# The Multi-Dimensional Decomposition with Constraints

## TL;DR

This paper introduces a novel constrained multi-dimensional matrix decomposition method that simplifies the optimization problem, enabling efficient gradient computation and effective convergence in three-way decomposition tasks.

## Contribution

It presents a new approach transforming a complex matrix approximation problem into a simpler one with fewer unknowns, improving computational efficiency and convergence.

## Key findings

- Gradient computation complexity is only four times the function evaluation.
- The new algorithm requires minimal additional memory.
- Successful application to three-way decomposition with good convergence results.

## Abstract

We search for the best fit in Frobenius norm of $A \in {\mathbb C}^{m \times n}$ by a matrix product $B C^*$, where $B \in {\mathbb C}^{m \times r}$ and $C \in {\mathbb C}^{n \times r}$, $r \le m$ so $B = \{b_{ij}\}$, ($i=1, \dots, m$,~ $j=1, \dots, r$) definite by some unknown parameters $\sigma_1, \dots, \sigma_k$, $k << mr$ and all partial derivatives of $\displaystyle \frac{\delta b_{ij}}{\delta \sigma_l}$ are definite, bounded and can be computed analytically. We show that this problem transforms to a new minimization problem with only $k$ unknowns, with analytical computation of gradient of minimized function by all $\sigma$. The complexity of computation of gradient is only 4 times bigger than the complexity of computation of the function, and this new algorithm needs only $3mr$ additional memory. We apply this approach for solution of the three-way decomposition problem and obtain good results of convergence of Broyden algorithm.