Robust Cognitive–Flexible Filtering
under Noisy Innovation Scores
Structure selection that stays reliable — even when the scores are noisy.
Submitted to IEEE Signal Processing Letters, 2026
slide:beamer-SPL.pdf
Julia source code:
-
lcss_experiments_documented.jl— Figs. 2&3, Table I (validates Theorems 1–3) -
fig_scaling_documented.jl— Fig. 4, bound tightness (Theorem 2)
Detect structural mismatch, Switch only when evidence is clear, and Stabilise under noise — guaranteed.
Belief \(\mathfrak{B}_t\) → Noisy Score \(\hat{\Phi}_t(s)\) → Margin Rule \(\delta>2\bar{\varepsilon}\) → Structure \(s_{t+1}\) → Belief Update \(\mathfrak{B}_{t+1}\)
A hysteresis-based switching rule that provably suppresses spurious transitions under bounded score noise.
What Is the Problem?
Cognitive Flexibility (CF) [Nuchkrua, Boonto & Liu, arXiv:2604.08130] selects, at each step, the latent structure \(s\in\mathcal{S}\) minimising an innovation-based predictive score \(\Phi_t(s):=-\log\ell_{\theta,s}(y_{t+1}|\mathfrak{B}_t,u_t)\). Under exact scores, three guarantees hold: descent in expectation, finite switching, and non-chattering.
In practice, scores are estimated from a particle filter and carry additive perturbations \(\epsilon_t(s)\) satisfying \(\mathbb{E}[\epsilon_t(s)\mid\mathcal{I}_t]=0\) and \(|\epsilon_t(s)|\leq\bar{\varepsilon}\) a.s. When score differences are of the same order as \(\bar{\varepsilon}\), direct score minimisation induces spurious switching — noise-driven transitions that degrade predictive accuracy and inflate computational cost. Whether CF remains stable under corrupted scores was entirely open.
⚙ The Margin-Based Solution
We introduce the margin-based switching rule:
\[ s_{t+1} = \begin{cases} \hat{s}_t, & \hat{\Phi}_t(s_t) - \hat{\Phi}_t(\hat{s}_t) > \delta, \\ s_t, & \text{otherwise,} \end{cases} \quad \hat{s}_t\in\arg\min_{s}\hat{\Phi}_t(s), \]
with \(\delta > 2\bar{\varepsilon}\). This single design choice restores all three stability properties of the noiseless theory, with no modification to the underlying filter recursion. Inspired by hysteresis switching control [Morse 1996; Hespanha & Morse 1999].
Lock-in and switching regimes (let \(\gamma_t=\Phi_t(s_t)-\Phi_t(s^\star)\)):
- \(\gamma_t > \delta+2\bar{\varepsilon}\): switching guaranteed
- \(\gamma_t-2\bar{\varepsilon}\leq\delta <\gamma_t+2\bar{\varepsilon}\): switching possible but not guaranteed
- \(\delta\geq\gamma_t+2\bar{\varepsilon}\): switching impossible (noise-only switch suppressed)
📐 Main Theoretical Guarantees
- Theorem 1 (Descent in Expectation): \(\mathbb{E}[\Phi_t(s_{t+1})\mid\mathcal{I}_t] \leq\mathbb{E}[\Phi_t(s_t)\mid\mathcal{I}_t]\) for all \(t\geq 0\) — the expected predictive score never increases after a switch, regardless of noise realisations.
- Theorem 2 (Finite Expected Switching): \[ \mathbb{E}[N_T]\leq\frac{1}{\delta-2\bar{\varepsilon}} \sum_{t=0}^{T-1} \mathbb{E}\!\left[\Phi_t(s_t)-\min_{s\in\mathcal{S}} \Phi_t(s)\right] \] — total switches scale as \((\delta-2\bar{\varepsilon})^{-1}\); finite whenever score suboptimality is summable.
