---
title: Nonnegative Matrix Factorization Using Dirichlet-Distribution-Based Regularization
tags:  #machine larning #nmf #sparse #smooth #regularization #optimization #drichlet distribution #audio signal processing  
author: [Kitamura Laboratory](https://www.docswell.com/user/8262029599)
site: [Docswell](https://www.docswell.com/)
thumbnail: https://bcdn.docswell.com/page/D7YZRZYPJM.jpg?width=480
description: Haru Ogawa, Daichi Kitamura, and Shoma Ayano, "Nonnegative matrix factorization using Dirichlet-distribution-based regularization," Proceedings of Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC 2025), pp. 1229–1234, Singapore, Singapore, October 2025.
published: October 24, 25
canonical: https://www.docswell.com/s/8262029599/K4471Q-2025-10-24-101158
---
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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


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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]


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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


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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


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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
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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


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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)
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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.
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