Composable Core-sets for Diversity and Coverage Maximization Piotr - PowerPoint PPT Presentation

Composable Core-sets for Diversity and Coverage Maximization Piotr Indyk (MIT) Sepideh Mahabadi (MIT) Mohammad Mahdian (Google) Vahab S. Mirrokni (Google)

Core-Set Definition Setup • Set of 𝑜 points 𝑸 in 𝑒 -dimensional – space Optimize a function 𝑔 –

Core-Set Definition Setup • Set of 𝑜 points 𝑸 in 𝑒 -dimensional – space Optimize a function 𝑔 – 𝒅 -Core-set: Small subset of points S ⊂ 𝑄 • which suffices to 𝑑 -approximate the optimal solution 𝑔 𝑝𝑝𝑝 𝑄 ≤ 𝑔 𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 ) • Maximization: 𝑑

Core-Set Definition Setup • Set of 𝑜 points 𝑸 in 𝑒 -dimensional – space Optimize a function 𝑔 – 𝒅 -Core-set: Small subset of points S ⊂ 𝑄 • which suffices to 𝑑 -approximate the optimal solution 𝑔 𝑝𝑝𝑝 𝑄 ≤ 𝑔 𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 ) • Maximization: 𝑑 Example • – Optimization Function: Distance of the two farthest points

Core-Set Definition Setup • Set of 𝑜 points 𝑸 in 𝑒 -dimensional – space Optimize a function 𝑔 – 𝒅 -Core-set: Small subset of points S ⊂ 𝑄 • which suffices to 𝑑 -approximate the optimal solution 𝑔 𝑝𝑝𝑝 𝑄 ≤ 𝑔 𝑝𝑝𝑝 𝑇 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 ) • Maximization: 𝑑 Example • – Optimization Function: Distance of the two farthest points 1 -Core-set: Points on the convex hull. –

Composable Core-sets Setup • 𝑸 𝟐 , 𝑸 𝟑 , … , 𝑸 𝒏 are set of points in – 𝑒 -dimensional space Optimize a function 𝑔 over their – union 𝑸 .

Composable Core-sets Setup • 𝑸 𝟐 , 𝑸 𝟑 , … , 𝑸 𝒏 are set of points in – 𝑒 -dimensional space Optimize a function 𝑔 over their – union 𝑸 . 𝒅 -Composable Core-sets: Subsets of • points S 1 ⊂ 𝑄 1 , S 2 ⊂ 𝑄 2 , … , S m ⊂ 𝑄 𝑛 points such that the solution of the union of the core-sets approximates the solution of the point sets. • Maximization : 1 𝑑 𝑔 𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔 opt S 1 ∪ ⋯ ∪ 𝑇 𝑛 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 )

Composable Core-sets Setup • 𝑸 𝟐 , 𝑸 𝟑 , … , 𝑸 𝒏 are set of points in – 𝑒 -dimensional space Optimize a function 𝑔 over their – union 𝑸 . 𝒅 -Composable Core-sets: Subsets of • points S 1 ⊂ 𝑄 1 , S 2 ⊂ 𝑄 2 , … , S m ⊂ 𝑄 𝑛 points such that the solution of the union of the core-sets approximates the solution of the point sets. • Maximization : 1 𝑑 𝑔 𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔 opt S 1 ∪ ⋯ ∪ 𝑇 𝑛 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ) • Example: two farthest points

Composable Core-sets Setup • 𝑸 𝟐 , 𝑸 𝟑 , … , 𝑸 𝒏 are set of points in – 𝑒 -dimensional space Optimize a function 𝑔 over their – union 𝑸 . 𝒅 -Composable Core-sets: Subsets of • points S 1 ⊂ 𝑄 1 , S 2 ⊂ 𝑄 2 , … , S m ⊂ 𝑄 𝑛 points such that the solution of the union of the core-sets approximates the solution of the point sets. • Maximization : 1 𝑑 𝑔 𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔 opt S 1 ∪ ⋯ ∪ 𝑇 𝑛 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ) • Example: two farthest points

Composable Core-sets Setup • 𝑸 𝟐 , 𝑸 𝟑 , … , 𝑸 𝒏 are set of points in – 𝑒 -dimensional space Optimize a function 𝑔 over their – union 𝑸 . 𝒅 -Composable Core-sets: Subsets of • points S 1 ⊂ 𝑄 1 , S 2 ⊂ 𝑄 2 , … , S m ⊂ 𝑄 𝑛 points such that the solution of the union of the core-sets approximates the solution of the point sets. • Maximization : 1 𝑑 𝑔 𝑝𝑝𝑝 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ≤ 𝑔 opt S 1 ∪ ⋯ ∪ 𝑇 𝑛 ≤ 𝑔 𝑝𝑝𝑝 ( 𝑄 1 ∪ ⋯ ∪ 𝑄 𝑛 ) • Example: two farthest points

Applications – Streaming Computation • Streaming Computation: Processing sequence of 𝑜 data elements “on the fly” – – limited Storage

