Randomized Filtering Overview

• Randomized Filtering is a collection of algorithmic strategies that integrates random operations into filtering processes for enhanced efficiency and robust statistical guarantees.
• It employs techniques like random embeddings, scenario-based filtering, and randomized consensus to preserve geometry in high-dimensional data and improve system performance.
• Applications span signal processing, distributed estimation, and neural network regularization, demonstrating low computational cost and practical benefits in real-time and adaptive contexts.

Randomized Filtering (RF) encompasses a spectrum of algorithmic strategies that leverage randomization within filtering or signal extraction procedures to achieve statistical guarantees, computational tractability, or robustness. Across signal processing, machine learning, dynamical system estimation, and neural network regularization, RF schemes exploit randomness either in the filter construction, application, or as part of a larger optimization and inference workflow. Approaches range from provable geometry-preserving embeddings for streaming high-dimensional data, scenario-based set approximations in nonlinear filtering, randomized consensus mechanisms in distributed Kalman filtering, spectral randomization for robustness, and random projection-based adaptive interference suppression.

1. Foundational Concepts and Formalisms

Randomized Filtering is characterized by integrating one or more randomly chosen operations—such as random transforms, randomized frequency selection, randomized consensus, randomized projections, or scenario-based sampling—into the filtering pipeline. The core aims are (a) statistical or deterministic control of estimation or classification risk, (b) computational efficiency, and/or (c) improved generalization and robustness. Mechanistically, randomization can appear in the embedding (as in RF embeddings for dimensionality reduction) (Bertrand et al., 13 Jan 2026), in the scenario selection for set-valued approximation (Dabbene et al., 2015), in randomized communication protocols within distributed estimation (Qin et al., 2018), or as a stochastic perturbation to input features/models for regularization (Islam et al., 2022, Zhang et al., 2024).

Mathematically, the prototypical RF transformation for dimension reduction is defined for $x \in \mathbb{R}^N$ as

$y = \Phi x, \qquad \Phi = S F D_\xi,$

where $D_\xi$ is a random diagonal Rademacher sign-flip matrix, $F$ is a unitary (typically DFT or Hadamard) transform, and $S$ is a random subsampling selector. The statistical properties of $\Phi$ can be tuned to control geometric distortion over a manifold $\mathcal{M} \subset \mathbb{R}^N$ (Bertrand et al., 13 Jan 2026).

In scenario-based filtering, randomization is introduced via stochastic selection of system and noise realizations to construct approximations of the propagated state under nonlinear dynamics, with sample-complexity bounds provided by scenario theory (Dabbene et al., 2015).

2. Algorithmic Instantiations of Randomized Filtering

2.1 Manifold-Preserving Embeddings

Randomized Filtering as described in "Stable Filtering for Efficient Dimensionality Reduction of Streaming Manifold Data" utilizes the $\Phi = S F D_\xi$ scheme to construct embeddings $y \in \mathbb{R}^M$ that possess the following property: for a fixed isometry constant $\delta$, all pairwise distances between points on a $D$-dimensional Riemannian manifold $\mathcal{M}$ are preserved to within a multiplicative factor of $(1 \pm \delta)$, with high probability, provided $M \gtrsim D\log N/\delta^2$ (up to log-log factors) (Bertrand et al., 13 Jan 2026). This mechanism enables geometry-preserving dimension reduction for streaming high-dimensional data without requiring data-dependent training or tuning.

2.2 Scenario-Based Randomized Filters

The randomized prediction-corrector filter for nonlinear state estimation constructs approximating sets (ellipsoids, hyperrectangles, or polynomial superlevel sets) for the posterior state via random sampling of process and measurement noise realizations, then solves a minimum-volume covering problem for these scenarios (Dabbene et al., 2015). Probabilistic bounds on containment violations are controlled by the number of scenarios $N$ relative to the desired violation $\varepsilon$ and confidence $\delta$, as quantified by scenario theory. The resulting recursive filter provides a convex set at each timestep, with explicit tradeoffs between conservatism and uncertainty.

