Spectral Identifiability Principle (SIP)

• SIP is a set of spectral conditions that guarantee the identifiability, stability, and interpretability of latent structures in high-dimensional settings.
• It uses specific metrics like eigengaps and minimal eigenvalues along with operator norms and concentration inequalities to predict phase transitions from reliable learning to collapse.
• The principle is broadly applied in neural representations, random matrix models, dynamical systems, and causal inference, offering practical diagnostic and regularization tools.

The Spectral Identifiability Principle (SIP) encapsulates a family of necessary and often sharp spectral conditions guaranteeing the identifiability, stability, and interpretability of latent structures or models in high-dimensional statistical, dynamical, and geometric settings. SIP asserts that when critical spectral characteristics—such as eigengaps, minimal eigenvalues, or geometric discrepancies—substantially exceed intrinsic sample-level fluctuations or structural degeneracies, stable identification and accurate downstream performance become possible. Conversely, breaches of these spectral thresholds induce phase transitions to statistical or algorithmic failure. The notion generalizes across settings ranging from neural representation probing, random matrix models, and dynamical system identification to information geometry and causal inference, formalizing when spectral data suffice to reliably recover or interpret underlying latent objects.

1. Foundational Formulation and Mathematical Framework

The core instantiation of SIP is as a finite-sample, operator-norm–based criterion for identifiability of informative subspaces or parameters. In the canonical linear-probe scenario (Huang, 20 Nov 2025), denote $X$ an input, $Y \in \{\pm1\}$ a label, and $h(X) \in \mathbb{R}^d$ a fixed representation. Define the population Fisher operator $\Gamma = \mathbb{E}[h(X)h(X)^\top]$ with ordered eigenvalues $\lambda_1 \geq \cdots \geq \lambda_d$ and choose $r$ top directions. The critical eigengap is

$$\gamma := \lambda_r(\Gamma) - \lambda_{r+1}(\Gamma)$$

and the empirical estimation error,

$\epsilon := \|\hat\Gamma - \Gamma\|_{op}$

where $\hat\Gamma$ is the empirical Fisher estimate from $n$ samples.

The principle posits a sharp threshold:

$\gamma > C \epsilon$

for some universal $C$ (often $C=1$), succinctly, if $\epsilon < \gamma$, both the subspace estimate and the probe's misclassification risk are uniformly controlled and stable. The geometric and probabilistic structure ensuring this concentration is made explicit by operator-perturbation bounds and matrix concentration inequalities. The SIP also admits direct operationalization: splitting held-out data estimates the spectral quantities, and passing or failing SIP predicts probe reliability.

Table: Key Quantities in Linear Probe SIP

Symbol	Description	Mathematical Expression
$\Gamma$	Population Fisher operator	$\mathbb{E}[h(X)h(X)^\top]$
$\gamma$	Eigengap (task-relevant gap)	$\lambda_r(\Gamma)-\lambda_{r+1}(\Gamma)$
$\epsilon$	Empirical Fisher error	$\|\hat\Gamma - \Gamma\|_{op}$

SIP's ramifications and associated concentration phenomena are universal across spectral statistical estimation, with variants based on minimal eigenvalues, geometric multiplicities, and information-theoretic divergence.

2. Spectral Phase Transitions and Finite-Sample Stability

SIP predicts abrupt phase transitions for stability and identifiability as spectral signals cross sample-level fluctuations (Huang, 20 Nov 2025, Huang, 4 Oct 2025). Specifically, in high-dimensional learning, failing to maintain $\lambda_{\min}(\Gamma)$ above a critical threshold proportional to $\sqrt{d/n}$ (the sample Fisher fluctuation scale) results in a qualitative transition from reliable learning and subspace concentration to collapse and algorithmic instability. This is evidenced in both linear and nonlinear models: as $n$ increases past the critical value $n^* \propto d/\gamma^2$, subspace estimates and risk measures improve rapidly.

The transition can be visualized as:

• For $\epsilon / \gamma \ll 1$, subspace estimation is stable and probe risk matches Bayes-optimal error.
• At $\epsilon / \gamma \approx 1$, risk and subspace error undergo sharp increases—interpreted as an information-theoretic or geometric phase change.
• For $\epsilon / \gamma > 1$, estimation becomes unreliable, and any further statistical guarantees are lost (Huang, 20 Nov 2025).

This finite-sample phase transition is model-agnostic and extends to covariance, Fisher information, and dynamical operators.

3. Spectrum-Driven Identifiability in Linear Dynamical and Random Matrix Models

In system identification and random matrix theory, SIP manifests through relationships between the algebraic and geometric multiplicity of eigenvalues and identifiable substructures (Naeem et al., 2023, Hayase, 2018). For an $n$-dimensional stable linear system $x_{t+1} = A x_t + w_t$ with matrix $A$, denote $D_i = AM(\lambda_i) - GM(\lambda_i)$ the discrepancy for each eigenvalue $\lambda_i$. SIP implies the following:

• Small $D_i$ for all $i$ ensures statistical independence across invariant subspaces, mode separability, and concentration of OLS estimates at non-asymptotic $O(1/\sqrt{N})$ error rates.
• Large $D_i$, such as a single nontrivial Jordan block, induces inseparable dynamics, loss of identifiability, and an exponential blow-up—the curse of dimensionality (Naeem et al., 2023).

