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A code C ∶ {0,1}k → {0,1}n is a q-locally decodable code (q-LDC) 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 2-LDCs 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 near-cubic lower bound of n ≥ Ω(k3/log6(k)) on the blocklength of 3-query 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. 

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 = (1-o(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/polylog(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 dimension-independent constants. In contrast, all previous methods that analyze such matrices lose logarithmic factors in the dimension. 

We design new polynomial-time algorithms for recovering planted cliques in the semi-random 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 planted-clique sizes approaching n1/2 -- the information-theoretic threshold in the semi-random model (Steinhardt 2017) and a conjectured computational threshold even in the easier fully-random model. This result comes close to resolving open questions by Feige 2019 and Steinhardt 2017. Our algorithms are based on higher constant degree sum-of-squares relaxation and rely on a new conceptual connection that translates certificates of upper bounds on biclique numbers in unbalanced bipartite Erdős--Rényi random graphs into algorithms for semi-random planted clique. The use of a higher-constant degree sum-of-squares 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 information-computation trade-off of our current algorithms may be inherent by proving an average-case lower bound for unbalanced bicliques in the low-degree-polynomials model. 

We give efficient algorithms for finding power-sum decomposition of an input polynomial with component s. The case of linear s is equivalent to the well-studied tensor decomposition problem while the quadratic case occurs naturally in studying identifiability of non-spherical Gaussian mixtures from low-order moments. Unlike tensor decomposition, both the unique identifiability and algorithms for this problem are not well-understood. For the simplest setting of quadratic s and , prior work of [GHK15] yields an algorithm only when . On the other hand, the more general recent result of [GKS20] builds an algebraic approach to handle any components but only when is large enough (while yielding no bounds for or even ) and only handles an inverse exponential noise. Our results obtain a substantial quantitative improvement on both the prior works above even in the base case of and quadratic s. Specifically, our algorithm succeeds in decomposing a sum of generic quadratic s for and more generally the th power-sum of generic degree- polynomials for any . Our algorithm relies only on basic numerical linear algebraic primitives, is exact (i.e., obtain arbitrarily tiny error up to numerical precision), and handles an inverse polynomial noise when the s have random Gaussian coefficients.