” by Feldmanand Langberg.
![Page 11: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/11.jpg)
Clustering in High Dimension (cont.)
• Stochastic model for uncertain and noisy data.
1 Existential uncertainty model: each data point has a fixedlocation but with an existential probability.
2 Locational uncertainty model: each data point has a set oflocation candidates.
• Current state-of-the-art techniques can only handle lowdimensional case. “Stochastic k-Center and j-Flat-CenterProblems” by Huang and Li; “epsilon-Kernel Coresets forStochastic Points” by Huang et al.
![Page 12: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/12.jpg)
Clustering in High Dimension (cont.)
• Stochastic model for uncertain and noisy data.
1 Existential uncertainty model: each data point has a fixedlocation but with an existential probability.
2 Locational uncertainty model: each data point has a set oflocation candidates.
• Current state-of-the-art techniques can only handle lowdimensional case. “Stochastic k-Center and j-Flat-CenterProblems” by Huang and Li; “epsilon-Kernel Coresets forStochastic Points” by Huang et al.
![Page 13: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/13.jpg)
Clustering in High Dimension (cont.)
• Stochastic model for uncertain and noisy data.
1 Existential uncertainty model: each data point has a fixedlocation but with an existential probability.
2 Locational uncertainty model: each data point has a set oflocation candidates.
• Current state-of-the-art techniques can only handle lowdimensional case. “Stochastic k-Center and j-Flat-CenterProblems” by Huang and Li; “epsilon-Kernel Coresets forStochastic Points” by Huang et al.
![Page 14: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/14.jpg)
Clustering in High Dimension (cont.)
• Stochastic model for uncertain and noisy data.
1 Existential uncertainty model: each data point has a fixedlocation but with an existential probability.
2 Locational uncertainty model: each data point has a set oflocation candidates.
• Current state-of-the-art techniques can only handle lowdimensional case. “Stochastic k-Center and j-Flat-CenterProblems” by Huang and Li; “epsilon-Kernel Coresets forStochastic Points” by Huang et al.
![Page 15: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/15.jpg)
SVM And Sparse Representation
• SVM is equivalent to a polytope distance problem, and wehave learned Gilbert algorithm for polytope distance.“Coresets for polytope distance” by Gartner and Jaggi.
• But what about non-separable case? “Random GradientDescent Tree: A Combinatorial Approach for SVM withOutliers” by Ding and Xu.
![Page 16: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/16.jpg)
SVM And Sparse Representation
• SVM is equivalent to a polytope distance problem, and wehave learned Gilbert algorithm for polytope distance.“Coresets for polytope distance” by Gartner and Jaggi.
• But what about non-separable case? “Random GradientDescent Tree: A Combinatorial Approach for SVM withOutliers” by Ding and Xu.
![Page 17: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/17.jpg)
Our Main Ideas
Key Observation: It is not necessary to select the single point in a deterministic way at each step of Gilbert algorithm.
Gradient Descent + Randomness
• Preserve fast convergence via gradient descent.
• Explicitly remove the influence of outliers via randomness.
![Page 18: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/18.jpg)
Our Main Ideas
Key Observation: It is not necessary to select the single point in a deterministic way at each step of Gilbert algorithm.
Gradient Descent + Randomness
• Preserve fast convergence via gradient descent.
• Explicitly remove the influence of outliers via randomness.
![Page 19: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/19.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 20: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/20.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 21: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/21.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 22: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/22.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 23: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/23.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 24: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/24.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 25: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/25.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 26: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/26.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 27: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/27.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 28: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/28.jpg)
Our Main Ideas
Gilbert Algorithm Our idea for the case with outliers
• Induce a data structure called Random Gradient Descent-Tree.
• Quality guarantee: (1− ε)-approximation with respect to
separating margin.
![Page 29: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/29.jpg)
How to Achieve Sub-linear Time Complexity
• From linear tosub-linear time:
![Page 30: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/30.jpg)
How to Achieve Sub-linear Time Complexity
• From linear tosub-linear time:
Top k selection Sampling
![Page 31: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/31.jpg)
How to Achieve Sub-linear Time Complexity
• From linear tosub-linear time:
Top k selection Sampling
Sampling Top k’ selection
x0x0
![Page 32: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/32.jpg)
How to Achieve Sub-linear Time Complexity
• From linear tosub-linear time:
Top k selection Sampling
Sampling Top k’ selection
x0x0
![Page 33: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/33.jpg)
How to Achieve Sub-linear Space Complexity
• Build the datastructure viabreadth-first search.
• Only two levels areneeded to store =⇒sub-linear extra spacecomplexity.
![Page 34: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/34.jpg)
SVM (cont.)
• Actually Gilbert algorithm is a special case of Frank-Wolfealgorithm. “ Coresets, sparse greedy approximation, and theFrank-Wolfe algorithm” by Clarkson.
• Another typical case of Frank-Wolfe algorithm is computingMinimum Enclosing Ball in high dimension.
• The result of Frank-Wolfe algorithm usually is a sparserepresentation, with potential applications in dictionarylearning.
![Page 35: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/35.jpg)
SVM (cont.)