2.3 Randomized Smoothing and Spectral Filtering (Deep Learning Contexts)

Filtered Randomized Smoothing (FRS) combines classical randomized smoothing (RS)—which injects Gaussian noise at inference time to enable certified $\ell_2$ robustness—with domain-specific spectral filtering. In the context of modulation classification for RF signals, FRS uses low-pass filtering in the spectral domain to exploit the empirical heterogeneity between signal (localized, low-frequency) and adversarial perturbation (broadband) spectra (Zhang et al., 2024). Post-smoothing and pre-smoothing variants establish certified radii via the inherent Lipschitz properties of the pipeline.

Frequency Dropout (FD), in turn, introduces random filtering at the feature-map level during deep neural network training, stochastically applying Gaussian, Laplacian of Gaussian, or Gabor filters to random channel subsets within intermediate feature maps (Islam et al., 2022). This prevents the overfitting of frequency-specific artifacts or spurious correlations, empirically improving out-of-distribution robustness and domain adaptation performance.

2.4 Randomized Gossip and Consensus in Distributed Kalman Filtering

Randomized consensus-based distributed Kalman filtering, particularly in sensor networks, achieves global state estimation via gossip protocols, where each node probabilistically selects neighbors for local information exchange (Qin et al., 2018). The update process is random both in agent activation and neighbor selection, resulting in a time-varying, stochastic evolution of the global error covariance for which mean square convergence—under mild detectability and controllability assumptions—can be established. This consensus-based randomization enhances estimation accuracy relative to noncooperative (local only) filters.

2.5 Randomized Projections for Adaptive Filtering

Within adaptive filtering for signal extraction in the presence of structured (low-rank) interference, randomized filter design employs random projections (either via Gaussian matrices or random column selection) to construct reduced-rank approximations of interference subspaces (Besson, 2022). This approach avoids expensive SVD computations, yielding computationally efficient implementations whose output SNR and interference-rejection performance closely approach that of principal component–based filters.

3. Theoretical Guarantees and Analysis

3.1 Manifold Embedding Isometry

Theoretical analysis of RF embeddings establishes that for any $x, y \in \mathcal{M}$,

$$(1 - \delta)\|x - y\|_2^2 \leq \|\Phi x - \Phi y\|_2^2 \leq (1 + \delta)\|x - y\|_2^2$$

with probability at least $1 - C_2 \rho$ as long as

$$M \ge (C_1 / \delta^2) D [\log(R N / (\tau \delta)) + \log(V/\rho)] \log^4 N \log(1/\rho)$$

where $V$ is the manifold volume, $R$ geodesic regularity, $1/\tau$ the condition number (Bertrand et al., 13 Jan 2026).

3.2 Certified Robustness in Deep Learning

In FRS, let $f$ denote the base classifier. The post-filtering randomized smoothing radius is

$$R_{\rm post} = \frac{\sigma}{2}[\Phi^{-1}(p_A) - \Phi^{-1}(p_B)],$$

where $p_A$ and $p_B$ are the probabilities of the most probable and runner-up classes, and $\sigma$ is the variance of the smoothing noise (Zhang et al., 2024). When smoothing is preceded by a filter of Lipschitz constant $L_{\rm lip}$, the certification radius shrinks as $R_{\rm pre} = R_{\rm post} / L_{\rm lip}$.

3.3 Scenario-Theoretic Probabilistic Bounds

For set-valued randomized filters, the probability $\mathrm{Viol}(A)$ that a random scenario is not covered by the set $A$ constructed from $N$ samples is bounded by

$\Pr\{\mathrm{Viol}(A) > \varepsilon\} \leq \delta$

provided $$N \geq \frac{e}{e-1} \cdot \frac{1}{\varepsilon}(d + \ln(1/\delta))$$, where $d$ is the number of decision variables in the set-approximation problem (Dabbene et al., 2015).

3.4 Mean-Square Convergence in Distributed Filtering

The global estimation error's covariance in randomized consensus Kalman filtering converges to a unique fixed point $\bar{P}_{\rm err}$ under a contraction mapping $T(X)$, following

$$E[\widetilde{x}_{k|k} \widetilde{x}_{k|k}^T] \to \bar{P}_{\rm err}$$

at an exponential rate, provided the prediction–update matrices and consensus step satisfy the required spectral radius constraints (Qin et al., 2018).

