Covariance Scattering Transforms

• Covariance Scattering Transforms (CSTs) are unsupervised methods that cascade multiscale spectral wavelets on covariance matrices to construct stable, hierarchical data representations.
• They leverage untrained, parameter-free architectures to achieve reliable performance in low-sample and high-noise regimes by controlling error propagation.
• CSTs have been applied in areas like cortical thickness age prediction and physical parameter inference, offering a robust alternative to PCA and conventional neural networks.

Covariance Scattering Transforms (CSTs) are a class of deep, untrained, unsupervised architectures that construct hierarchical data representations by cascading spectral filters defined on the covariance structure of multivariate data. CSTs generalize the principle of scattering transforms to covariance-domain analysis by sequentially applying multiscale spectral “wavelets” to a data sample, extracting stable, expressive features across both high- and low-variance directions. Designed to bridge the advantages of classical spectral decompositions such as Principal Component Analysis (PCA) and of expressive, trainable architectures such as coVariance Neural Networks (VNNs), CSTs achieve robust representations in low-sample and high-noise regimes while remaining completely label-free and requiring no learned parameters (Cavallo et al., 12 Nov 2025).

1. Formal Framework of Covariance Scattering Transforms

Given a mean-zero data matrix $X \in \mathbb{R}^{N \times T}$ (with $N$ features and $T$ samples), the covariance matrix is $C = \mathbb{E}[xx^T]$ with eigendecomposition $C = V W V^T$, $W = \operatorname{diag}(w_1, \ldots, w_N)$, $w_1 \geq \ldots \geq w_N$. The sample covariance $\widehat{C}$ and its eigendecomposition $(\widehat{V}, \widehat{W})$ are estimated from $X$.

CSTs rely on covariance wavelet operators: $T = V \Lambda V^T, \quad \Lambda = f(W)$ where $f(\cdot)$ preserves positive semidefiniteness. Commonly used normalizations are

$$T_N = \gamma \frac{C}{w_1}, \qquad T_I = \gamma (I - \frac{C}{w_1})$$

where $\gamma > 0$ enforces $\lambda_i \leq 1$ for all $i$.

At each scale $j=1,\ldots,J-1$, define a wavelet bandpass $h_j(\lambda)$ (with $h_0$ low-pass), and associated spectral filters: $$H_j(T) = V \operatorname{diag}\left(h_j(\lambda_1), \ldots, h_j(\lambda_N)\right) V^T$$ Wavelet designs include diffusion wavelets: $$h_0(\lambda) = 1 - \lambda, \quad h_j(\lambda) = \lambda^{2^{j-1}} - \lambda^{2^j}, \quad j \geq 1$$ which can be computed recursively in powers of $T$.

The CST architecture recursively applies these wavelet operators: $$x^{(0)} = x;\qquad x^{(\ell)}_{(j_\ell,\ldots,j_1)} = \rho\left(H_{j_\ell}(T) x^{(\ell-1)}_{(j_{\ell-1},\ldots,j_1)}\right)$$ with $\rho$ a pointwise nonlinearity (e.g., $|\cdot|$). Scattering coefficients at each path are obtained via an aggregation $U(\cdot)$ (e.g., vector mean): $$\varphi_{j_\ell \cdots j_1}(x) = U(x^{(\ell)}_{(j_\ell,\ldots,j_1)})$$ All path outputs up to layer $L-1$ form the CST feature vector $\Phi(T,x) \in \mathbb{R}^M$, with $M = \sum_{\ell=0}^{L-1} F_\ell$ and $F_\ell$ the number of retained branches per layer.

2. Computation and Pruning Strategies

Algorithmic Procedure

The CST embedding is efficiently constructed via a tree-like procedure (summarized in the following pseudocode):

tree = [{ 'signal': x, 'norm': np.linalg.norm(x) }] Φ = [ U(x) ] for ℓ in range(1, L): next_tree = [] for s in tree: for j in range(J): y = ρ(H_j(T) @ s['signal']) if np.linalg.norm(y) / s['norm'] > τ: next_tree.append({ 'signal': y, 'norm': np.linalg.norm(y) }) Φ.append(U(y)) tree = next_tree return Φ

Pruning

Due to the exponentially growing number of paths ($(J^L-1)/(J-1)$), CSTs prune any branch $x_{(\ldots)}$ for which

$$\frac{ \| x_{(\ldots)} \| }{ \| \text{parent} \| } < \tau$$

for some $\tau \in [0.5, 0.9]$. This reduces computation and embedding size by several orders of magnitude, with empirical performance maintaining stability and predictive power.

3. Theoretical Stability to Covariance Estimation

Perturbation Model

CSTs are proven to be robust to errors in the estimated covariance matrix $\widehat{C} = T + E_C$ and in signal noise. Analysis considers the high-probability control of spectral filter deviations and the propagation of error through depths.

Spectral Filter Stability

For a Lipschitz bound $P$ on $h_j(\lambda)$: $$\| H_j(T) - H_j(\widehat{T}) \| \leq \Delta := \frac{PN}{\sqrt{T}(k_{\max} e^{\epsilon/2} + \ldots)} + O(T^{-1})$$ where $\Delta = O(T^{-1/2})$ is uniform—independent of eigengaps in the covariance spectrum.

