Spectral Supersaturation Problem

• Spectral supersaturation is defined as the phenomenon where surpassing a critical spectral threshold enforces a fixed error reduction rate in regularization or forces the abundance of specific substructures in graphs.
• In inverse problems, the global saturation function sets the optimal convergence limit, as seen in Tikhonov regularization where error reduces at a rate of δ^(2/3) under polynomial qualifications.
• In extremal graph theory, exceeding spectral radii compels the emergence of numerous forbidden subgraphs, providing sharp bounds and insights into graph structure.

The spectral supersaturation problem characterizes the precise phenomenon where, for regularization methods defined by spectral filtering or for spectral bounds in discrete extremal structures, exceeding a spectral threshold induces not just the presence but abundance of a target structure—whether small subgraphs in combinatorics or rapid convergence in functional inverse problems. In spectral regularization of inverse problems, this is formalized by the existence of a global saturation set and function: for solutions in a specified source space, no greater uniform rate of error reduction is possible, no matter the smoothness of the underlying solution; in extremal graph theory, surpassing key spectral radii forces numerous instances of forbidden subgraphs.

1. Spectral Regularization and Supersaturation: Core Definitions

In regularization for ill-posed linear inverse problems, operators $T:X \to Y$ (with $X, Y$ Hilbert, $R(T)$ nonclosed, $T$ injective) are inverted via spectral regularization methods (SRMs). An SRM is defined by a family of real-valued functions $\{g_\alpha\}$ satisfying continuity, uniform boundedness, and the limiting property $g_\alpha(\lambda) \to 1/\lambda$ as $\alpha \to 0^+$ for $\lambda > 0$. The corresponding regularization operators take the form $R_\alpha = g_\alpha(T^*T) T^*$.

Qualification quantifies the link between residual behavior and attainable error rates: a function $p \in \mathcal{O}$ is an optimal qualification if for some $s \in \mathcal{S}$, the pair exhibits strong and order source–order properties. For noisy data $y^\delta$, the total error of the method is

$$E_\text{tot}(x, \delta) = \inf_{\alpha > 0} \sup_{\|T x - y^\delta\| \leq \delta} \|R_\alpha y^\delta - x\|.$$

A global saturation function $\varphi$ on a set $M$ exists when $E_\text{tot}(x, \delta) = O(\varphi(x, \delta))$ for $x \in M$, and this rate cannot be improved on any strictly larger set while maintaining invariance (uniformity) and optimality (Mazzieri et al., 2011, Herdman et al., 2010).

2. Existence and Structure of Saturation in Spectral Regularization

The existence theorem underpins the spectral supersaturation phenomenon in operator regularization. If $\{g_\alpha\}$ has optimal qualification $p$, fulfills uniform boundedness, monotonicity, and spectral gap conditions, then the total error saturates globally on the source set $M = \text{Range}(s_p(T^*T)) \setminus \{0\}$, with saturation function

$$\varphi(\delta) = p(\psi^{-1}(\delta)), \quad \text{where}~\psi(t) = \sqrt{t} p(t).$$

The parameter $\alpha = \psi^{-1}(\delta)$ solves $\sqrt{\alpha} p(\alpha) = \delta$. The saturation exponent is determined by the qualification order:

• Polynomial qualification $p(\alpha) = \alpha^k$ yields $\varphi(\delta) = \delta^{2k/(2k+1)}$ on $M = \text{Range}((T^*T)^k) \setminus \{0\}$.
• For Tikhonov regularization ($k=1$), $\varphi(\delta) = \delta^{2/3}$, $M = \text{Range}(T^*T) \setminus \{0\}$ (Mazzieri et al., 2011, Herdman et al., 2010).

No additional smoothness of $x$ outside $M$ yields a uniformly faster rate; the saturation "plateau" is strict.

Table: Characteristic Saturation Exponents and Sets in Classical SRMs

Regularization Type	Qualification $p(\alpha)$	Saturation Function $\varphi(\delta)$	Saturation Set $M$
Tikhonov ($k=1$)	$\alpha$	$\delta^{2/3}$	Range$(T^*T)\setminus\{0\}$
Order-$k$ Tikhonov	$\alpha^k$	$\delta^{2k/(2k+1)}$	Range$((T^*T)^k)\setminus\{0\}$
Exponential filter	$\alpha^k$	$\delta^{2k/(2k+1)}$	Range$((T^*T)^k)\setminus\{0\}$

3. Generalizations: Maximal and Non-Polynomial Qualification

Spectral regularization theory admits methods with non-polynomial or maximal qualification. For a general qualification $\rho(\alpha)$ (e.g., $\rho(\alpha) = 1/|\ln \alpha|$), the saturation function takes the form

$$\psi(x,\delta) = \rho(\Theta^{-1}(\delta)),\quad \text{where}~\Theta(t) = \sqrt{t} \rho(t),$$

and the saturation set is $X^\rho = \text{Range}(\rho(T^*T))\setminus\{0\}$. This includes logarithmic and subexponential decays, accommodating ultradifferentiable or analytic source conditions. Classical truncated SVD yields maximal qualification ($\mu_0=+\infty$), saturating at $\varphi(\delta) = \delta$ (Herdman et al., 2010).

