Task-Directed Residual AddUNet:Perfect-Reconstruction Routing for Full-Rate Representations

Summary

This paper reframes additive U-Nets via perfect-reconstruction (PR) multirate filter banks. It first argues that a constrained additive U-Net's survivor–skip structure is exactly equivalent to a critically sampled two-channel PR filter bank, then removes critical sampling to obtain a full-rate formulation free of quadrature and complementary-subband constraints, and finally specializes to a residual form where each stage routes ql = Ql(p_{l-1}) away and propagates pl = p_{l-1} − ql. Proposition 1 shows the joint representation (pL, q1..qL) reconstructs the input exactly by telescoping, for arbitrary shape-compatible and possibly nonlinear Ql, without invertibility, a synthesis bank, reconstruction loss, or decoder; identity-shortcut ResNet blocks are likewise recoverable when their residual is retained. Experiments: four orthogonal sinusoids routed with relative errors below 3.2×10⁻¹³ and reconstruction at 1.1×10⁻¹⁶; ResNet-34 layer1 recovery at ~10⁻⁸ (float32); TIMIT phone recognition where a six-stage front-end with a fixed BiLSTM-CTC backend lowers mean test PER from 28.60±2.09% to 25.76±0.41% (C=2, 9×5 kernel, three seeds), with capacity and context showing non-monotonic effects and speaker probes dropping only ~3 points. Key limitations: Proposition 1 is a trivial telescoping identity; the authors concede no capacity-matched non-PR front-end control was run, so gains cannot be attributed to PR/routing structure specifically; absolute PERs sit well above TIMIT state of the art; no code release or significance testing.