# A theoretical guarantee for data completion via geometric separation

## TL;DR

This paper provides a theoretical guarantee for data completion methods that leverage geometric separation, especially when data can be modeled as a mixture of different structural components.

## Contribution

It introduces a unified theoretical framework that guarantees successful data recovery using combined separation and completion techniques for structured, incomplete data.

## Key findings

- Theoretical proof of success for combined separation and completion methods.
- Generalization of previous proofs to more complex data structures.
- Applicable to data modeled as superpositions of multiple structures.

## Abstract

Scientific and commercial data is often incomplete. Recovery of the missing information is an important pre-processing step in data analysis. Real-world data can in many cases be represented as a superposition of two or more different types of structures. For example, images may often be decomposed into texture and cartoon-like components. When incomplete data comes from a distribution well-represented as a mixture of different structures, a sparsity-based method combining concepts from data completion and data separation can successfully recover the missing data. This short note presents a theoretical guarantee for success of the combined separation and completion approach which generalizes proofs from the distinct problems.