Joint Estimation of Common-Slope Decay Rates and Spatial Amplitudes Using Parameterized Nonnegative Matrix Factorization
Authors: Jeremy B. Bai, Filip Elvander, Sebastian J. Schlecht
Organizations: Multimedia Comms. & Signal Proc., Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany · Dept. of Information and Communications Engineering, Aalto University, Finland
We formulate joint estimation of common-slope decay rates and amplitudes from room impulse responses (RIRs) as parameterized nonnegative matrix factorization with the Itakura--Saito divergence as the loss function (IS-NMF). Estimation at each short-time Fourier transform frequency bin produces detailed reverberation time (RT) curves directly from RIR powers with no backward integration needed. Standard space-alternating generalized expectation-maximization (SAGE) algorithm yields closed-form amplitude updates and a convex subproblem for each decay rate update. To accelerate estimation, we introduce contribution-weighted SAGE, which emphasizes observations where each component contributes strongly to the modeled power. Experiments with synthetic data show accurate recovery of well-separated decays and faster loss reduction than standard SAGE. Application to measured coupled-room RIRs yields frequency-dependent RT curves and reveals complementary space-time contributions of the shared decay components.
Figures & tables
Figure 1: Parameterized NMF for common-slope estimation. A fixed-frequency slice of all the stacked RIR spectrograms (in blue frame) forms a space–time power matrix Y . Its expected power V is factored into nonnegative spatial amplitudes A and exponential temporal features Φ with decay rates shared across RIRs. The right panel shows an individual observed response (in green frame) and the two fitted exponential decay components shown as straight lines in dB scale.
Figure 2: Excess IS loss (solid, left axis) and L2 RT-pair error (dashed, right axis) over SAGE iterations. Loss is shown relative to the lowest value attained by the displayed methods within 1000 iterations.
Figure 3: Amplitude-profiled observed-data IS loss and estimated RT trajectories from a pooled linear-fit initialization. The gray region is excluded due to parameter symmetry between the two RTs.
Figure 4: Estimated RTs for the room-to-hallway data. Blue and orange distinguish shorter and longer RTs.
Figure 5: Space–time component contributions near 2 kHz for the two configurations. The blending colors show the distribution of modeled variance shares between the decay components and noise floor. The dashed line marks the doorway between the room and hallway.