4. Applications Across Domains

Domain	RF Mechanism	Notable Outcomes
Streaming dimensionality reduction	Subsampled DFT sign-randomized embedding	$\delta$-isometric geometry, no need for training; robust to manifold drift (Bertrand et al., 13 Jan 2026)
Nonlinear state filtering	Scenario-based set approximations	Probabilistic containment guarantees, convex program per step (Dabbene et al., 2015)
Distributed sensor estimation	Randomized gossip/protocol selection	Mean-square convergence, tractable sensor scheduling (Qin et al., 2018)
Adversarially robust learning	Filtering combined with smoothing	Certified robustness, improved clean and adversarial accuracy (Zhang et al., 2024)
Neural network regularization	Stochastic filter application (FD)	Better generalization, domain adaptation, and segmentation (Islam et al., 2022)
Adaptive signal filtering	Randomized projections	Principal component–like performance with reduced complexity (Besson, 2022)

Applications include streaming manifold data analysis (calcium imaging, fMRI, turbulence, neural signals) (Bertrand et al., 13 Jan 2026); adversarially robust RF modulation classification (RadioML 2016.10a) (Zhang et al., 2024); unsupervised domain adaptation and medical image segmentation in computer vision (Islam et al., 2022); wireless sensor network estimation (Qin et al., 2018); and adaptive beamforming/interference rejection (Besson, 2022).

5. Computational and Practical Considerations

Randomized Filtering methods achieve computational and storage advantages over data-dependent or classical approaches:

• Dimensionality-reducing RF embeddings require only $O(N \log N)$ time per sample and $O(N)$ storage, with zero training overhead (Bertrand et al., 13 Jan 2026).
• Scenario-based recursive filters have per-step convex-program complexity dominated by the number of constraints $N$ and decision variables $d$ (Dabbene et al., 2015).
• Randomized projection adaptive filters bypass matrix decompositions, yielding $O(N K R + K R^2 + R^3)$ complexity versus $O(N K^2)$ for principal component methods (Besson, 2022).
• Distributed filters with randomized consensus avoid global synchronization, at the expense of stochastic convergence analysis; optimization of neighbor selection under energy constraints can be relaxed to convex programs (Qin et al., 2018).

Parameter selection is critical: for RF embeddings, $M$ must be chosen in accordance with the intrinsic manifold dimension $D$ and desired distortion $\delta$ (Bertrand et al., 13 Jan 2026). In scenario-based filters, controlling violation levels $\varepsilon, \delta$ determines $N$.

6. Comparative Analysis and Related Approaches

Randomized Filtering differs from classical random projections, standard data-dependent learning, and deterministic filtering in the following ways:

• JL-based random projections guarantee geometry only on finite sets; manifold RF schemes extend these properties to infinite-dimensional or continuous manifolds, and exploit the structure of the data for greater efficiency (Bertrand et al., 13 Jan 2026).
• Traditional PCA and manifold learning methods require access to all data and do not scale to streaming or high-throughput applications; RF is streaming and data-independent.
• Particle filters represent uncertainty with a weighted empirical distribution, leading to sample degeneracy and computational bottlenecks; scenario-based RF maintains a convex set summary at each step (Dabbene et al., 2015).
• In distributed estimation, deterministic consensus approaches may have higher communication demands and slower convergence compared to randomized gossip strategies (Qin et al., 2018).
• In machine learning, deterministic frequency filtering or curriculum smoothing lacks the stochastic diversity and persistent regularization provided by FD approaches (Islam et al., 2022).

7. Limitations, Practical Extensions, and Open Directions

Randomized Filtering methodologies, though provably effective, present limitations and areas for extension. For scenario-based set filtering, the resetting distribution at each step may lose multimodal features unless PAS (polynomial superlevel sets) or unions of sets are incorporated. In distributed settings, convergence is in mean-square only and depends on random communication schedules; tradeoffs between communication overhead, estimation performance, and energy consumption remain active areas of study. For spectral regularization and smoothing, the effectiveness of randomization depends on spectral structure and context, motivating targeted filter design or adaptation in nonstationary environments.

Extensions include adaptive scenario sizing, importance sampling strategies, combinations of convex approximating families, multi-modal set representations, and data-driven hybridizations with Gaussian or moment-based filters (Dabbene et al., 2015, Zhang et al., 2024). In manifold RF, leveraging sparse FFTs or structured transforms further reduces costs for highly sparse or structured data streams (Bertrand et al., 13 Jan 2026).

Ongoing work explores the application of RF in online learning, real-time control, adaptive streaming analytics, and scalable, robust neural networks.