Tselil Schramm

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It is thought that speciation in phytophagous insects is often due to colonization of novel host plants, because radiations of plant and insect lineages are typically asynchronous. Recent phylogenetic comparisons have supported this model of diversification for both insect herbivores and specialized pollinators. An exceptional case where contemporaneous(More)
We give new rounding schemes for the standard linear programming relaxation of the correlation clustering problem, achieving approximation factors almost matching the integrality gaps: For complete graphs our approximation is 2.06 - ε, which almost matches the previously known integrality gap of 2. For complete k-partite graphs our approximation is 3.(More)
The stochastic block model is a classical cluster exhibiting random graph model that has been widely studied in statistics, physics, and computer science. In its simplest form, the model is a random graph with two equal-sized clusters, with intracluster edge probability p, and intercluster edge probability q. We focus on the sparse case, i.e., p, q =(More)
In this work, we achieve gap amplification for the Small-Set Expansion problem. Specifically, we show that an instance of the Small-Set Expansion Problem with completeness ε and soundness 12 is at least as difficult as Small-Set Expansion with completeness ε and soundness f (ε), for any function f (ε) which grows faster than √ ε. We achieve this(More)
We consider two problems that arise in machine learning applications: the problem of recovering a planted sparse vector in a random linear subspace and the problem of decomposing a random low-rank overcomplete 3-tensor. For both problems, the best known guarantees are based on the sum-of-squares method. We develop new algorithms inspired by analyses of the(More)
Random constraint satisfaction problems (CSPs) are known to exhibit threshold phenomena: given a uniformly random instance of a CSP with <i>n</i> variables and <i>m</i> clauses, there is a value of <i>m</i> = &#206;&#169;(<i>n</i>) beyond which the CSP will be unsatisfiable with high probability. Strong refutation is the problem of certifying that no(More)
We give a lower bound of Ω̃( √ n) for the degree-4 Sum-of-Squares SDP relaxation for the planted clique problem. Specifically, we show that on an Erdös-Rényi graph G(n, 1 2 ), with high probability there is a feasible point for the degree-4 SOS relaxation of the clique problem with an objective value of Ω̃( √ n), so that the program cannot distinguish(More)
We develop fast spectral algorithms for tensor decomposition that match the robustness guarantees of the best known polynomial-time algorithms for this problem based on the sum-of-squares (SOS) semidefinite programming hierarchy. Our algorithms can decompose a 4-tensor with n-dimensional orthonormal components in the presence of error with constant spectral(More)
We study planted problems&amp;#x2014;finding hidden structures in random noisy inputs&amp;#x2014;through the lens of the sum-of-squares semidefinite programming hierarchy (SoS). This family of powerful semidefinite programs has recently yielded many new algorithms for planted problems, often achieving the best known polynomial-time guarantees in terms of(More)