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A code C ∶ {0,1}k → {0,1}n is a qlocally decodable code (qLDC) if one can recover any chosen bit bi of the message b ∈ {0,1}k with good confidence by randomly querying the encoding x = C(b) on at most q coordinates. Existing constructions of 2LDCs achieve n = exp(O(k)), and lower bounds show that this is in fact tight. However, when q = 3, far less is known: the best constructions achieve n = exp(ko(1)), while the best known results only show a quadratic lower bound n ≥ Ω(k2/log(k)) on the blocklength. In this paper, we prove a nearcubic lower bound of n ≥ Ω(k3/log6(k)) on the blocklength of 3query LDCs. This improves on the best known prior works by a polynomial factor in k. Our proof relies on a new connection between LDCs and refuting constraint satisfaction problems with limited randomness. Our quantitative improvement builds on the new techniques for refuting semirandom instances of CSPs and, in particular, relies on bounding the spectral norm of appropriate Kikuchi matrices.more » « lessFree, publiclyaccessible full text available July 1, 2024

In [Saunderson, 2011; Saunderson et al., 2013], Saunderson, Parrilo, and Willsky asked the following elegant geometric question: what is the largest m = m(d) such that there is an ellipsoid in ℝ^d that passes through v_1, v_2, …, v_m with high probability when the v_is are chosen independently from the standard Gaussian distribution N(0,I_d)? The existence of such an ellipsoid is equivalent to the existence of a positive semidefinite matrix X such that v_i^⊤ X v_i = 1 for every 1 ⩽ i ⩽ m  a natural example of a random semidefinite program. SPW conjectured that m = (1o(1)) d²/4 with high probability. Very recently, Potechin, Turner, Venkat and Wein [Potechin et al., 2022] and Kane and Diakonikolas [Kane and Diakonikolas, 2022] proved that m ≳ d²/log^O(1) d via a certain natural, explicit construction. In this work, we give a substantially tighter analysis of their construction to prove that m ≳ d²/C for an absolute constant C > 0. This resolves one direction of the SPW conjecture up to a constant. Our analysis proceeds via the method of Graphical Matrix Decomposition that has recently been used to analyze correlated random matrices arising in various areas [Barak et al., 2019; Bafna et al., 2022]. Our key new technical tool is a refined method to prove singular value upper bounds on certain correlated random matrices that are tight up to absolute dimensionindependent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension.more » « lessFree, publiclyaccessible full text available July 1, 2024

We design new polynomialtime algorithms for recovering planted cliques in the semirandom graph model introduced by Feige and Kilian 2001. The previous best algorithms for this model succeed if the planted clique has size at least n2/3 in a graph with n vertices (Mehta, Mckenzie, Trevisan 2019 and Charikar, Steinhardt, Valiant 2017). Our algorithms work for plantedclique sizes approaching n1/2  the informationtheoretic threshold in the semirandom model (Steinhardt 2017) and a conjectured computational threshold even in the easier fullyrandom model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017. Our algorithms are based on higher constant degree sumofsquares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite ErdősRényi random graphs into algorithms for semirandom planted clique. The use of a higherconstant degree sumofsquares is essential in our setting: we prove a lower bound on the basic SDP for certifying bicliques that shows that the basic SDP cannot succeed for planted cliques of size k=o(n2/3). We also provide some evidence that the informationcomputation tradeoff of our current algorithms may be inherent by proving an averagecase lower bound for unbalanced bicliques in the lowdegreepolynomials model.more » « less