  In data-driven control and machine learning, a common requirement involves
breaking down large matrices into smaller, low-rank factors that possess
specific levels of sparsity. This paper introduces an innovative solution to
the orthogonal nonnegative matrix factorization (ONMF) problem. The objective
is to approximate input data by using two low-rank nonnegative matrices,
adhering to both orthogonality and $\ell_0$-norm sparsity constraints. the
proposed maximum-entropy-principle based framework ensures orthogonality and
sparsity of features or the mixing matrix, while maintaining nonnegativity in
both. Additionally, the methodology offers a quantitative determination of the
``true'' number of underlying features, a crucial hyperparameter for ONMF.
Experimental evaluation on synthetic and a standard datasets highlights the
method's superiority in terms of sparsity, orthogonality, and computational
speed compared to existing approaches. Notably, the proposed method achieves
comparable or improved reconstruction errors in line with the literature.
