  The Matrix Decomposition techniques have been a vital computational approach
to analyzing the hierarchy of functional connectivity in the human brain.
However, there are still four shortcomings of these methodologies: 1). Large
training samples; 2). Manually tuning hyperparameters; 3). Time-consuming and
require extensive computational source; 4). It cannot guarantee convergence to
a unique fixed point.
  Therefore, we propose a novel deep matrix factorization technique called Deep
Linear Matrix Approximate Reconstruction (DELMAR) to bridge the abovementioned
gaps. The advantages of the proposed method are: at first, proposed DELMAR can
estimate the important hyperparameters automatically; furthermore, DELMAR
employs the matrix backpropagation to reduce the potential accumulative errors;
finally, an orthogonal projection is introduced to update all variables of
DELMAR rather than directly calculating the inverse matrices.
  The validation experiments of three peer methods and DELMAR using real
functional MRI signal of the human brain demonstrates that our proposed method
can efficiently identify the spatial feature in fMRI signal even faster and
more accurately than other peer methods. Moreover, the theoretical analyses
indicate that DELMAR can converge to the unique fixed point and even enable the
accurate approximation of original input as DNNs.
