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In this paper, we construct general machinery for proving Sum-of-Squares lower bounds on certification problems by generalizing the techniques used by [BHK + 16] to prove Sum-of-Squares lower bounds for planted clique. Using this machinery, we prove degree n ε Sum-of-Squares lower bounds for tensor PCA, the Wishart model of sparse PCA, and a variant of planted clique which we call planted slightly denser subgraph.

Given a graph and an integer k, Densest k-Subgraph is the algorithmic task of finding the subgraph on k vertices with the maximum number of edges. This is a fundamental problem that has been subject to intense study for decades, with applications spanning a wide variety of fields. The state-of-the-art algorithm is an O(n 1/4+ε )-factor approximation (for any ε > 0) due to Bhaskara et al. [STOC '10]. Moreover, the so-called log-density framework predicts that this is optimal, i.e. it is impossible for an efficient algorithm to achieve an O(n 1/4−ε )-factor approximation. In the average case, Densest k-Subgraph is a prototypical noisy inference task which is conjectured to exhibit a statistical-computational gap.In this work, we provide the strongest evidence yet of hardness for Densest k-Subgraph by showing matching lower bounds against the powerful Sum-of-Squares (SoS) algorithm, a meta-algorithm based on convex programming that achieves stateof-art algorithmic guarantees for many optimization and inference problems. For k ≤ n

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We study the problem of reconstructing a causal graphical model from data in the presence of latent variables. The main problem of interest is recovering the causal structure over the latent variables while allowing for general, potentially nonlinear dependence between the variables. In many practical problems, the dependence between raw observations (e.g. pixels in an image) is much less relevant than the dependence between certain high-level, latent features (e.g. concepts or objects), and this is the setting of interest. We provide conditions under which both the latent representations and the underlying latent causal model are identifiable by a reduction to a mixture oracle. The proof is constructive, and leads to several algorithms for explicitly reconstructing the full graphical model. We discuss efficient algorithms and provide experiments illustrating the algorithms in practice.