Identifiability of Kronecker-structured Dictionaries for Tensor Data
This provides theoretical guarantees for dictionary learning in tensor data analysis, which is incremental to existing sparse coding theory.
This paper establishes sufficient conditions for locally recovering coordinate dictionaries in Kronecker-structured dictionaries used for representing tensor data, showing that the sample complexity to recover K coordinate dictionaries up to error ε_k is max_{k∈[K]} O(m_k p_k^3 ε_k^{-2}) with high probability.
This paper derives sufficient conditions for local recovery of coordinate dictionaries comprising a Kronecker-structured dictionary that is used for representing $K$th-order tensor data. Tensor observations are assumed to be generated from a Kronecker-structured dictionary multiplied by sparse coefficient tensors that follow the separable sparsity model. This work provides sufficient conditions on the underlying coordinate dictionaries, coefficient and noise distributions, and number of samples that guarantee recovery of the individual coordinate dictionaries up to a specified error, as a local minimum of the objective function, with high probability. In particular, the sample complexity to recover $K$ coordinate dictionaries with dimensions $m_k \times p_k$ up to estimation error $\varepsilon_k$ is shown to be $\max_{k \in [K]}\mathcal{O}(m_kp_k^3\varepsilon_k^{-2})$.