For random Gaussian matrices, no online algorithm can beat the incremental greedy procedure's dense-submatrix value, with tight factor 4/(3√2), and 2√p/(p+1) for p-tensors.
Gamarnik, The overlap gap property: A topological barrier to optimizing over random structures , Proceedings of the National Academy of Sciences 118 (2021), no
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Finding a dense submatrix of a random matrix. Sharp bounds for online algorithms
For random Gaussian matrices, no online algorithm can beat the incremental greedy procedure's dense-submatrix value, with tight factor 4/(3√2), and 2√p/(p+1) for p-tensors.