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Great SDP relaxation of max-k-cut and max-bisection: very inspiring! ### Complicated analysis: The analysis to prove the approximation guarantee is quite complicated though. Much more complicated than the one of [the Goemans-Williamson algorithm](https://en.wikipedia.org/wiki/Semidefinite_programming#Example_3) for max-cut. Is a simpler analysis possible? Or another relaxation leading to a simpler analysis having similar or better approximation guarantees? ### Implementing the algorithms in a scalable way: How can we implement such an algorithm in a scalable way? Say for a sparse graph with 1M nodes and 100M edges? For Goemans-Williamson, this is a try: https://github.com/maxdan94/spingraphSDP
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## Looks good. The first impression: the paper is similar to Kleinberg's [masterpiece](https://www.cs.cornell.edu/home/kleinber/nips15.pdf). Also, one column format is more agreable to read for normal humans than 2-colums.
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### Concerns about the time complexity given in Table 1: Section 3.3.5. Graph Factorization (GF). "To the best of our knowledge, Graph Factorization [21] was the first method to obtain a graph embedding in $O(|E|)$ time". This contradicts the time complexities given in Table 1 "$O(|E|d)$" for GF and earlier complexities in $O(|E|d^2)$. ### Dimensionality reduction is different from graph embedding: Some of the given references are actually for dimensionality reductions: the input is a set of vectors in dimension $D$ and the output is a set of vectors in dimension $d$ (such that $d<<D$). The problem where the input is a graph and the output is a set of vectors of small dimension is actually different. LLE and Laplacian Eigenmaps are methods starting from a set of vectors, then forming a graph out of the vectors and then forming vectors of small dimension out of the graph. Maybe this can be emphasized. In particular, maybe using these techniques on social networks (for instance) may not be adapted. The input graph is in most of the cases unweighted and the structure might be very different from the structure of a graph obtained from vectors. ### Cauchy graph embedding: - What is the complexity of Cauchy graph embedding (not given in table 1)? - How is the solution obtained? From eigenvectors like the two previous methods (LLE and Laplacian Eigenmaps)? - How to tune the parameter $\sigma$? After reading the paper [33], the Cauchy graph embedding method is more complex and seems less scalable. ### GF and SDP matrices: Page 4, section 3.3.5. Graph Factorization (GF). "Note that as the adjacency matrix is often not positive semidefinite, the minimum of the loss function is greater than 0 even if the dimensionality of embedding is |V|.". - The loss function is never 0 if $\lambda>0$. - If $W$ is SDP (and different from the null matrix) then the associated graph needs to contain self-loops. If self-loops are forbidden, then the matrix is never SDP. - If $W$ is SDP, then there exist a set of vectors $\{Y_i\}$ such that $\forall i,j$, $W_{ij}=<Y_i,Y_j>$. Is it true that such vectors can always be of dimension $n$ (the size of $W$) or less? A reference or proof? ### Minors: - Page 3, section 3.1.1. Locally Linear Embedding (LLE). It should be specified that the rows of the weight matrix need sum to one: $\sum_j W_{ij}=1$. - Page 3, section 3.1.1. Locally Linear Embedding (LLE). "the sparse matrix $(I-W)^T(I-W)$" this matrix is often not sparse (in the sense that it has a lot more non-zero values compared to the adjacency matrix $W$). - Page 3, section 3.1.3. Cauchy Graph Embedding. Is there a solution of the optimization using eigenvectors as in the two previous methods? - What is "positive pointwise mutual information (PPMI)"? - Figure 1 refers to CPE (Community Preserving Embedding), but CPE is not discussed in the survey. - The section on GraRep refers to the section on HOPE which is after, maybe swapping the two sections? ### Typos: - Many text-overflows
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