CST Feature Stability

Under identical pruning and layer structure, the feature deviation obeys: $$\| \Phi(T, x) - \Phi(\widehat{T}, x) \| \leq B_U \Delta \|x\| \sqrt{ \sum_{\ell=1}^{L-1} \ell^2 B^{2\ell-2} F_\ell }$$ where $B$ is the largest wavelet frame bound and $B_U$ the operator norm of $U$.

Comparison to PCA

PCA’s projection error for $k$ components,

$$\| \widehat{V}_{1..k}^T x - V_{1..k}^T x \| = O\left( \frac{1}{\min_{i \neq j \leq k}|w_i - w_j|} \right)$$

diverges as eigenvalues approach and eigengaps vanish. In contrast, CST’s $\Delta$ remains bounded and decays as $T^{-1/2}$, which crucially improves estimation reliability in the low-sample and small-eigengap regime.

4. Relationship to PCA and CoVariance Neural Networks

Method	Nature	Filtering	Training	Stability	Computational Cost
PCA	Untrained	Top-$k$ spectrum	None	Degrades with eigengap	$O(N^3)$ (eigendecomp.)
VNN	Trainable, Superv	Polynomial	Gradient descend	Improved, weight-dependent	$O(K N^2 L)$
CST	Untrained	Multiscale wavelet	None	Uniform $O(T^{-1/2})$	$O(J L N^2)$ (w/ prune)

CSTs constitute a multiscale alternative to PCA, capturing both high- and low-variance covariance modes by constructing feature cascades since the wavelet filters span the entire spectrum. VNNs, in contrast, permit adaptive spectral filtering but require labeled data and suffer potential overfitting under small labeled sets. CSTs combine the expressivity of VNNs with the unsupervised, untrained, and parameter-free benefits of PCA, while achieving more stable behavior in finite-sample regimes.

5. Empirical Performance: Cortical Thickness Age Prediction

Evaluation is performed on cortical thickness measurements from the ADNI1, ADNI2 (Alzheimer’s datasets, $T=801,1142$), PPMI (Parkinson’s, $T=1704$), and ABIDE (Autism, $T=1035$) datasets with $N=62$ or $68$ cortical regions. The task is to predict chronological age from thickness profiles using ridge regression atop the embeddings.

Protocol:

• 50% of data is held out for unsupervised feature estimation.
• Embeddings computed on union of unlabeled + train split.
• Ridge regression trained on 10% train split, hyperparameters via 20% validation split, and evaluated on 20% test.
• Covariance perturbations are simulated by subsampling the pool for covariance estimation; same regressors are used for embedding robustness studies.

Results:

• Stability: Under decreasing $T$, PCA’s MAE and embedding MSE escalate, whereas CST’s remain nearly constant. CST is robust against covariance estimation noise even when eigengaps vanish.
• Accuracy: CST+ridge matches or exceeds the best results by PCA+ridge, VNN, raw-ridge, and small MLP across all datasets.
• Pruning: As $\tau$ increases (from 0 to 0.9), embedding time and size drop orders of magnitude with negligible effect on MAE, demonstrating computational scalability.
• Label efficiency: CST features provide strong predictive performance even when downstream labels constitute as little as 1% of the total data; aggregation by $U=\text{mean}$ is beneficial in extreme low-label settings.
• Interpretation: CST features extract patterns informative of brain-age and are robust to the small-sample conditions typical in medical studies ($T \ll N$), supporting hypothesis-free analyses.

6. CSTs in the Context of Scattering for Physics and High-Order Structure

Covariance scattering models are distinguished from classical wavelet scattering for random fields (Cheng et al., 2023). In the latter, scattering channels are constructed from means and covariances of wavelet-modulus coefficients over stationary fields, summarizing up to fourth-order moments via a combination of first-order ($S_1$), power-spectrum ($S_2$), bispectrum ($S_3$), and trispectrum ($S_4$) analogues. Compared to full polyspectral descriptions, CST-based scattering achieves significant dimensionality reduction ($O(\log^3 L)$ for field size $L$) via group averaging and Fourier thresholding, while remaining sensitive to non-Gaussian interactions and interpretable at the level of scale-orientation combinations.

Applications include physical parameter inference, classification of turbulent and astrophysical regimes, symmetry detection, and component separation. These physically-motivated CST variants further illustrate the breadth of the covariance scattering paradigm across unsupervised representation learning for both vector-valued and spatial data.

7. Significance and Implications

Covariance Scattering Transforms address longstanding deficiencies of covariance-based representations in unsupervised data analysis by achieving provable stability under sampling noise and eigengap collapse, multiscale sensitivity in feature construction, and computational efficiency through principled pruning strategies. Their performance in both theoretical and empirical regimes, especially in high-noise or sample-poor settings, places CSTs as a robust alternative or complement to PCA and supervised spectral architectures, with a broad scope spanning medical data analysis and physical sciences.

A plausible implication is that CSTs will facilitate hypothesis-free or exploratory analyses in domains where labeled data is scarce or unreliable, while their proven stability may drive their adoption in scientific settings dependent on interpretable and reproducible spectral representations.