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PI-AMFM: Permutation-Invariant Learning for Variable-Cardinality AM-FM Mode Decomposition in Biomedical Signal Analysis
Youngsun Kong, Ki H. Chon
TL;DR
Physiological signals contain nonstationary oscillatory modes whose number and dynamics vary, while neural decomposition approaches often assume a predefined component structure or rely on separate selection mechanisms. PI–AMFM uses permutation-invariant learning to predict a variable number of AM–FM modes. It outperformed compared methods across evaluated synthetic metrics and recovered cardiac and respiratory dynamics from PPG despite synthetic-only training.
Problem
Neural-network decomposition approaches often assume predefined component structures or prior frequency information, despite physiological signals containing varying numbers of time-varying oscillatory modes.
Source: §1 · p. 1Method
PI–AMFM combines a multiscale temporal encoder, Mamba backbone, component-presence estimation, and permutation-invariant training to predict a variable number of AM–FM modes.
Source: §1 · p. 1, §2.2 · p. 2Results
PI–AMFM outperformed all compared methods across evaluated metrics on in-distribution synthetic tests and recovered physiologically meaningful time-varying components from PPG despite synthetic-only training.
Source: §3.1 · p. 3, §4 · p. 4Takeaways & Limitations
The PPG results support the feasibility of recovering physiologically meaningful, time-varying components from real signals using a model trained only on synthetic AM–FM mixtures.
Source: §4 · p. 4The maximum mode count must be predefined; larger K may increase ambiguity among slots and make inactive-slot identification more difficult.
Source: §4 · p. 4
Abstract
from arXiv · showhide
Physiological recordings often contain nonstationary oscillatory components whose number and dynamics vary across signals. Amplitude- and frequency-modulated (AM-FM) representations are well suited to characterizing such dynamics and have shown broad utility in biomedical signal analysis. Recent approaches have incorporated neural networks to learn mode decomposition patterns from data, but component cardinality is often predefined or determined through separate stopping or selection mechanisms. We propose a permutation-invariant neural framework for variable-cardinality AM-FM mode decomposition (PI-AMFM). PI-AMFM combines a multiscale temporal encoder, Mamba backbone, and component-presence estimation, with permutation-invariant Hungarian matching during training. On synthetic AM-FM signals, PI-AMFM achieved lower decomposition, instantaneous-frequency, reconstruction, and mode-count errors than the compared methods while preserving the overall trajectory pattern in a crossing-chirp example. On photoplethysmographic recordings, recovered modes captured cardiac and respiratory dynamics despite training only on synthetic signals. These results support the feasibility of PI-AMFM for variable-cardinality decomposition of nonstationary biomedical signals.
1. INTRODUCTION
Physiological recordings mix oscillations with time-varying amplitudes and frequencies, motivating AM–FM decomposition. PI–AMFM addresses the challenge of predicting an unknown number of modes without predefined frequency trajectories or extraction order.
1. INTRODUCTION
- Physiological signals can mix nonstationary oscillations from processes operating at different, time-varying scales, as in PPG’s cardiac oscillation and respiratory modulation.Noise and motion artifacts further complicate component separation.Source: §1 · p. 1
- Existing approaches include methods that estimate AM–FM amplitude and instantaneous frequency, methods requiring a predefined mode count, and methods that extract modes adaptively or sequentially.VFCDM characterizes time-varying amplitude and IF, while VNCMD requires the number of modes in advance; ACMP and PDS–SSA do not require a prior component count.Source: §1 · p. 1
- PI–AMFM directly predicts a variable number of AM–FM modes without predefined IF trajectories or extraction order, combining learning-based representation with permutation-invariant prediction.The paper identifies predefined component structures or prior frequency information as common in neural-network formulations.Source: §1 · p. 1
2. METHODOLOGY
PI–AMFM uses a multiscale Mamba encoder and permutation-invariant slots to predict AM–FM modes. The methodology trains and evaluates the model on synthetic signals and assesses physiological recovery from PPG recordings.