• Actually Gilbert algorithm is a special case of Frank-Wolfealgorithm. “ Coresets, sparse greedy approximation, and theFrank-Wolfe algorithm” by Clarkson.
• Another typical case of Frank-Wolfe algorithm is computingMinimum Enclosing Ball in high dimension.
• The result of Frank-Wolfe algorithm usually is a sparserepresentation, with potential applications in dictionarylearning.
![Page 36: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/36.jpg)
SVM (cont.)
• Actually Gilbert algorithm is a special case of Frank-Wolfealgorithm. “ Coresets, sparse greedy approximation, and theFrank-Wolfe algorithm” by Clarkson.
• Another typical case of Frank-Wolfe algorithm is computingMinimum Enclosing Ball in high dimension.
• The result of Frank-Wolfe algorithm usually is a sparserepresentation, with potential applications in dictionarylearning.
![Page 37: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/37.jpg)
Other Applications of Core-set
• Compress GPS/LiDAR data, and help data mining on GPSdatasets with applications in traffic management, autopilot,etc. “The Single Pixel GPS: Learning Big Data Signals fromTiny Coresets” by Feldman et al.
• Dictionary learning and de-noise for big data. “Learning Big(Image) Data via Coresets for Dictionaries” by Feldman et al.
• Large scale training for Gaussian Mixture Model (GMM).“Scalable Training of Mixture Models via Coresets” byFeldman et al.
• Speedup alignment/registration for computer vision androbotics. “Low-cost and Faster Tracking Systems UsingCore-sets for Pose-Estimation” by Nasser et al.
![Page 38: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/38.jpg)
Other Applications of Core-set
• Compress GPS/LiDAR data, and help data mining on GPSdatasets with applications in traffic management, autopilot,etc. “The Single Pixel GPS: Learning Big Data Signals fromTiny Coresets” by Feldman et al.
• Dictionary learning and de-noise for big data. “Learning Big(Image) Data via Coresets for Dictionaries” by Feldman et al.
• Large scale training for Gaussian Mixture Model (GMM).“Scalable Training of Mixture Models via Coresets” byFeldman et al.
• Speedup alignment/registration for computer vision androbotics. “Low-cost and Faster Tracking Systems UsingCore-sets for Pose-Estimation” by Nasser et al.
![Page 39: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/39.jpg)
Other Applications of Core-set
• Compress GPS/LiDAR data, and help data mining on GPSdatasets with applications in traffic management, autopilot,etc. “The Single Pixel GPS: Learning Big Data Signals fromTiny Coresets” by Feldman et al.
• Dictionary learning and de-noise for big data. “Learning Big(Image) Data via Coresets for Dictionaries” by Feldman et al.
• Large scale training for Gaussian Mixture Model (GMM).“Scalable Training of Mixture Models via Coresets” byFeldman et al.
• Speedup alignment/registration for computer vision androbotics. “Low-cost and Faster Tracking Systems UsingCore-sets for Pose-Estimation” by Nasser et al.
![Page 40: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/40.jpg)
Other Applications of Core-set
• Compress GPS/LiDAR data, and help data mining on GPSdatasets with applications in traffic management, autopilot,etc. “The Single Pixel GPS: Learning Big Data Signals fromTiny Coresets” by Feldman et al.
• Dictionary learning and de-noise for big data. “Learning Big(Image) Data via Coresets for Dictionaries” by Feldman et al.
• Large scale training for Gaussian Mixture Model (GMM).“Scalable Training of Mixture Models via Coresets” byFeldman et al.
• Speedup alignment/registration for computer vision androbotics. “Low-cost and Faster Tracking Systems UsingCore-sets for Pose-Estimation” by Nasser et al.
![Page 41: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/41.jpg)
Open Problems
• Constrained Core-set, such as the applications in socialnetwork (e.g., users or products may have mutual conflict).
• Stochastic model in high dimension.
• Algorithm for computing Core-set via GPU (highly parallel).
• Any other new applications, or theoretical improvement.
![Page 42: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/42.jpg)
Open Problems
• Constrained Core-set, such as the applications in socialnetwork (e.g., users or products may have mutual conflict).
• Stochastic model in high dimension.
• Algorithm for computing Core-set via GPU (highly parallel).
• Any other new applications, or theoretical improvement.
![Page 43: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/43.jpg)
Open Problems
• Constrained Core-set, such as the applications in socialnetwork (e.g., users or products may have mutual conflict).
• Stochastic model in high dimension.
• Algorithm for computing Core-set via GPU (highly parallel).
• Any other new applications, or theoretical improvement.
![Page 44: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/44.jpg)
Open Problems
• Constrained Core-set, such as the applications in socialnetwork (e.g., users or products may have mutual conflict).
• Stochastic model in high dimension.
• Algorithm for computing Core-set via GPU (highly parallel).
• Any other new applications, or theoretical improvement.
![Page 45: Core-set: Summary and Open Problems](https://reader035.vdocuments.us/reader035/viewer/2022062216/62a3d75b18e52b77df610cf8/html5/thumbnails/45.jpg)
Thank You!
Any Question?