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On using affine sketches for multiple-response dynamic graph regression

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Abstract

In the multiple-response dynamic graph regression problem, given a \(n\times d\) matrix embedding \({\varvec{M}}\) of a graph G and a \(n \times d'\) response matrix \({\varvec{B}}\), we want to update the solution of \(argmin_{\varvec{X}} ||{\varvec{M}} \cdot {\varvec{X}} - {\varvec{B}}||_F^2\), after update operations in the graph. In this paper, we present new theoretical results for the update time of this problem. More precisely, we show that using an affine embedding defined as subsampled randomized Hadamard transform and after an edge insertion or an edge deletion, a \(1\pm \epsilon \) approximation to the optimal solution can be updated in \( O\left( d' \epsilon ^{-2}\log ^2 n \right) \) time. Moreover, using an affine embedding defined as CountSketch and after a node insertion or a node deletion or an edge insertion or an edge deletion, an approximate solution can be updated in \(O\left( d' \epsilon ^{-2} \log ^6 \epsilon ^{-1} \right) \) time. To the best of our knowledge, these are the best theoretical results obtained for the update time of approximate multiple-response dynamic graph regression.

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Notes

  1. Note that there are other notions of embeddings in the field of machine learning with graphs, such as embeddings in frequent pattern discovery [26, 27], that are different from affine (and subspace) embeddings used in this paper.

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  1. Department of Computer Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran, Iran

    Mostafa Haghir Chehreghani

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Haghir Chehreghani, M. On using affine sketches for multiple-response dynamic graph regression. J Supercomput 79, 5139–5153 (2023). https://doi.org/10.1007/s11227-022-04865-x

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