High Dimensional Structured Superposition Models
This work addresses a foundational challenge in statistical learning for researchers dealing with complex structured data, though it appears incremental as it generalizes prior results.
The paper tackles the problem of estimating parameters in high-dimensional superposition models, where parameters are sums of multiple structured components, by presenting a simple estimator and providing non-asymptotic error bounds and sample complexity analysis.
High dimensional superposition models characterize observations using parameters which can be written as a sum of multiple component parameters, each with its own structure, e.g., sum of low rank and sparse matrices, sum of sparse and rotated sparse vectors, etc. In this paper, we consider general superposition models which allow sum of any number of component parameters, and each component structure can be characterized by any norm. We present a simple estimator for such models, give a geometric condition under which the components can be accurately estimated, characterize sample complexity of the estimator, and give high probability non-asymptotic bounds on the componentwise estimation error. We use tools from empirical processes and generic chaining for the statistical analysis, and our results, which substantially generalize prior work on superposition models, are in terms of Gaussian widths of suitable sets.