Intra-Phase Representation (IPR)

• IPR is a method that independently extracts and encodes phase-specific features from periodic data, preserving distinct morphology and variation cues.
• It employs tailored architectures such as CNNs, Gabor filters, graph-attention, and contrastive losses to enhance signal discrimination and feature stability.
• IPR improves model performance in biosignal analysis and quantum state localization by mitigating inter-phase interference and preserving inherent structural details.

Intra-Phase Representation (IPR) encompasses a set of methodologies in signal processing and quantum physics, in which the structure, morphology, or localization properties within distinct, well-defined intervals—termed phases or periods—are extracted, quantified, and encoded for downstream tasks. In biosignal analysis, IPR refers to independent encoding of physiologically or temporally defined segments; in quantum many-body theory, it denotes the quantification of wavefunction localization within a discrete basis. The term appears in contemporary literature across biometry and self-supervised learning, as well as in studies of quantum phase transitions.

1. Phase-Specific Feature Extraction and Motivation

IPR in the context of biosignal processing is designed to independently encode information within each defined segment of a periodic cycle, for example, within an ECG heartbeat cycle (P, QRS, ST, T/U phases). By isolating and separately mapping these intra-phase signals to learned representations, the method prevents cross-phase entanglement—avoiding interference between features that pertain to distinct physiological events (Huang et al., 1 Jan 2026). The rationale is that morphological cues critical for discrimination (identity, disease, etc.) often manifest in a phase-specific manner; their integrity and interpretability can be compromised by joint modeling. This independent processing facilitates subsequent structured fusion, yielding robust composite embeddings.

In self-supervised and semi-supervised settings, intra-phase representation learning provides a mechanism to extract stable morphology or invariant structure from repeated segments within a noisy or variably annotated recording. This is evidenced in inter-intra period-aware ECG learning, where the aggregation of single-beat segments into a prototypical representation enables discrimination irrespective of inter-beat variability (Zhu et al., 2024).

2. Formal Definitions and Architectures

Biometric Signal Analysis

Given a periodic signal (e.g., ECG), define the set of $m$ segments, indexed by $i$:

$x_i \in \mathbb{R}^{L_i}$

Each phase-specific encoder $f_i$ is applied:

$\mathbf{h}_i = f_i(x_i) \in \mathbb{R}^D$

In "Hear the Heartbeat in Phases" (Huang et al., 1 Jan 2026), $f_i$ consists of a dual-branch Morphology–Variation Feature Extractor (MVFE):

• Morphology branch (MFEB): 1D CNN stack for coarse waveform capture.
• Variation branch (VFEB): Gabor-convolutional filters for fine-grained variation, zero-mean enforced to remove DC components. The resulting vectors $\mathbf{z}_m^i, \mathbf{z}_v^i$ are fused by a phase-specific Graph-Attention (PR-GAT) mechanism and attention pooling to yield $\mathbf{h}_i$.

Self-Supervised Representation Learning

In "Self-supervised inter-intra period-aware ECG representation learning" (Zhu et al., 2024), intra-phase representation is defined via:

• R-peak detection to segment every single beat.
• Windowed extraction and alignment:

$x_j = X[:, t_j-\Delta : t_j+\Delta]$

• Element-wise median to yield the stable, denoised morphology:

$$\widetilde{x} = \operatorname{median}\left\{x_1, \dots, x_n\right\}$$

Feature extractors $f_{\text{multi}}$ and $f_{\text{single}}$ (first 33 layers of ResNeXt-34; 3-layer dilated CNN, respectively) encode multi-beat and single-beat (median) views. Both outputs are projected to a low-dimensional representation via a shared MLP.

Quantum Many-Body Physics

The Inverse Participation Ratio (IPR), frequently denoted $P(\lambda)$, is defined for a state $|\psi\rangle$ decomposed as $|\psi\rangle = \sum_n c_n | \phi_n \rangle$:

$$\text{IPR}(\lambda) = \frac{1}{ \sum_n |c_n(\lambda)|^4 }$$

This metric quantifies the spread ("delocalization") of a quantum state over a chosen basis; minimal for localized, maximal for delocalized states (Baena et al., 2022).

