Controllability-Based Interpretability Framework

• Controllability-based interpretability is a framework that quantitatively links a model’s internal control mechanisms with its ability to be interpreted via targeted interventions.
• It employs control theory metrics such as Gramians, Hankel singular values, and intervention pipelines to assess and rank neuron importance and steerability.
• Practical implications include enhanced model diagnostics, safer intervention strategies, and improved human-in-the-loop control in complex systems.

A controllability-based interpretability framework is a class of methods that rigorously link a model’s internal representations or functional units to their power to steer output behavior under targeted interventions. Interpretability, in this perspective, is evaluated by the model’s amenability to intervention and the degree to which model explanations correspond to directions or structures that reliably afford control. This paradigm unifies mechanistic insight and practical steering, offering metrics and pipelines for quantitatively monitoring, diagnosing, and optimizing model controllability (via interventions) alongside or as a prerequisite for interpretability.

1. Theoretical Foundations: Controllability and Interpretability

The mathematical roots derive from control theory, with core constructs such as the controllability and observability Gramians, Kalman rank conditions, and state space localizations. In the context of neural networks, controllability quantifies which neurons or directions in hidden state space can be effectively excited or steered by external signals (including input perturbations or model-internal interventions), and how such changes propagate to outputs. Observability complements this with a measure of how effectively internal changes are externally manifest.

In static or feedforward architectures, the input-to-state Jacobian $B$ and state-to-output Jacobian $C$ form the basis for constructing Gramians:

• Controllability Gramian: $W_c = B B^\top$
• Observability Gramian: $W_o = C^\top C$

Modal analysis via their Hankel product $M=W_c W_o$ identifies dominant directions—those simultaneously excitable and observable—yielding interpretable rankings of neurons and internal pathways. Neuron-importance metrics based on $\mathrm{diag}(W_c)$ and $\mathrm{diag}(W_o)$ quantitatively attribute “excitation” and “influence,” and Hankel singular values $\sigma_i$ and their associated eigenvectors $v_i$ structure internal modes by effective input–output energy (Moon, 17 Nov 2025).

These principles are further extended to complex networks, where strong structural controllability (SSC) is algorithmically characterized via a structural dissection process. Here, node and edge roles are classified using local leaf-removal and core-percolation criteria: “spareable” (topology-only) vs “effective” (weight-dependent) links, and critical vs. redundant vs. intermittent nodes, thereby mapping structural units to their essentiality for controllability (Shen et al., 2015).

2. Mechanistic Interventions and Emergence of Steerability

Controllability-based interpretability in deep models often centers on direct interventions into hidden representations to diagnose and induce semantic control over behaviors. The Intervention Detector (ID) pipeline provides a canonical instantiation in LLMs (She et al., 3 Aug 2025):

• Linear steerability is defined via a “concept direction” $v_c$ in hidden space: $h' = h + \alpha v_c$ for scalar $\alpha$.
• Effective control (steering output toward/away from concept $c$) depends upon linear separability of $c$ in hidden space, quantified by a separability score $S(t)$ and centroid-cosine metrics.
• The ID pipeline tracks these metrics over training checkpoints and layers, extracting emergence signatures of steerability, such as entropy troughs and heatmaps of alignment scores $I_{l,t}(c)$.
• Key empirical findings include staged emergence of steerability, dependence on concept, and model-family generality. Notably, separability is a necessary precursor for successful intervention; steerability crystallizes only when $S(t) \gtrsim 0.6$.

This approach goes beyond heuristic “add-vector” interventions, furnishing interpretable, quantitative monitoring for when and where control—and thus faithful interpretability—becomes possible.

3. Encoder–Decoder Abstractions and Feature-Space Interventions

A general encoder–decoder abstraction systematizes interventions on human-interpretable features, both for interpretability and control (Bhalla et al., 2024):

• Encoder: $z = f_{\mathrm{enc}}(x)$ projects high-dimensional activations to a latent feature basis (e.g., via a dictionary, sparse code, token-unembedding, or probe weights).
• Decoder: $x̂ = f_{\mathrm{dec}}(z)$ reconstructs original space; correctness of the explanation is measured by reconstruction error.
• Intervention: Direct manipulation $z_i' = \alpha$ (or similar) for interpretable dimension $i$, then decode to $x̂'$ and propagate through the model.
• Evaluation: Intervention Success Rate (ISR) quantifies how reliably intervening on $z_i$ induces target behavior; Coherence–Intervention Tradeoff (CIT) quantifies the best attainable ISR for a specified drop in generative coherence.

Empirically, logit/tuned lens methods achieve high ISR (ISR $\approx 0.5-0.6$ within $\pm 1$ point coherence drop), while direct prompting remains superior for simple concepts (Bhalla et al., 2024).

