Nonnegative Matrix Factorization Using Dirichlet-Distribution-Based Regularization

17th Asia Pacific Signal and Information Processing Association Annual Summit and Conference Session: Deep Learning: Algorithm, Implementations, and Applications Time: Wed., 23. Oct., 10:30-12:00 (UTC +7) Nonnegative Matrix Factorization Using Dirichlet-Distribution-Based Regularization Haru Ogawa*, Daichi Kitamura*, Shoma Ayano* *National Institute of Technology, Kagawa College, Japan

2 Background • Nonnegative matrix factorization: NMF [Lee+, 1999] – A method to approximate a nonnegative matrix by the product of two low-rank nonnegative matrices • Applications ⁃ Audio source separation ⁃ Image feature extraction ⁃ Topic modeling ⁃ Bioinformatics ⁃ Recommender Systems etc. [Lee+, 1999]

Background • Regularized NMF – Regularized NMF is an extension of standard NMF that introduces additional constraints to improve the quality of the results • Representative Regularized NMF – Sparse NMF[Hoyer, 2004], [Le Roux+, 2015], [Marmin+, 2023], etc • L1-norm regularized NMF that induces sparsity in the factor matrices – Smooth NMF [Virtanen+, 2007], etc • Inducing smoothness via adjacency-difference regularization • Limitations of Existing Regularized NMF – No unified regularization for sparsity and smoothness – scale indeterminacy between and 3

Background • Regularized NMF – Regularized NMF is an extension of standard NMF that introduces additional constraints to improve the quality of the results • Representative Regularized NMF – Sparse NMF[Hoyer, 2004], [Le Roux+, 2015], [Marmin+, 2023], etc • L1-norm regularized NMF that induces sparsity in the factor matrices Novelty Dirichlet-regularized NMF Unifies sparsity and smoothness scale ambiguity – Smooth NMF [Virtanen+, resolves 2007], etc • Inducing smoothness via adjacency-difference regularization • Limitations of Existing Regularized NMF – No unified regularization for sparsity and smoothness – scale indeterminacy between and 4

Dirichlet NMF • Dirichlet distribution – The Dirichlet distribution has a parameter that controls the concentration of each vector – By tuning this parameter, the resulting basis can become either sparse or smooth Smooth Sparse Sparse Sparse • Objective Function of Dirichlet NMF – Each basis vector is sampled from the Dirichlet distribution to balance sparsity and smoothness Regularization term from Dirichlet prior 5

Application to Howling Suppression(1/2) 6 • Experiment overview – The howling suppression performance was evaluated using speech signals with simulated feedback noise Regularize to be sparse Observed spectrogram Basis matrix Howling Extract bases and coefficients except for howling components Coefficient matrix Howling is suppressed

Application to Howling Suppression(2/2) • Results – Average SDR over 3 signals × 50 initializations Higher is better SDR [dB] L1-sparse NMF Dirichlet NMF (proposed) 7

Conclusion • Problems – Conventional NMF cannot handle both sparsity and smoothness,and suffers from scale indeterminacy between W and H • Proposed Method – Proposed Dirichlet NMF using a Dirichlet prior on each basis vector – It unifies sparsity and smoothness control and resolves scale indeterminacy • Experiments – Applied to speech signals with simulated feedback noise. – Outperformed L₁-sparse NMF in SDR and speech preservation • The proposed Dirichlet NMF provides a unified and effective framework for regularized matrix factorization Thank you for your attention. 8