3. Loss Functions and Regularization

In biosignal-based IPR modules, no explicit phase-level loss is imposed; rather, the entire hierarchical architecture (IPR → PGHF → GRF) is optimized end-to-end using a margin-based contrastive loss at the beat-embedding level:

$$\mathcal{L}_{\mathrm{CL}} = \frac{1}{B} \sum_i \left[ m + s(\mathbf{u}_i, \mathbf{u}^N_i) - s(\mathbf{u}_i, \mathbf{u}^P_i) \right]_+$$

where $s(\cdot, \cdot)$ is cosine similarity; $P,N$ denote positive/negative pairs (Huang et al., 1 Jan 2026). The zero-mean constraint on Gabor kernels explicitly regularizes the variation branch, enforcing physiological plausibility.

For self-supervised intraperiod learning, the dominant loss is NT-Xent contrastive:

$$\mathcal{L}_{\text{intra}} = \frac{1}{2N} \sum_{k=1}^N \bigl[\ell(2k-1, 2k) + \ell(2k, 2k-1)\bigr]$$

with

$$\ell(i, j) = -\log \frac{\exp(\operatorname{sim}(\boldsymbol{z}_i, \boldsymbol{z}_j)/\tau)}{\sum_{k\neq i} \exp(\operatorname{sim}(\boldsymbol{z}_i, \boldsymbol{z}_k)/\tau)}$$

$\operatorname{sim}(\cdot, \cdot)$ denotes normalized cosine similarity (Zhu et al., 2024).

4. Downstream Fusion and Information Integration

The outputs of intra-phase module(s) are used as atomic units for higher-order integration. In hierarchical ECG systems, four phase outputs $$\{\mathbf{h}_\mathrm{P}, \mathbf{h}_\mathrm{QRS}, \mathbf{h}_\mathrm{ST}, \mathbf{h}_\mathrm{TU}\}$$ are fed to phase-grouped fusion (PGHF), followed by global fusion (GRF) using attention mechanisms on grouped embeddings. This structured approach ensures that distinct phase information is preserved and subsequently combined in a manner respecting physiological relationships (Huang et al., 1 Jan 2026).

In the self-supervised scenario, the concordance between single-beat and multi-beat features is enforced by the contrastive loss, while, when combined with temporal interval prediction (interperiod loss), the final representation jointly encodes intra-beat morphology and inter-beat dynamics (Zhu et al., 2024).

5. Empirical Evaluation and Comparative Analysis

Ablation studies and empirical analyses indicate that the IPR (intra-phase) representation, even in isolation, delivers superior discriminative power compared to standard self-supervised approaches such as SimCLR or MCL. For example, intra-period contrastive loss alone achieves AUC = 0.945 on BTCH, exceeding conventional baselines; fusion with interperiod objectives further increases performance to AUC = 0.953 (Zhu et al., 2024). In biometric systems, phase-aware fusion architectures based on IPR attain state-of-the-art results in closed and open-set identification tasks (Huang et al., 1 Jan 2026).

6. Theoretical Interpretation and Physical Significance

The core principle underlying IPR is the preservation and explicit modeling of phase-localized structures. In biophysical signals, this enables robust extraction of morphological and variation cues resistant to phase mixing. In quantum systems, the IPR quantifies state localization, sharply signaling phase transitions—such as the abrupt change in ground-state structure at the critical point $\lambda_c$ in the two-level atom–diatomic molecule model (Baena et al., 2022). In all cases, the phase-aware design enhances interpretability, stability, and discriminative capacity of downstream representations.

7. Broader Context and Generalizations

The explicit consideration of intra-phase or intra-period representations is broadly applicable to periodic or phase-structured data domains, including but not limited to biosignals and quantum systems. The methodology provides a systematic paradigm for leveraging inherent phase structure in both supervised and unsupervised settings, with empirical evidence supporting gains in robustness and task performance across biometric authentication and arrhythmia detection (Huang et al., 1 Jan 2026, Zhu et al., 2024). A plausible implication is that similar phase-aware representation schemes may provide advantages in other domains characterized by structured repeating phenomena.