Reducibility and Statistical-Computational Gaps from Secret Leakage

Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science, statistical physics and discrete probability. While there has been success evidencing these gaps from the failure of restricted classes of algorithms, progress towards a more traditional reduction-based approach to computational complexity in statistical inference has been limited. These average-case problems are each tied to a different natural distribution, high-dimensional structure and conjecturally hard parameter regime, leaving reductions among them technically challenging. Despite a flurry of recent success in developing such techniques, existing reductions have largely been limited to inference problems with similar structure – primarily mapping among problems representable as a sparse submatrix signal plus a noise matrix, which is similar to the common starting hardness assumption of planted clique ($\textsc{pc}$). The insight in this work is that a slight generalization of the planted clique conjecture – secret leakage planted clique ($\textsc{pc}_\rho$), wherein a small amount of information about the hidden clique is revealed – gives rise to a variety of new average-case reduction techniques, yielding a web of reductions relating statistical problems with very different structure. Based on generalizations of the planted clique conjecture to specific forms of $\textsc{pc}_\rho$, we deduce tight statistical-computational tradeoffs for a diverse range of problems including robust sparse mean estimation, mixtures of sparse linear regressions, robust sparse linear regression, tensor PCA, variants of dense $k$-block stochastic block models, negatively correlated sparse PCA, semirandom planted dense subgraph, detection in hidden partition models and a universality principle for learning sparse mixtures. This gives the first reduction-based evidence for a number of conjectured statistical-computational gaps. We introduce a number of n