Factorization with Delay Embedding
For any partially observed vector 
with observed index set 
, one can consider the following matrix factorization for reconstructing unobserved entries:
)\|_{F}^{2},)
where 
denotes the orthogonal projection supported on the observed index set , which is defined as follows,
]_{t}=\left\{\begin{array}{ll} y_{t}, & \text{if $t\in\Omega$,} \\ 0, & \text{otherwise}. \\ \end{array}\right.)
In the matrix factorization, 
are the rank-
latent factor matrices. 
is the circulant delay embedding operator with the kernel size 
(see Section 3.2 & Equation (14) of this paper for definition), while 
is the pseudoinverse of the circulant delay embedding.
Given any vector , its circulant deley embedding with kernel size is defined as follows,
\triangleq\begin{bmatrix} y_1 & y_2 & \cdots & y_T \\ y_2 & y_3 & \cdots & y_1 \\ \vdots & \vdots & \ddots & \vdots \\ y_\tau & y_{\tau+1} & \cdots & y_{\tau-1} \end{bmatrix}\in\mathbb{R}^{\tau\times T}.)
The question is how to utilize discrete Fourier transform to solve this optimization problem?
