Statistical-Computational Tradeoffs in Planted Problems and Submatrix Localization with a Growing Number of Clusters and Submatrices (1402.1267v3)

Abstract: We consider two closely related problems: planted clustering and submatrix localization. The planted clustering problem assumes that a random graph is generated based on some underlying clusters of the nodes; the task is to recover these clusters given the graph. The submatrix localization problem concerns locating hidden submatrices with elevated means inside a large real-valued random matrix. Of particular interest is the setting where the number of clusters/submatrices is allowed to grow unbounded with the problem size. These formulations cover several classical models such as planted clique, planted densest subgraph, planted partition, planted coloring, and stochastic block model, which are widely used for studying community detection and clustering/bi-clustering. For both problems, we show that the space of the model parameters (cluster/submatrix size, cluster density, and submatrix mean) can be partitioned into four disjoint regions corresponding to decreasing statistical and computational complexities: (1) the \emph{impossible} regime, where all algorithms fail; (2) the \emph{hard} regime, where the computationally expensive Maximum Likelihood Estimator (MLE) succeeds; (3) the \emph{easy} regime, where the polynomial-time convexified MLE succeeds; (4) the \emph{simple} regime, where a simple counting/thresholding procedure succeeds. Moreover, we show that each of these algorithms provably fails in the previous harder regimes. Our theorems establish the minimax recovery limit, which are tight up to constants and hold with a growing number of clusters/submatrices, and provide a stronger performance guarantee than previously known for polynomial-time algorithms. Our study demonstrates the tradeoffs between statistical and computational considerations, and suggests that the minimax recovery limit may not be achievable by polynomial-time algorithms.

• The paper defines four distinct regimes that capture the interplay between statistical recoverability and computational feasibility in planted clustering and submatrix localization.
• The paper shows that maximum likelihood estimators reach minimax optimality while efficient convex relaxations are hindered by a spectral barrier.
• The paper highlights a significant gap between theoretical recovery limits and practical algorithms, guiding future research on scalable statistical methods.

Statistical-Computational Tradeoffs in Planted Problems and Submatrix Localization

The paper under consideration examines the intricate balance between statistical and computational aspects involved in solving two closely related problems: the planted clustering problem and the submatrix localization problem. These problems have significant implications across various fields, including computer science and statistical physics, due to their applications in modeling community structures in networks and capturing block structures in data matrices.

Planted Clustering and Submatrix Localization

The planted clustering problem involves recovering underlying clusters in a graph that is randomly generated while favoring certain node groupings. Similarly, submatrix localization focuses on identifying hidden submatrices with elevated mean values within a large random matrix. Both problems share the common feature of dealing with structured noise and manifest a high-dimensional setting where the number of clusters or submatrices grows unbounded with problem size.

Four Distinct Regimes

For both problems, the authors delineate four regimes based on varying levels of statistical and computational difficulty. These regimes represent distinct tradeoffs between the ability to statistically recover the hidden structures and the computational resources required:

1. Impossible Regime: No algorithm can succeed, regardless of its complexity. Conditions exist where even with infinite computation, recovery is statistically impossible due to inherent limitations in the data's informativeness.
2. Hard Regime: Only computationally expensive algorithms, such as the Maximum Likelihood Estimator (MLE), can successfully recover the structures. This regime represents the limits of what is theoretically possible with undefined computational budgets.
3. Easy Regime: Efficient polynomial-time algorithms, particularly convex relaxations of the MLE, are able to recover the structures. These algorithms are computationally feasible, though they may not achieve the theoretical limits of recovery.
4. Simple Regime: Extremely simple algorithms, often linear time or close, like counting or thresholding, suffice for recovery. While computationally attractive, these algorithms succeed only when the problem is inherently simple from a statistical standpoint.

Theoretical and Computational Boundaries

The paper sets out rigorous theoretical boundaries for each of the regimes, showing sharp conditions under which recovery is feasible or infeasible. The results are non-asymptotic and hence applicable to finite problem sizes, capturing the detailed boundary conditions for cluster recovery in large datasets.

The Hard Regime and Minimax Optimality: The MLE-based approaches achieve minimax recovery limits, meaning they can recover clusters or submatrices up to the point where it is statistically impossible to do better given the data.

The Easy Regime’s Spectral Barrier: Convex relaxations, while efficient, are shown to encounter a "spectral barrier", which limits their ability to resolve more complex structures that might be recoverable by intractable algorithms.

Gap Between Statistical and Computational Efficiency: A notable gap exists between the theoretical recovery limits and those achievable by polynomial-time algorithms, suggesting that certain problems may inherently resist efficient solutions. This aligns with the conjectures of information-computation tradeoffs in average-case complexity.

Future Directions and Implications

The insights from this paper are broad, touching upon key questions around the feasibility of computational methods in statistics and machine learning. Extending this framework to more complex models, such as those with overlapping structures or unknown parameters, remains an open area for research.

The paper suggests a movement toward understanding computational complexity not as an isolated concern but as an integrated aspect of statistical problem solving. This has potential implications for algorithm design, particularly in fields relying on large-scale network data and complex matrix models.

The findings and methodologies discussed are instrumental for researchers aiming to push the limits of what can be computationally achieved in large-scale statistical problems, providing a foundation for balancing between computational tractability and statistical optimality.