Persistent Homology-guided Frequency Selection

• Persistent Homology-guided Frequency Selection is a methodology that transforms temporal data streams into compact frequency metadescriptors using DFT and variance-based selection to capture key dynamic features.
• It employs chunking, batch averaging, and clustering techniques to identify frequency components reflecting significant concept shifts, enhancing visualization and unsupervised analysis.
• In quantum optics, the approach adapts the Mandel parameter to filter photon correlations, optimizing experimental setups and revealing nonclassical phenomena such as antibunching.

The frequency-filtering metadescriptor (FFM) is a formalism for extracting, characterizing, and leveraging informative frequency-domain features from temporal or sequential data, primarily for nonstationary data streams or quantum signal analysis. It encompasses both algorithmic procedures for streaming data and physical measures for photon correlation analysis, manifesting as a set of indices or parameterized quantities capturing the most informative, discriminative, or physically meaningful frequency components across time, space, or spectral domain (Komorniczak, 7 Feb 2025, Gonzalez-Tudela et al., 2015).

1. Mathematical Foundations

For data stream analysis, FFM is defined by transforming real-valued sample vectors $x\in\mathbb{R}^d$ into the frequency domain using the discrete Fourier transform (DFT):

$$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$

Exploiting conjugate symmetry, only the first $d/2$ real Fourier coefficients are retained to derive the reduced representation:

$$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$

Averaging these representations over batches (chunks) $B_j$ yields chunk-wise mean spectra $f_j\in\mathbb{R}^{d/2}$. Stacking these for all $k$ chunks forms $F\in\mathbb{R}^{k\times(d/2)}$. The variance of each frequency component $v_l$ is computed:

$$\bar f_l = \frac{1}{k}\sum_{j=1}^k f_{j,l}, \qquad v_l = \frac{1}{k}\sum_{j=1}^k(f_{j,l}-\bar f_l)^2$$

FFM identifies the set $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$0 of indices with the top $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$1 values $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$2, yielding per-chunk $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$3-dimensional metadescriptors $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$4.

In quantum optics, the FFM is instantiated by the frequency-resolved Mandel parameter $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$5:

$$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$6

where $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$7 is the filtered field, $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$8 is the one-photon spectrum, and $$\hat x_k = \sum_{m=0}^{d-1} x_m \, e^{-2\pi i\,k\,m/d}, \qquad 0\leq k<d$$9 is the filtered second-order photon correlation.

2. Construction and Algorithmic Workflow

The FFM construction for data streams consists of the following main steps (Komorniczak, 7 Feb 2025):

1. Chunking: Partition the data stream into $d/2$0 non-overlapping batches $d/2$1.
2. Sample Transformation: For each sample $d/2$2 in $d/2$3, compute the real-part DFT vector $d/2$4.
3. Batch Averaging: Compute mean frequency vector $d/2$5 for each chunk.
4. Stacking and Variance Calculation: Form $d/2$6 and compute per-frequency variances $d/2$7.
5. Frequency Selection: Identify top $d/2$8 frequency indices $d/2$9 by sorting $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$0.
6. Descriptor Extraction: Construct per-chunk metadescriptors $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$1 from frequencies in $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$2.
7. (Optional) Normalization: Subtract mean and divide by standard deviation per column.
8. (Optional) Clustering: Apply $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$3-means (or similar) to $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$4 to discover concept clusters.
9. (Optional) Visualization: For each frequency in $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$5, reconstruct spatial patterns via inverse FFT by sparse spectrum insertion.

Pseudocode formalizing this procedure is directly available, ensuring full reproducibility and algorithmic transparency.

3. Physical Interpretation and Informativeness

The FFM quantifies the degree to which specific frequency components vary across data partitions, capturing concept drift or dynamical changes. High-variance frequencies reflect features most sensitive to structural or distributional changes. By reducing high-dimensional signal data to a compact set of informative frequency features, FFM enables downstream clustering and pattern discovery while minimizing loss of semantic meaning owing to the invertibility of the Fourier transform for visualization.

In quantum correlation analysis, the Mandel FFM transcends pure $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$6 analysis by weighting photon correlations according to filtered signal intensity, thus suppressing misleading artifacts in regions of vanishing photon flux. Negative $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$7 unequivocally signals nonclassicality (e.g., antibunching), and its construction highlights only those frequency regions where both correlation and signal strength are experimentally accessible (Gonzalez-Tudela et al., 2015).

