rPCEN: Scale-Equivariant Rapid-Release Per-Channel Energy Normalization via Parallel Adjoint Block Scans

Learnable audio front-ends, like LEAF, utilize Per-Channel Energy Normalization (PCEN) for noise whitening and automatic gain control. However, the standard symmetric version of this technique (sPCEN) depends on a first-order Linear Time-Invariant (LTI) smoother, creating three primary limitations. First, its sequential, frame-by-frame processing is highly inefficient on GPUs, operating up to 53 times slower than parallelized layers. Second, because attack and release times are shared, loud sounds such as vowels can over-inflate the system's internal state. This phenomenon, termed Post-Vowel Forward-Masking Dropout, suppresses subsequent quieter sounds below the background noise level. Third, this swollen internal state causes Filterbank Jacobian Choking, which shrinks the backpropagated gradients sent to learnable filterbanks by a factor of up to 18.6. While non-causal workarounds like the temporal batch normalization (LogTBN) used in EfficientLEAF bypass the sequential recurrence issue, their performance suffers in the presence of short, localized bursts of noise. To resolve these challenges, this work introduces rPCEN, a scale-equivariant, rapid-release Linear Time-Varying (LTV) front-end. This new architecture is driven by a highly efficient Two-Level Adjoint Block Scan. The proposed method achieves nearly identical precision to sequential IIR recurrence (within 3 ULPs in FP32, or a 4.1 x 10^-7 relative error) while speeding up CUDA forward and backward passes by 14.5x to 53.0x. Furthermore, rPCEN restores the raw-waveform filterbank gradients by 18.6x. This improvement boosts filter-migration accuracy from 72.9% to 99.8% and cuts the zero-shot Word Error Rate on the Free Spoken Digit Dataset by 86.1% compared to both LEAF and EfficientLEAF baselines.