Applications – Streaming Computation • Streaming Computation: Processing sequence of 𝑜 data elements “on the fly” – – limited Storage 𝒅 -Composable Core-set of size 𝒍 • Chunks of size 𝑜𝑜 , thus number of chunks = 𝑜 / 𝑜 –

Applications – Streaming Computation • Streaming Computation: Processing sequence of 𝑜 data elements “on the fly” – – limited Storage 𝒅 -Composable Core-set of size 𝒍 • Chunks of size 𝑜𝑜 , thus number of chunks = 𝑜 / 𝑜 – – Core-set for each chunk Total Space: 𝑜 𝑜 / 𝑜 + 𝑜𝑜 = 𝑃 ( 𝑜𝑜 ) – Approximation Factor: 𝑑 –

Applications – Distributed Systems • Streaming Computation Distributed System: • Each machine holds a block of data. – A composable core-set is computed and sent to the server –

Applications – Distributed Systems • Streaming Computation Distributed System: • Each machine holds a block of data. – A composable core-set is computed and sent to the server – • Map-Reduce Model: • One round of Map-Reduce 𝑜 / 𝑜 mappers each getting 𝑜𝑜 points • Mapper computes a composable core-set of size 𝑜 • Will be passed to a single reducer •

Applications – Similarity Search • Streaming Computation Distributed System • • Similarity Search: Small output size

Applications – Similarity Search • Streaming Computation Distributed System • • Similarity Search: Small output size Good to have result from each • cluster: relevant and diverse

Applications – Similarity Search • Streaming Computation Distributed System • • Similarity Search: Small output size Good to have result from each • cluster: relevant and diverse • Diverse Near Neighbor Problem [ Abbar, Amer-Yahia, Indyk, Mahabadi WWW’13] [Abbar, Amer-Yahia, Indyk, Mahabadi, Varadarajan, SoCG’13]

Applications – Similarity Search • Streaming Computation Distributed System • • Similarity Search: Small output size Good to have result from each • cluster: relevant and diverse • Diverse Near Neighbor Problem [ Abbar, Amer-Yahia, Indyk, Mahabadi WWW’13] [Abbar, Amer-Yahia, Indyk, Mahabadi, Varadarajan, SoCG’13] uses Locality Sensitive Hashing – (LSH) and Composable Core- sets techniques.

Diversity Maximization Problem A set of 𝑜 points 𝑄 in metric space • ( Δ , 𝑒𝑒𝑒𝑒 ) Optimization Problem: • Find a subset of 𝑜 points 𝑇 which – maximizes Diversity k=4 n = 6

Diversity Maximization Problem A set of 𝑜 points 𝑄 in metric space • ( Δ , 𝑒𝑒𝑒𝑒 ) Optimization Problem: • Find a subset of 𝑜 points 𝑇 which – maximizes Diversity Diversity: • – Minimum pairwise distance (Remote Edge) k=4 n = 6

Diversity Maximization Problem A set of 𝑜 points 𝑄 in metric space • ( Δ , 𝑒𝑒𝑒𝑒 ) Optimization Problem: • Find a subset of 𝑜 points 𝑇 which – maximizes Diversity Diversity: • – Minimum pairwise distance (Remote Edge) Sum of Pairwise distances (Remote – k=4 Clique) n = 6

Diversity Maximization Problem A set of 𝑜 points 𝑄 in metric space • ( Δ , 𝑒𝑒𝑒𝑒 ) Optimization Problem: • Find a subset of 𝑜 points 𝑇 which – maximizes Diversity Diversity: • – Minimum pairwise distance (Remote Edge) Sum of Pairwise distances (Remote – k=4 Clique) n = 6 Long list of variants [Chandra and • Halldorsson ‘01]

Diversity Functions Diversity function over Description a set 𝑇 of 𝑜 point Minimum Pairwise Distance: min 𝑝 , 𝑟∈𝑇 𝑒𝑒𝑒𝑒 ( 𝑞 , 𝑟 ) Remote-edge Sum of Pairwise Distances : ∑ 𝑒𝑒𝑒𝑒 ( 𝑞 , 𝑟 ) Remote-clique 𝑝 , 𝑟∈𝑇 Weight of Minimum Spanning Tree (MST) of the set 𝑇 Remote-tree Weight of minimum Traveling Salesman Tour (TSP) of the set 𝑇 Remote-cycle 𝑝∈𝑇 ∑ Weight of minimum star: min 𝑒𝑒𝑒𝑒 ( 𝑞 , 𝑟 ) Remote-star 𝑟∈𝑇 Remote-Pseudoforest Sum of the distance of each point to its nearest neighbor ∑ min 𝑟∈𝑇 𝑒𝑒𝑒𝑒 ( 𝑞 , 𝑟 ) 𝑝∈𝑇 Weight of minimum perfect Matching of the set 𝑇 Remote-Matching Max-Coverage How well the points cover each coordinate 𝑒 � max 𝑝∈𝑇 𝑞 𝑗 𝑗=1