4. Spectral Supersaturation in Extremal Graph Theory

In extremal graph theory, the concept of spectral supersaturation is applied to discrete structures. Given an $n$-vertex graph $G$ with adjacency matrix $A(G)$ and spectral radius $\lambda(G)$, thresholding $\lambda(G)$ above that of a forbidden extremal graph compels the existence of many target subgraphs:

• Erdős–Faudree–Rousseau spectral variant: If $\lambda(G) \geq \sqrt{\lfloor n^2/4 \rfloor}$, then either $G$ is the balanced bipartite extremal or has at least $2\lfloor n/2\rfloor-1$ triangular edges (Li et al., 2024).
• Triangles and bowties: If $$\lambda(G) \geq \lambda(K_{\lceil n/2\rceil, \lfloor n/2\rfloor}^{+2})$$, then $G$ contains at least $\lfloor n/2\rfloor$ bowties; extremality is uniquely realized by $K_{\lceil n/2\rceil, \lfloor n/2\rfloor}^{+2}$ (Li et al., 2024).
• Color-critical graphs: For $F$ with chromatic number $r+1$, if $$\lambda(G) \geq \min_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$$ for $q = O(\sqrt{n})$, $G$ must have at least $q c(n,F)$ copies of $F$; if $$\lambda(G) \geq \max_{T \in \mathcal{T}_{n,r,q}} \lambda(T)$$ for $q = O(n)$, an extremal structure is uniquely identified (Fang et al., 27 Dec 2025).

Table: Spectral Supersaturation Results in Graphs

Target Subgraph	Spectral Threshold	Minimum # of Subgraphs	Extremal Graph Structure
Triangle	$\sqrt{\lfloor n^2/4 \rfloor}$	$2\lfloor n/2\rfloor-1$	Balanced bipartite or extensions
Bowtie	$$\lambda(K_{\lceil n/2\rceil, \lfloor n/2\rfloor}^{+2})$$	$\lfloor n/2\rfloor$	$K_{\lceil n/2\rceil, \lfloor n/2\rfloor}^{+2}$
$F$ color-critical	$\min \lambda(T_{n,r,q})$	$q c(n, F)$	$\mathcal{T}_{n,r,q}$ family

5. Saturation Boundaries, Plateau, and Practical Implications

In both operator regularization and discrete spectral extremal theory, the saturation phenomenon identifies intrinsic limitations: for a solution $x$ lying within the source space prescribed by qualification, the error convergence cannot be better than the saturation function—even if $x$ is smoother. In graph theory, once the spectral radius surpasses a critical extremal value, subgraph counts jump discontinuously, and the structural features are sharply constrained.

For regularization, the practical implication is the necessity to select regularization methods whose qualification matches or exceeds the smoothness class of the solution; otherwise, the error rate plateaus at the saturation exponent. In combinatorics, spectral tools yield sharp subgraph count bounds under spectral, rather than purely edge-based, thresholding—often admitting stronger or tighter results.

6. Open Problems and Research Directions

Current research investigates several directions:

• Relaxation of hypotheses: The global saturation theorem relies on monotonicity and a spectral gap condition. Analyses study extending saturation existence to settings with continuous spectra or weaker qualifications (Mazzieri et al., 2011).
• Nonlinear and Banach-space extensions: The theory for nonlinear inverse problems and Banach spaces remains less developed; the structure of saturation and supersaturation functions in these cases is a subject of inquiry.
• Generalizations in graph theory: Extending supersaturation results to spectral analogues for hypergraphs and non-color-critical structures (books, quadrilaterals, etc.), quantifying minimal spectral increments for prescribed subgraph counts (Fang et al., 27 Dec 2025, Li et al., 2024).

A plausible implication is that a systematic spectral approach may unify and strengthen bounds for forbidden and mandated substructures across both continuous and discrete domains, and may provide new extremal configurations beyond classical Turán-theoretic settings. The saturation plateau remains a defining threshold for the effectiveness of both regularization algorithms and spectral extremal phenomena.