for directed and residual reverberations. Juliano G. C. Ribeiro1, Shoichi Koyama2, and Hiroshi Saruwatari1 1Graduate School of Information Science and Technology, The University of Tokyo 2Digital Content and Media Sciences Research Division, National Institute of Informatics (NII) 1 / 20

propagation has several practical applications ñ acoustic transfer function (ATF). Our objective: region-to-region interpolation ñ continuous variation within constrained regions. 2 / 20

using spherical wave functions. [Ribeiro+, 2020]: Embed physical properties using reproducing kernel Hilbert space (RKHS). [Ribeiro+, 2022]: Extend kernel function to include basic directionality. ñThe kernel method outperformed wave function expansion Better performance frequency-by-frequency and spatially. More broadly applicable: no need for truncation orders, agnostic to geometry of array. Can be generalized with plane wave expansion weighting. 3 / 20

Ω Ă R3. Distribute L loudspeakers in source region ΩS Ă Ω and M microphones in receiver region ΩR Ă Ω. Interpolate ATF between regions from the N “ LM measurements. Source region: Receiver region: Region 4 / 20

kq “ hDpr|s, kq ` hRpr|s, kq. Direct component hD: known, equivalent to point source in free-field. hDpr|s, kq “ G0pr|s, kq “ eik}r´s} 4π}r ´ s} Reverberant component hR: unknown, but satisfies Helmholtz equation on position variables. p∇2 r ` k2qhRpr|s, kq “ p∇2 s ` k2qhRpr|s, kq “ 0 Reciprocity @r|s: hpr|s, kq “ hps|r, kq 5 / 20

ˆ hR “ argmin gPH N ÿ n“1 |yn ´ gpqnq|2 ` λ}g}2 H Feature space H holds properties of reverberant field. If H is a RKHS: optimization has closed form solution called kernel ridge regression. Must define H to be a RKHS with flexible form. Herglotz wave function generalized model. 6 / 20

kernel capable of learning the specific properties of the environment from the data. Weight function w divided into two components: w “ wdir ` wres. wdir represents directed components: plane wave components with high amplitude but sparse angular representation. wres represents residual components: plane wave components with lower amplitudes, but dense angular representation. Adaptive kernel κ superposition of the component kernels: κ “ κdir ` κres. 9 / 20

in order to freely learn patterns. Neural network parameters: θ θ θ. Simple architecture: 2 fully connected hidden layers with 20 neurons each and tanh activation. Capable of learning irregular shapes and patterns. 12 / 20

No closed form ñ integral operator is approximated numerically. Reciprocity is learned by data augmentation. Since the properties are enforced by the integrator and training process, θ θ θ learns freely. 13 / 20

θ θ θ chosen as to minimize the leave-one-out cross validation error. ELOOpα, β, θq “ 1 N N ÿ n“1 |ˆ hRpqn; ˘ Qn, ˘ yn, wq ´ yn|2, β β β and θ θ θ optimized using the gradient descent method and α α α using the reduced gradient method with line search. Unlike a deep learning model, this model needs no outside measurements. It learns from the N recorded ATFs alone. 14 / 20

on shoebox-shaped room with dimensions 3.2 m ˆ 4.0 m ˆ 2.7 m. Reverberation time T60 “ 0.45 s. Radii of ΩR and ΩS both 0.2 m. Center of ΩR; r0 “ r´0.65, ´0.80, ´0.48sT m. Center of ΩS; s0 “ r0.65, 0.80, 0.48sT m. L “ M “ 41. Noise was added so SNR “ 20dB. Compared Uniform, Sunkern sphere, Residual only, Directed only, and Proposed. 15 / 20

directed and residual reverberations in order to derive an adaptive kernel. By guaranteeing the general physical properties of the ATF are respected using the Herglotz wave function, the weight function associated with the adaptive kernel is free to learn without further restrictions. The formulated model learns the optimal model parameters using internal data only, with no need to experiment outside the derivation data set. The proposed method outperformed the previously established methods in both a frequency-by-frequency and on a spatial basis. We also evaluated the proposed method against the directed and residual weight components separately, confirming the advantages of optimizing both together. 20 / 20