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1. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar ; Kalhan, Sanchit ; Makarychev, Konstantin ; Makarychev, Yury ( January 2021 , International Conference on Machine Learning)

   null (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector is a vector indexed by the vertices of G such that the v-th coordinate of the disagreements vector equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α ≤ 1 and w > 0 is a scaling parameter). We give an O((1/α)^{1/2−1/(2p)} log 1/α) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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2. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar ; Kalhan, Sanchit ; Makarychev, Konstantin ; Makarychev, Yury ( July 2021 , International Conference on Machine Learning)

   null (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph G with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering C of graph G, a similar edge is in disagreement with C, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with C if its endpoints belong to the same cluster. The disagreements vector, Agree, is a vector indexed by the vertices of G such that the v-th coordinate Disagre equals the weight of all disagreeing edges incident on v. The goal is to produce a clustering that minimizes the ℓp norm of the disagreements vector for p≥1. We study the ℓ_p objective in Correlation Clustering under the following assumption: Every similar edge has weight in [αw,w] and every dissimilar edge has weight at least αw (where α≤1 and w>0 is a scaling parameter). We give an O((1/α)^{1/2−1/2p}⋅log(1/α)) approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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3. Evaluating Stability in Massive Social Networks: Efficient Streaming Algorithms for Structural Balance

   https://doi.org/10.4230/LIPICS.APPROX/RANDOM.2023.58  

   Ashvinkumar, Vikrant ; Assadi, Sepehr ; Deng, Chengyuan ; Gao, Jie ; Wang, Chen ( January 2023 , APPROX-RANDOM)

   Megow, Nicole ; Smith, Adam (Ed.)

   Structural balance theory studies stability in networks. Given a n-vertex complete graph G = (V,E) whose edges are labeled positive or negative, the graph is considered balanced if every triangle either consists of three positive edges (three mutual "friends"), or one positive edge and two negative edges (two "friends" with a common "enemy"). From a computational perspective, structural balance turns out to be a special case of correlation clustering with the number of clusters at most two. The two main algorithmic problems of interest are: (i) detecting whether a given graph is balanced, or (ii) finding a partition that approximates the frustration index, i.e., the minimum number of edge flips that turn the graph balanced. We study these problems in the streaming model where edges are given one by one and focus on memory efficiency. We provide randomized single-pass algorithms for: (i) determining whether an input graph is balanced with O(log n) memory, and (ii) finding a partition that induces a (1 + ε)-approximation to the frustration index with O(n ⋅ polylog(n)) memory. We further provide several new lower bounds, complementing different aspects of our algorithms such as the need for randomization or approximation. To obtain our main results, we develop a method using pseudorandom generators (PRGs) to sample edges between independently-chosen vertices in graph streaming. Furthermore, our algorithm that approximates the frustration index improves the running time of the state-of-the-art correlation clustering with two clusters (Giotis-Guruswami algorithm [SODA 2006]) from n^O(1/ε²) to O(n²log³n/ε² + n log n ⋅ (1/ε)^O(1/ε⁴)) time for (1+ε)-approximation. These results may be of independent interest. 

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4. Local Correlation Clustering with Asymmetric Classification Errors

   Jafarov, Jafar ; Kalhan, Sanchit ; Makarychev, Konstantin ; Makarychev, Yury ( January 2021 , Proceedings of Machine Learning Research)

   Meila, Marina ; Zhang, Tong (Ed.)

   In the Correlation Clustering problem, we are given a complete weighted graph $G$ with its edges labeled as “similar" and “dissimilar" by a noisy binary classifier. For a clustering $\mathcal{C}$ of graph $G$, a similar edge is in disagreement with $\mathcal{C}$, if its endpoints belong to distinct clusters; and a dissimilar edge is in disagreement with $\mathcal{C}$ if its endpoints belong to the same cluster. The disagreements vector, $\mathbf{disagree}$, is a vector indexed by the vertices of $G$ such that the $v$-th coordinate $\mathbf{disagree}_v$ equals the weight of all disagreeing edges incident on $v$. The goal is to produce a clustering that minimizes the $\ell_p$ norm of the disagreements vector for $p\geq 1$. We study the $\ell_p$ objective in Correlation Clustering under the following assumption: Every similar edge has weight in $[\alpha\mathbf{w},\mathbf{w}]$ and every dissimilar edge has weight at least $\alpha\mathbf{w}$ (where $\alpha \leq 1$ and $\mathbf{w}>0$ is a scaling parameter). We give an $O\left((\frac{1}{\alpha})^{\frac{1}{2}-\frac{1}{2p}}\cdot \log\frac{1}{\alpha}\right)$ approximation algorithm for this problem. Furthermore, we show an almost matching convex programming integrality gap. 

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5. Approximation Algorithms for Orthogonal Non-negative Matrix Factorization

   Charikar, Moses ; Hu, Lunjia ( January 2021 , Proeedings of the International Workshop on Artificial Intelligence and Statistics)

   null (Ed.)

   In the non-negative matrix factorization (NMF) problem, the input is an m×n matrix M with non-negative entries and the goal is to factorize it as M ≈ AW. The m × k matrix A and the k × n matrix W are both constrained to have non-negative entries. This is in contrast to singular value decomposition, where the matrices A and W can have negative entries but must satisfy the orthogonality constraint: the columns of A are orthogonal and the rows of W are also orthogonal. The orthogonal non-negative matrix factorization (ONMF) problem imposes both the non-negativity and the orthogonality constraints, and previous work showed that it leads to better performances than NMF on many clustering tasks. We give the first constant-factor approximation algorithm for ONMF when one or both of A and W are subject to the orthogonality constraint. We also show an interesting connection to the correlation clustering problem on bipartite graphs. Our experiments on synthetic and real-world data show that our algorithm achieves similar or smaller errors compared to previous ONMF algorithms while ensuring perfect orthogonality (many previous algorithms do not satisfy the hard orthogonality constraint). 

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