2.1. PI–AMFM Architecture
- The architecture combines parallel multiscale convolutions, six bidirectional Mamba blocks, and eight slots that predict complex modes and presence scores; Hungarian matching handles arbitrary mode order during training.Slots with presence scores of at least 0.5 are active, allowing up to eight modes without requiring the component count at inference.Source: §2.1 · p. 1, Figure 1 · p. 2
2.2. Synthetic AM–FM Data and Model Training
- Training mixtures contain one to eight AM–FM modes with varied amplitude envelopes and frequency trajectories, additive Gaussian noise, and trajectories that may overlap or cross.Signal lengths range from 512 to 4096 samples, with 100,000 training examples and 10,000 each for validation and testing.Source: §2.2 · p. 2
- The composite loss combines matched-mode complex error, active/inactive-slot classification, mixture reconstruction error, and a penalty on unmatched-slot outputs.Training used AdamW for 50 epochs with validation-based model selection.Source: §2.2 · p. 2
2.3. PPG Respiratory Data and Evaluation
- The physiological evaluation uses 53 eight-minute BIDMC recordings, resampling PPG from 125 Hz to 10 Hz before baseline centering and RMS normalization, without bandpass filtering.Reference HR comes from ECG; RR is derived from manually annotated breaths.Source: §2.3 · p. 2
2.4. Benchmarks
- Benchmarks include DD–ACMD, IDSD, EMD, CEEMDAN, and FIF; methods that extract more than 20 modes use a 20-mode cap.IDSD was adapted to variable-length synthetic data and retrained on the same synthetic dataset using PI–AMFM’s optimization protocol while retaining its original loss.Source: §2.4 · p. 2
2.4.1. Synthetic Signal Decomposition
- Synthetic evaluation matches predicted and true modes using normalized complex squared error, then measures mode, instantaneous-frequency, reconstruction, and mode-count errors.IF error is evaluated only where adjacent reference samples are active and exceed 10^-3 of the reference-mode peak; crossing behavior is tested with two equal-amplitude chirps crossing at the midpoint.Source: §2.4.1 · pp. 2–3
2.4.2. PPG Respiratory Component Extraction
- PPG cardiac- and respiratory-related modes are selected by maximum integrated Welch power in their respective frequency bands, with an additional rule for possible cardiac harmonics.No reference measurements are used for mode selection, and the analysis is intended to demonstrate physiologically meaningful recovery rather than general-purpose attribution.Source: §2.4.2 · p. 3
- Heart- and respiratory-rate estimates use local phase slopes, followed by clipping and median filtering, and are compared with reference measurements using pooled MAE and RMSE.The analysis excludes the first and last 10 seconds; HR is interpolated to reference timestamps, while RR is averaged over valid inter-breath intervals.Source: §2.4.2 · p. 3
3. RESULTS
On synthetic signals, PI–AMFM outperformed all compared methods across evaluated metrics and preserved the overall crossing-chirp pattern despite IF fluctuations. On BIDMC PPG, it achieved the lowest respiratory-rate errors, while DD–ACMD was slightly better for heart rate.
3.1. Synthetic Signal Decomposition
- PI–AMFM identified two active modes while compared methods extracted 4–20, preserving the overall crossing-chirp pattern despite noticeable IF fluctuations.The IF fluctuations were most evident early in the signal and near the frequency crossing.Source: §3.1 · p. 3
- Greater flexibility in PI–AMFM’s recovered AM–FM representation may improve source-specific trajectory recovery at the cost of weaker local IF smoothing.The authors suggest explicit frequency-selective mechanisms in DD–ACMD, FIF, and IDSD may favor smoother modes and suppress local fluctuations.Source: §3.1 · p. 3
- PI–AMFM outperformed all compared methods, including same-training-set IDSD, across evaluated metrics on the in-distribution synthetic test set.The test followed the training signal model and enabled controlled evaluation with component-level ground truth unavailable in real-world recordings.Source: §3.1 · pp. 3–4, Table 1 · p. 4
3.2. PPG Respiratory Component Extraction
- On BIDMC PPG, PI–AMFM achieved the lowest respiratory-rate errors, while DD–ACMD was slightly better for heart rate by 0.2 beats/min MAE and 0.8 beats/min RMSE.DD–ACMD’s RR RMSE was approximately 2.7 breaths/min (33–34%) higher than PI–AMFM’s; Fig. 3 illustrates recovered cardiac and respiratory IF trajectories.Source: §3.2 · p. 4, Table 2 · p. 4
4. DISCUSSION AND CONCLUSION
PI–AMFM estimates mode presence jointly across candidate slots, avoiding sequential extraction but requiring a predefined maximum mode count. The authors frame PPG as a feasibility demonstration and identify broader signal types and recording conditions for future evaluation.
4. DISCUSSION AND CONCLUSION
- Mamba was selected empirically after preliminary chirp-crossing comparisons with CNNs, LSTMs, Transformers, TCNs, and S4D.The authors suggest its input-dependent selective state-space mechanism may adaptively propagate and update temporally relevant information across long sequences.Source: §4 · p. 4
- PI–AMFM estimates mode presence jointly across candidate slots, avoiding sequential extraction and residual-update error propagation, but requires a predefined maximum mode count.The study used K = 8 based on empirical observations; larger K may increase slot ambiguity and complicate inactive-slot identification.Source: §4 · p. 4
- Unlike SignalNet and SlotFlow, which estimate fixed sinusoidal parameters, PI–AMFM predicts time-varying AM–FM modes with per-slot presence estimates.The passage also contrasts PI–AMFM’s direct, parallel permutation-invariant prediction with sequential residual-based extraction.Source: §4 · p. 4
- The PPG evaluation demonstrates recovery of physiologically meaningful, time-varying components after training only on synthetic mixtures, not a new HR or RR estimation method.The authors propose evaluating electrodermal activity, ECG-derived skin nerve activity, and EEG, as well as robustness across signal types and recording conditions.Source: §4 · p. 4
5. COMPLIANCE WITH ETHICAL STANDARDS
The study used synthetic signals and public, de-identified BIDMC data from PhysioNet; the dataset’s public status meant institutional IRB review was not required.
5. COMPLIANCE WITH ETHICAL STANDARDS
- Public, de-identified BIDMC data from PhysioNet were used, so institutional IRB review was not required.Source: §5 · p. 5