A new framework for tensor PCA based on trace invariants

We consider the Principal Component Analysis (PCA) problem for tensors $T \in (\mathbb{R}^n)^{\otimes k}$ of large dimension $n$ and of arbitrary order $k\geq 3$. It consists in recovering a spike $v_0^{\otimes k}$ (related to a signal vector $v_0 \in \mathbb{R}^n$) corrupted by a Gaussian noise tensor $Z \in (\mathbb{R}^n)^{\otimes k}$ such that $T=\beta v_0^{\otimes k} + Z$ where $\beta$ is the signal-to-noise ratio. In this paper, we propose a new framework based on tools developed by the theoretical physics community to address this important problem. They consist in trace invariants of tensors built by judicious contractions (extension of matrix product) of the indices of the tensor $T$. Inspired by these tools, we introduce a new process that builds for each invariant a matrix whose top eigenvector is correlated to the signal for $\beta$ sufficiently large. Then, we give examples of classes of invariants for which we demonstrate that this correlation happens above the best algorithmic threshold ($\beta\geq n^{k/4}$) known so far. This method has many algorithmic advantages: (i) it provides a detection algorithm linear in time and that has only $\mathcal{O}(1)$ memory requirements (ii) the algorithms are very suitable for parallel architectures and have a lot of potential of optimization given the simplicity of the mathematical tools involved (iii) experimental results show an improvement of the state of the art for the symmetric tensor PCA. Furthermore, this framework allows more general applications by being able to theoretically study the recovery of a spike in the form of $v_1 \otimes \dots \otimes v_k$ with different dimensions ($T \in \mathbb{R}^{n_1\times n_2\times \dots \times n_k}$ with $n_1,\dots, n_k \in \mathbb{N}$) as well as the recovery of a sum of different orthogonal spikes. We provide experimental results to these different cases that match well with our theoretical findings.