4. Use Cases: Concept Clustering, Visualization, and Quantum Metrology

Data Stream Analysis

• Concept identification and grouping: FFM enables unsupervised clustering of data stream chunks by treating each metadescriptor $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$8 as a sample in reduced frequency space, supporting identification of distinct “concepts” or regimes in the stream, especially under concept drift.
• Visualization: FFM supports visualization by reconstructing spatial-domain patterns for selected frequencies, providing an interpretable “fingerprint” for each chunk based on the inverse FFT of sparse frequency vectors.

Quantum Optics

• Photon correlation mapping: FFM, as instantiated in $$x^{-} = \Re[\hat x_0, \hat x_1, \ldots, \hat x_{d/2-1}]^T \in \mathbb{R}^{d/2}$$9, produces fine-grained maps of photon correlations across frequency and time windows, exposing “valleys of accessible correlations” and distinguishing real versus virtual emission processes.
• Experimental optimization: FFM guides experimentalists in choosing filter bandwidths, time delays, and detector configurations to maximize observable quantum features such as cascade bunching or virtual leapfrog processes.

5. Comparative Experimental Performance

Quantitative evaluation of FFM against baseline and state-of-the-art descriptor schemes in data streaming yields the following outcomes (for $B_j$0 frequencies and post-hoc clustering into $B_j$1 known concepts):

Drift Scenario	FFM (NMI)	PCA (NMI)	ICI (NMI)	CED (NMI)
Sudden (4 concepts)	0.991±0.014	0.960±0.053	0.548±0.203	0.595±0.145
Gradual	0.876±0.032	0.834±0.068	0.318±0.127	0.190±0.061
Incremental	0.906±0.023	0.851±0.078	0.347±0.142	0.276±0.115

Adjusted Rand Index (ARI), Completeness, and Homogeneity follow similar trends, with FFM outperforming or matching PCA, and significantly exceeding ICI and CED under all drift types. Standard deviations indicate more stable clustering performance for FFM relative to PCA. Statistical significance tests ($B_j$2) confirm these comparisons (Komorniczak, 7 Feb 2025).

In quantum optics, the FFM-based Mandel parameter collapses ghost correlations and optimizes photon flux–correlation trade-offs. It reveals accessible quantum-optical features and guides parameter sweeps for maximizing observable phenomena such as real or virtual photon cascades (Gonzalez-Tudela et al., 2015).

6. Practical Considerations and Extensions

• Parameterization: The number of selected frequencies $B_j$3 and the chunking strategy should be tuned to the timescale and expected granularity of regime shifts.
• Extensibility: The FFM methodology extends directly to higher-order statistics or temporally-structured quantum correlators, by generalizing the filtering and aggregation steps to $B_j$4 and joint $B_j$5-photon spectral densities.
• Visualization and interpretability: The inverse-FFT visualization pipeline provides domain-expert interpretable insights into which features drive chunk differentiation.
• Experimental guidelines: In quantum experiments, optimal settings for frequency filtering, delay, and windowing are dictated by the interplay of system emission linewidths, pumping, and physical transition energies; FFM provides quantitative recipes for these choices.

7. Connections to Related Methodologies

FFM occupies a distinct methodological niche: it compresses high-dimensional, nonstationary data (classical or quantum) to a concise set of frequency features of maximal discriminative or physically meaningful power across observed temporal structure. It improves upon designed metafeature schemes (e.g., CED, ICI) by leveraging domain-general frequency analysis, and achieves or surpasses performance parity with unsupervised dimensionality reduction methods (PCA) while retaining spatial-domain invertibility.

In quantum photonics, FFM as realized by the frequency-resolved Mandel parameter supersedes traditional $B_j$6 analysis by introducing signal-aware correlation metrics, ensuring only physically relevant (high-flux, observable) correlations are retained. The approach is compatible with a variety of theoretical and experimental contexts, including time-resolved sensor formalisms and regression analyses.

---

The frequency-filtering metadescriptor thus provides a unified statistical and physical characterization tool for both nonstationary data streams and quantum correlation measurements, distinguished by its principled selection of informative frequencies, downstream clusterability, and direct interpretability in both the frequency and spatial/time domains (Komorniczak, 7 Feb 2025, Gonzalez-Tudela et al., 2015).