- Theorem 3 (Non-Chattering): Once \(\limsup_{t\to\infty}[\Phi_t(s_t)-\min_s\Phi_t(s)] <\delta-2\bar{\varepsilon}\) a.s., no further switching occurs a.s. — CF stabilises after finitely many transitions.
Numerical Experiments
Three experiments validate Theorems 1–3 on the canonical nonlinear stochastic growth model [Gordon 1993; Arulampalam 2002]: \[ z_{t+1}=\tfrac{1}{2}z_t+\tfrac{25z_t}{1+z_t^2} +8\cos(1.2t)+w_t, \quad y_t=\tfrac{1}{20}z_t^2+v_t, \] with \(w_t\sim\mathcal{N}(0,10)\), \(v_t\sim\mathcal{N}(0,1)\), \(z_0\sim\mathcal{N}(0,5)\), \(N_p=500\) particles, \(M=100\) runs, \(T=200\). Candidate structures: \(\mathcal{S}=\{s_\mathrm{nl}, s_\mathrm{lin}\}\). Score perturbations \(\epsilon_t(s)\sim\mathrm{Uniform}(-\bar{\varepsilon}, \bar{\varepsilon})\). Three methods: Exact CF (oracle), CF without margin (\(\delta=0\)), and Robust CF (\(\delta=2.5\bar{\varepsilon}\)).
Table I: Switching and Score Results
| Method | \(\bar{\varepsilon}=0.5\) | \(\bar{\varepsilon}=1.5\) | \(\bar{\varepsilon}=3.0\) | |||
|---|---|---|---|---|---|---|
| \(\mathbb{E}[N_T]\) | \(\bar{\Phi}_T\) | \(\mathbb{E}[N_T]\) | \(\bar{\Phi}_T\) | \(\mathbb{E}[N_T]\) | \(\bar{\Phi}_T\) | |
| Exact CF (oracle) | 0.3 | 2.71 | 0.3 | 2.71 | 0.3 | 2.71 |
| CF w/o margin (\(\delta=0\)) | 83.7 | 9.60 | 81.1 | 9.58 | 79.2 | 9.55 |
| Robust CF (proposed) | 0.3 | 2.83 | 1.3 | 2.95 | 7.9 | 3.48 |
| Thm. 2 bound | 8.2 | — | 11.4 | — | 17.6 | — |
Lower \(\mathbb{E}[N_T]\) and \(\bar{\Phi}_T\) indicate better performance. Robust CF stays well below Theorem 2 bound ✓
Experiment Results
- Exp. I (Descent, Fig. 2): Running-average \(\bar{\Phi}_t\) of Robust CF tracks the oracle throughout; CF without margin remains elevated by \(\approx3\times\), confirming Theorem 1.
- Exp. II (Finite switching, Table I): Robust CF empirical \(\mathbb{E}[N_T]\) lies well below the Theorem 2 bound at all noise levels, confirming Theorem 2.
- Exp. III (Non-chattering, Fig. 3): Robust CF maintains \(\hat{p}_t<0.05\); CF without margin exhibits \(\hat{p}_t\approx0.4\) persistently, confirming Theorem 3.
- Bound tightness (Fig. 4): Empirical \(\mathbb{E}[N_T]\) decays monotonically with \(\alpha=\delta/\bar{\varepsilon}\) and lies strictly below the Theorem 2 bound for all \(\alpha>2.5\).
Impact and Applications
- Adaptive state estimation under sensor noise and model uncertainty
- Particle filter-based systems where score approximation errors are unavoidable
- Neural-network-aided filtering (KalmanNet, Split-KalmanNet, Latent-KalmanNet) where learned scores carry structured errors
- Supervisory control with noisy performance metrics
- Fault detection in drifting systems
Reproducibility
All Julia source code is publicly available:
-
lcss_experiments_documented.jl— Figs. 2&3, Table I (validates Theorems 1–3) -
fig_scaling_documented.jl— Fig. 4, bound tightness (Theorem 2)