Beyond Unfolding: Exact Recovery of Latent Convex Tensor Decomposition under Reshuffling

TL;DR

This paper establishes conditions for exact recovery in convex tensor decomposition models, introduces a generalized reshuffling operation, and demonstrates improved performance in image steganography applications.

Contribution

It provides a rigorous proof of exact recovery conditions for latent convex tensor decomposition and extends the model with reshuffling operations for broader applications.

Findings

Proves sufficient conditions for exact recovery in LCTD.

Introduces a generalized reshuffling operation for tensor modeling.

Outperforms state-of-the-art methods in image steganography.

Abstract

Exact recovery of tensor decomposition (TD) methods is a desirable property in both unsupervised learning and scientific data analysis. The numerical defects of TD methods, however, limit their practical applications on real-world data. As an alternative, convex tensor decomposition (CTD) was proposed to alleviate these problems, but its exact-recovery property is not properly addressed so far. To this end, we focus on latent convex tensor decomposition (LCTD), a practically widely-used CTD model, and rigorously prove a sufficient condition for its exact-recovery property. Furthermore, we show that such property can be also achieved by a more general model than LCTD. In the new model, we generalize the classic tensor (un-)folding into reshuffling operation, a more flexible mapping to relocate the entries of the matrix into a tensor. Armed with the reshuffling operations and exact-recovery property, we explore a totally novel application for (generalized) LCTD, i.e., image steganography. Experimental results on synthetic data validate our theory, and results on image steganography show that our method outperforms the state-of-the-art methods.