Propagation Vector Classifiers

• Propagation vector classifiers are algorithms that iteratively propagate feature vectors to classify and infer in high-dimensional, structured domains.
• They integrate methods like wavelet scattering, message-passing (VASP/KVASP), and vector-label propagation to handle invariance and model mismatches.
• Applications range from image recognition and community detection to quantum classification, offering improved robustness and reduced error rates.

Propagation vector classifiers denote a broad class of algorithms that utilize iterative, vector-based propagation mechanisms to perform classification and inference across high-dimensional, structured domains. The main technical motifs underlying these methods include the propagation of feature- or message-vectors in contexts such as survey/message passing, modularity optimization, co-occurrence descriptor aggregation, and metric learning in both classical and quantum settings. Notable instantiations range from wavelet-scattering PCA classifiers in vision to advanced message-passing schemes such as Vector Approximate Survey Propagation (VASP) and its K-step RSB extension (KVASP), as well as vector-label propagation algorithms for community detection.

1. Mathematical Principles of Vector-Based Propagation

Propagation vector classifiers aggregate local or global information using structured vectors and propagate these through the underlying data, graph, or feature space according to mathematically principled update rules.

• Scattering Vector Classifiers: These compute a multiscale descriptor for images using cascades of wavelet transforms, modulus nonlinearities, and local averaging. The resulting scattering vector $S_J f(x)$ concatenates coefficients from multiple orders, encoding translation invariance and linearized deformation sensitivity. Classification is achieved by building PCA-based affine models for each class and assigning test samples via minimum projection error (Bruna et al., 2010).
• Message and Survey Propagation: AMP, VAMP, GASP, VASP, and KVASP algorithms implement iterative message-passing between variable and factor nodes in graphical models. Messages are typically Gaussian or vector-valued distributions, updated via moment matching, with vectorization enabling richer modeling of correlations and degeneracies in structured data (Chen et al., 2023, Chen et al., 28 Oct 2024).
• Vector Label Propagation: VLPA and sVLPA assign each node a continuous vector-label $v_i \in \mathcal{U}^+(\mathbb{R}^n)$ representing soft membership across communities, with updates via gradient steps optimizing a vector-modularity objective function (Fang et al., 2020).

2. Algorithmic Frameworks and Update Rules

The implementations of propagation vector classifiers span deterministic and stochastic, continuous and discrete, classical and quantum frameworks.

• Wavelet-Based Scattering: Given input $f(x)$, directional wavelets $\psi_{j,\gamma}$ and low-pass filters $\phi_J$ produce first-order coefficients as $$S_{1,J} f(x) = |f * \psi_{j_1, \gamma_1}| * \phi_J(x)$$ and repeated modulus-wavelet-averaging for higher orders. The total feature vector is $S_J f(x) = \{ S_{q,J} f(x) \}_{q=0}^m$ (Bruna et al., 2010).
• VASP and KVASP Message Updates: Iteratively compute forward/backward vector survey messages by projecting nonlinear functions to their closest Gaussian distributions (minimize KL divergence):

$$\text{Proj}[g(\tilde{x})] = \arg\min_{h \in \mathcal{N}} \mathrm{KL}(g \| h)$$

where each variable $x_i$ is replaced by $L$ replicas forming a vector $\vec{x}_i$, and the measurement matrix is lifted as $\tilde{H} = I_L \otimes H$ to model correlations (Chen et al., 2023, Chen et al., 28 Oct 2024).

• Vector-Label Propagation Updates: Use projected gradient ascent in modularity:

$$v_i \gets \mathcal{P}_{\mathcal{SU}^+(\mathbb{R}^n, d_e)}(p_i)$$

with $p_i$ being the positive part of the modularity gradient at node $i$, projected and normalized to sparse subset with essential dimension $d_e$ (Fang et al., 2020).

• Quantum Label Propagation: Quantum classifiers encode triplets (anchor, positive, negative) into superposition states, process them through parameterized quantum circuits $W(\theta)$, and apply hybrid measurement strategies (Hadamard and Z-basis) to extract similarity metrics. The embedding function $g(\cdot)$ and angular metric $D(\cdot, \cdot)$ govern the triplet loss (Hou et al., 2023).

3. Treatment of Invariance, Stability, and Robustness

A defining feature is the design of invariance and stability to deformations, context, and model mismatch.

• Scattering Representation: Enforces local translation invariance and linearizes small deformations. The stability is quantified as:

$$\Vert S_J(D_\tau f) - S_J f \Vert \leq C m \Vert f \Vert (2^{-J}|\tau|_\infty + J(|\nabla \tau|_\infty + |H\tau|_\infty))$$

enabling robust classification under image transformations (Bruna et al., 2010).

