Reducibility and Statistical-Computational Gaps from Secret Leakage

Inference problems with conjectured statistical-computational gaps are ubiquitous throughout modern statistics, computer science and statistical physics. 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. 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 are similar to the common hardness assumption of planted clique. The insight in this work is that a slight generalization of the planted clique conjecture – secret leakage planted clique – gives rise to a variety of new average-case reduction techniques, yielding a web of reductions among problems with very different structure. Using variants of the planted clique conjecture for specific forms of secret leakage planted clique, 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. In particular, a k-partite hypergraph variant of the planted clique conjecture is sufficient to establish all of our computational lower bounds. Our techniques also reveal novel connections to combinatorial designs and to random matrix theory. This work gives the first evidence that an expanded set of hardness assumptions, such as for secret leakage planted clique, may be a key first step towards a more complete theory of reductions among statistical problems.

• 11 publications
• 32 publications
06/19/2018

Reducibility and Computational Lower Bounds for Problems with Planted Sparse Structure

The prototypical high-dimensional statistics problem entails finding a s...
08/08/2019

Average-Case Lower Bounds for Learning Sparse Mixtures, Robust Estimation and Semirandom Adversaries

This paper develops several average-case reduction techniques to show ne...
07/25/2021

Logspace Reducibility From Secret Leakage Planted Clique

The planted clique problem is well-studied in the context of observing, ...
09/12/2020

Open Problem: Average-Case Hardness of Hypergraphic Planted Clique Detection

We note the significance of hypergraphic planted clique (HPC) detection ...
07/16/2018

Group Invariance and Computational Sufficiency

Statistical sufficiency formalizes the notion of data reduction. In the ...
05/19/2022

The Franz-Parisi Criterion and Computational Trade-offs in High Dimensional Statistics

Many high-dimensional statistical inference problems are believed to pos...
03/17/2018

Learning Mixtures of Product Distributions via Higher Multilinear Moments

Learning mixtures of k binary product distributions is a central problem...