4. Causal and System-Level Controllability Interventions

In generative or nonlinear models, causal intervention frameworks extend the scope of controllability:

• In VAEs, input, latent, and activation interventions robustly isolate “circuit motifs” controlling semantic factors (Roy, 6 May 2025).
• Causal effect strength, specificity, and modularity scores quantitatively evaluate the depth and exclusivity of controllability.
• In user-facing systems such as recommendation engines, counterfactual interventions offer both retrospective (removal of past behaviors) and prospective (addition of new actions) explanations and levers for control (Tan et al., 2023).
• Metrics of controllability—complexity and accuracy—objectively benchmark user influence over system outputs.

These frameworks elevate intervention not only as a test of interpretability, but as a vehicle for robust, actionable control, central to human-in-the-loop and safety-critical applications.

5. Monitoring, Diagnosing, and Optimizing Controllable Interpretability

Controllability-based interpretability frameworks enable both monitoring—for example, the ID pipeline’s visualization of steerability emergence over training steps and layers (She et al., 3 Aug 2025)—and diagnosis and remediation of failure modes. In concept bottleneck models, control-based “leakage” scores (CTL and ICL) rigorously measure the extent to which concepts harbor extraneous task or interconcept information, directly predicting breakdowns under human intervention (Parisini et al., 18 Apr 2025). High correlation between CTL/ICL and post-intervention accuracy loss enables these scores to serve as early-warning diagnostics.

Control-theoretic adaptivity can also regulate interpretability in real-time settings, as exemplified by the SCI framework for signal intelligence (Meesala, 15 Nov 2025): interpretability is treated as a regulated equilibrium (state variable “Surgical Precision”), driven toward target values via projected parameter updates, Lyapunov-stabilized feedback, and trust-region/rollback safeguards. This closed-loop scheme achieves substantial reductions in interpretive error variance and improved stability across biomedical, industrial, and environmental domains (mean $\Delta$SP reduction 38%, SP variance halved).

6. Domain-Specific Instantiations and Applications

Controllability-based interpretability is instantiated in diverse modalities:

• Style transfer: Fourier-based phase and amplitude manipulations provide interpretability aligned with controllability—preservation or modulation of content structure can be explicitly attributed to phase manipulation, and stylization control is realized via frequency blending (Jin et al., 2022).
• Mixture-of-experts and conditional computation: User-defined topic partitions yield explanations and controls aligned with user intent, with sparsity and topic-exclusion penalties guaranteeing faithfulness and per-instance actionability (Swamy et al., 2024).
• Complex networks: The role of network topology in global controllability is directly tied to interpretable, local structural units, supporting both theory-driven and empirical large-scale network diagnoses (Shen et al., 2015).

These instantiations confirm that controllability-principled frameworks are both general and adaptable, subsuming pre-existing mechanistic, structural, causal, and user-facing interpretability tools under a unified paradigm.

7. Limitations and Outlook

Despite their promising quantitative and conceptual advances, controllability-based interpretability frameworks exhibit open challenges:

• Tradeoff with task coherence and utility: In LLMs, mechanistic interventions often degrade output coherence before high intervention fidelity is achieved, and may ultimately underperform prompt-based control (Bhalla et al., 2024).
• Concept dependence and temporal dynamics: The emergence of steerable directions is concept- and layer-dependent—the underlying alignment dynamics resist simple generalizations and necessitate continual, context-specific monitoring (She et al., 3 Aug 2025).
• Dependence on explicit intervention pathways: In causal frameworks, only monosemantic or modular circuits guarantee reliable control; polysemanticity and circuit entanglement constrain the effectiveness of circuit-level interventions (Roy, 6 May 2025).

Future advances may target: joint optimization of controllability and faithfulness, explicit regularization for pathway modularity and effect strength, and tighter integration with safety constraints and human-in-the-loop protocols.

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References:

• "How Does Controllability Emerge In LLMs During Pretraining?" (She et al., 3 Aug 2025)
• "From Black-Box to White-Box: Control-Theoretic Neural Network Interpretability" (Moon, 17 Nov 2025)
• "Towards Unifying Interpretability and Control: Evaluation via Intervention" (Bhalla et al., 2024)
• "Causal Intervention Framework for Variational Auto Encoder Mechanistic Interpretability" (Roy, 6 May 2025)
• "User-Controllable Recommendation via Counterfactual Retrospective and Prospective Explanations" (Tan et al., 2023)
• "Leakage and Interpretability in Concept-Based Models" (Parisini et al., 18 Apr 2025)
• "SCI: An Equilibrium for Signal Intelligence" (Meesala, 15 Nov 2025)
• "Style Spectroscope: Improve Interpretability and Controllability through Fourier Analysis" (Jin et al., 2022)
• "Intrinsic User-Centric Interpretability through Global Mixture of Experts" (Swamy et al., 2024)
• "Fundamental building blocks of controlling complex networks: A universal controllability framework" (Shen et al., 2015)