• VASP/KVASP: The vector survey propagation and K-step RSB hierarchy grant robustness under non-differentiable priors and correlated (non-i.i.d.) measurement matrices, overcoming the limitations of VAMP and GASP. Empirically, VASP/KVASP yields significantly lower MSE in estimation tasks and aligns per-iteration SE fixed-points with the saddle-points of the free energy under 1RSB/KRSB (Chen et al., 2023, Chen et al., 28 Oct 2024).
• VLPA/sVLPA: Soft vector labels retain weak community signals, with stochastic projection steps helping to escape local optima and improving detection in networks with weak modular structure (Fang et al., 2020).
• Quantum Classifiers: The superposition and entanglement mechanisms, together with triplet adversarial training, yield increased robustness to adversarial perturbations (Hou et al., 2023).

4. Performance and Impact in Applications

Propagation vector classifiers have demonstrated empirical and theoretical advances across multiple domains.

Classifier Type	Benchmark Domains	Performance Characteristics
Scattering PCA	MNIST, CUReT	State-of-the-art recognition, strong error rates vs. deep networks with limited data (Bruna et al., 2010)
Fisher Vector NN	Pascal VOC 2007	Enhanced MAP over FV+SVM, end-to-end optimization, scalable with multi-GPU (Wieschollek et al., 2017)
VLPA/sVLPA	LFR, Real Networks	Higher modularity, improved NMI & global optimum acquisition vs. Louvain in weak community structure (Fang et al., 2020)
VASP/KVASP	MIMO, High-Dim. Linear	Lower MSE under discrete priors, SE/free energy agreement, robust to model mismatch and correlation (Chen et al., 2023, Chen et al., 28 Oct 2024)
Quantum Classifier	Iris, MNIST	Increased accuracy, adversarial robustness over swap test classifier (Hou et al., 2023)

In scattering classifiers, PCA model selection further reduces intra-class variation and enhances discriminative power. In message passing, even with NP-hard MAP estimation, VASP/KVASP efficiently approximate the solution with cubic or lower complexity under SVD availability.

5. Connections to Theoretical Frameworks

These classifiers are often tightly connected to statistical physics formulations and optimization theory.

• Replica Symmetry Breaking: VASP/KVASP directly implement MAP estimators whose complexity is structurally NP-hard, using SE analysis to track MSE and showing equivalence between SE fixed-points and 1RSB/KRSB saddle-points in the free energy. These connections clarify when and why multi-step vector surveys outperform classical AMP/VAMP (Chen et al., 2023, Chen et al., 28 Oct 2024).
• Vector Modularities and Gradient Descent: The conversion from discrete to continuous label propagation in VLPA/sVLPA allows for global optimization schemes beyond greedy local search, by optimizing objectives interpolated by inner products, rather than Kronecker delta functions (Fang et al., 2020).
• Quantum Metric Learning: Quantum classifiers leverage Hilbert-space embeddings and basis measurements to encode and propagate label information, invoking quantum analogs of classical kernels and metric margins (Hou et al., 2023).

6. Extensions, Generalizations, and Future Directions

Propagation vector classifiers define a general mathematical paradigm applicable to structured learning, clustering, signal recovery, and quantum information.

• High-Dimensional Extensions: VLPA/sVLPA generalize directly to networks with nodes possessing multiple feature vectors. Similarly, VASP/KVASP generalize to non-linear models and applications with highly structured, correlated input domains (Fang et al., 2020, Chen et al., 28 Oct 2024).
• Flexible Objective Functions: Vector-label propagation and message/survey passing can be adapted for objectives beyond modularity, e.g., modularity with resolution parameters, or general inference and partition measures.
• Adaptive Algorithms and Representation Learning: Future directions include rigorous analysis of convergence in finite dimensions, adaptive selection of RSB steps, and deployment in decentralized or online systems.
• Quantum-Classical Hybrid Models: The development of quantum propagation classifiers that process multiple data triplets in superposition, with efficient measurement and robustness to perturbation, suggests ongoing expansion into quantum machine learning (Hou et al., 2023).

A plausible implication is that the integration of vector-based propagation update schemes—whether implemented via wavelet scattering, survey/message passing, vector-label propagation, or quantum circuits—will increasingly dominate the design of robust, high-dimensional classifiers for both inference and structured data modeling, especially under model mismatch and complex energy landscapes.