In random matrix models, SIP takes the form: for compound Wishart matrices, the spectral law map is injective up to unitary conjugation (eigenvalue ordering), while for signal-plus-noise models identifiability is up to biunitary rotations and sign, as determined via operator-valued free probability and free deconvolution tools (Hayase, 2018).

4. Information-Theoretic and Causal Inference Perspectives

SIP admits extension to information geometry and causal discovery from time series. In the spectral independence criterion (SIC), the principle asserts that power spectral density (PSD) of inputs and the modulus-squared of the system's transfer function remain uncorrelated ("independent") in the true causal direction (Besserve et al., 2021). Formally, for $\{X_t\}$ and $\{Y_t\}$ with $Y_t = (h * X)_t$ and frequency response $H(\omega)$,

$$\langle S_{xx}(\omega) |H(\omega)|^2 \rangle = \langle S_{xx}(\omega) \rangle \langle |H(\omega)|^2 \rangle$$

The statistical dependency ratio $\rho_{X \to Y}$ then satisfies $\rho_{X \to Y} = 1$ if and only if SIC holds, which a high-dimensional generative model shows is almost surely true only in the correct causal direction for random, nontrivial $H$. This yields direct identifiability criteria and justifies spectral independence for directionality inference in dynamical systems.

5. Diagnostic Algorithms and Practical Utility

SIP's operational value arises from its verifiable spectral diagnostics. In neural representations, the recipe (Huang, 20 Nov 2025) proceeds as:

1. Extract fixed-layer representations $h_i$.
2. Compute empirical $\hat\Gamma$ and eigengap $\hat\gamma$ via SVD.
3. Estimate error proxy $\hat\epsilon$ by data split or bootstrapping.
4. Declare "SIP pass" if $\hat\epsilon < \hat\gamma$.
5. If heavy-tailed, adapt via feature clipping to optimize $\hat\epsilon/\hat\gamma$.

For Fisher-based loss landscapes, direct penalization of the minimal Fisher eigenvalue with a "Fisher floor" regularization enforces identifiability throughout training, with the critical threshold determined by $\lambda_{\min}(\Gamma) > C \sqrt{d/n}$ (Huang, 4 Oct 2025). These algorithms are robust to smoothing, scaling, and regularization and serve as ex-ante diagnostics to anticipate or prevent probe collapse and overfitting.

Table: SIP-Based Diagnostic Workflow

Step	Operation	Output
1. (Data Prep)	Extract $h_i$ from network	features $h_1,\ldots,h_n$
2. (Estimation)	Compute $\hat\Gamma$, $\hat\gamma$	spectral quantities
3. (Error Proxy)	Estimate $\hat\epsilon$	error estimate
4. (Decision)	Compare $\hat\epsilon$ to $\hat\gamma$	reliability verdict

6. Statistical, Geometric, and Algorithmic Implications

SIP unifies several classical and contemporary concepts:

• Rigorous, non-asymptotic sample-complexity bounds linking spectral geometry (eigengap, minimal eigenvalue) to finite-sample identifiability (Huang, 4 Oct 2025, Huang, 20 Nov 2025).
• Phase-transition phenomena and impossibility results for estimation below critical spectra.
• Minimal or rotation-invariant parameter identifiability in random matrix ensembles, signal-plus-noise models, and dynamical systems (Hayase, 2018, Naeem et al., 2023).
• Information-orthogonality and independence of irregularities for causal inference in Gaussian processes (Besserve et al., 2021).
• Model-agnostic diagnostics and regularization strategies across deep learning, system identification, and high-dimensional statistics.

A plausible implication is that SIP provides a fundamental, verifiable boundary demarcating when high-dimensional inference is theoretically and algorithmically tractable, and when it suffers inevitable instability or nonidentifiability due to spectral degeneracy.

7. Limitations, Failure Modes, and Extensions

SIP rests on critical spectral conditions that are both necessary and sufficient in many regimes. Limitations include:

• Absence or vanishing of eigengap or minimal eigenvalue: SIP predicts and confirms collapse of identifiability and stability, e.g., via phase transitions or the curse of dimensionality (Naeem et al., 2023, Huang, 4 Oct 2025).
• Pathological spectral structure: large Jordan block discrepancies or constant transfer modulus in causal models violate SIP and nullify its guarantees (Hayase, 2018, Besserve et al., 2021).
• Heavy-tailed, non-subGaussian features may require preprocessing to meet SIP regularity conditions (Huang, 20 Nov 2025).
• Application scope: nonlinear, non-LTI, or hidden confounder settings may escape SIP's coverage and require further generalization.

Extensions incorporate robustification under smoothing, invariance via group actions, and generalizations to nonparametric or nonstationary models, but these remain bounded by the presence of tractable and nondegenerate spectral structure.

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SIP constitutes a foundational principle in contemporary high-dimensional inference, providing both stringent mathematical guarantees and practical diagnostic criteria, encapsulating the geometric phase transition from identifiability to collapse as a function of underlying spectral structure. Its applicability is broad, encompassing statistics, machine learning, system identification, and causal inference, and is substantiated by sharp non-asymptotic theorems and algorithmic tools across diverse research domains (Huang, 20 Nov 2025, Huang, 4 Oct 2025, Hayase, 2018, Naeem et al., 2023, Besserve et al., 2021).