
A Tight Degree 4 SumofSquares Lower Bound for the SherringtonKirkpatrick Hamiltonian
We show that, if W∈R^N × N_sym is drawn from the gaussian orthogonal ens...
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Highdimensional estimation via sumofsquares proofs
Estimation is the computational task of recovering a hidden parameter x ...
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Machinery for Proving SumofSquares Lower Bounds on Certification Problems
In this paper, we construct general machinery for proving SumofSquares...
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Sparse PCA: Algorithms, Adversarial Perturbations and Certificates
We study efficient algorithms for Sparse PCA in standard statistical mod...
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Counterexamples to the LowDegree Conjecture
A conjecture of Hopkins (2018) posits that for certain highdimensional ...
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Sum of squares bounds for the total ordering principle
In this paper, we analyze the sum of squares hierarchy (SOS) on the tota...
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SOS lower bounds with hard constraints: think global, act local
Many previous SumofSquares (SOS) lower bounds for CSPs had two deficie...
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Lifting SumofSquares Lower Bounds: Degree2 to Degree4
The degree4 SumofSquares (SoS) SDP relaxation is a powerful algorithm that captures the best known polynomial time algorithms for a broad range of problems including MaxCut, Sparsest Cut, all MaxCSPs and tensor PCA. Despite being an explicit algorithm with relatively low computational complexity, the limits of degree4 SoS SDP are not well understood. For example, existing integrality gaps do not rule out a (2ε)algorithm for Vertex Cover or a (0.878+ε)algorithm for MaxCut via degree4 SoS SDPs, each of which would refute the notorious Unique Games Conjecture. We exhibit an explicit mapping from solutions for degree2 SumofSquares SDP (GoemansWilliamson SDP) to solutions for the degree4 SumofSquares SDP relaxation on boolean variables. By virtue of this mapping, one can lift lower bounds for degree2 SoS SDP relaxation to corresponding lower bounds for degree4 SoS SDPs. We use this approach to obtain degree4 SoS SDP lower bounds for MaxCut on random dregular graphs, SheringtonKirkpatrick model from statistical physics and PSD Grothendieck problem. Our constructions use the idea of pseudocalibration towards candidate SDP vectors, while it was previously only used to produce the candidate matrix which one would show is PSD using much technical work. In addition, we develop a different technique to bound the spectral norms of _graphical matrices_ that arise in the context of SoS SDPs. The technique is much simpler and yields better bounds in many cases than the _trace method_ – which was the sole technique for this purpose.
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