Joint Higher-Order Low-Rank Tensor Decompositions for Multichannel Acoustic System Compression
Multichannel room impulse responses (RIRs) in acoustics are often long and therefore the sample count becomes large if seen as finite impulse response (FIR) filters, making storage and real-time convolution costly. Pole-zero (PZ) models are an established way to represent RIRs compactly as infinite impulse response (IIR) transfer functions. We show that a PZ model implies a low-rank canonical polyadic decomposition (CPD) of tensorized RIRs: the early reflections form a sparse rank-1 sum, while the late reverberation admits a CPD with factors determined by the poles. This motivates higher-order-tensor-based compression of RIRs via the block term decomposition (BTD), a generalization of the CPD. We propose two multichannel compression schemes: (i) a stacked multichannel block term decomposition (SMCBTD) applied to a (D+1)-mode tensor of RIRs, and (ii) a joint multichannel block term decomposition (JMCBTD) that shares subsets of factors across channels discovered by spectral clustering on factor-subspace affinity graphs. Both are optimized by an alternating higher-order orthogonal iteration (HOOI) procedure. On simulated rooms as well as measured datasets, the proposed methods achieve consistently lower reconstruction error at equal compression ratios compared to separate single-channel decompositions and a generalized low-rank approximation of matrices (GLRAM) baseline. The largest gains occur with scalar cores (i.e., CPD), supporting the PZ-to-CPD modelling link. We also find D = 3 to be a practical sweet spot, offering most of the achievable gain at moderate compute/memory cost. The compressed models enable efficient downstream multichannel convolution.