Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities

Abstract: We develop a tensor-amplification framework for Sidorenko-type inequalities in graphon classes. The framework applies to any admissible class, meaning a class closed under tensor powers and normalized principal restrictions. These two closure properties isolate the structural input needed for the amplification arguments, while preserving natural positivity constraints such as the doubly nonnegative constraint. For every admissible class $\mathcal{C}$, we prove two transfer principles. First, equality cases regularize optimally: if a non-matching graph $H$ is $\mathcal{C}$-Sidorenko, then every equality case $t(H,W)=p(W)<sup>{e(H)}$ with $W\in\mathcal{C}$ is regular. Consequently, relative forcing is equivalent to relative regular-forcing for every non-matching $\mathcal{C}$-Sidorenko graph. Second, in the range $v(H)\le e(H)$, ordinary $\mathcal{C}$-Sidorenko is equivalent, as a universal property over $\mathcal{C}$, to the spectral inequality $$t(H,W)\ge ρ(W)<sup>{2e(H)-v(H)}p(W)<sup>{v(H)-e(H)}$$ for every non-zero $W\in\mathcal{C}$. The spectral transfer is obtained from a Perron-biased tensor regularization theorem detecting the Perron spectral radius on the exponential scale. We also prove quantitative near-equality variants and apply the framework to doubly nonnegative graphons and bounded doubly nonnegative kernels. This yields spectral equivalences for Sidorenko-good graphs in the range $v(F)\le e(F)$, and identifies Sidorenko-good forcing with regular-KNRS forcing for non-matching Sidorenko-good graphs.

• The paper introduces a novel tensor amplification framework for Sidorenko-type inequalities to analyze subgraph densities.
• It establishes key equality and spectral transfer principles linking regular graphons with non-matching Sidorenko properties.
• Quantitative stability results and applications to doubly nonnegative kernels highlight practical implications for extremal graph theory.

Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities

Introduction and Context

The paper "Tensor Amplification and Spectral Transfer for Sidorenko-Type Inequalities" (2607.02260) develops a unifying tensor-power amplification framework for Sidorenko-type inequalities within various classes of graphons. Sidorenko's conjecture concerns the minimization of subgraph densities in large dense graphs and, equivalently, inequalities for homomorphism densities of bipartite graphs in graphon models. Despite significant progress, the general conjecture remains unresolved. Numerous techniques exist—dependant random choice, entropy methods, blow-ups, among others—but tensor powers have emerged as a structural mechanism underlying many previous Sidorenko-type results.

This work isolates the algebraic operations required for amplification arguments—tensor powers and normalized principal restrictions—by introducing the notion of admissible graphon classes: those closed under these two operations. The key contribution is to formulate, prove, and apply powerful transfer principles and regularization theorems for Sidorenko-type inequalities in such admissible classes, with considerable emphasis on both equality and spectral strengthening phenomena. The framework also accommodates positivity-preserving notions, including doubly nonnegative kernels.

Main Results and Theoretical Advances

Tensor Amplification Framework

The central innovation is the "tensor amplification" method. For a graphon $W$, tensor powers $W^{\otimes k}$ and principal restrictions to suitable subsets in the product space allow for systematic amplification of non-uniformities in $W$. This can be leveraged to detect and transfer key structural properties, notably via:

• Degree-biased amplification: Targets failure of regularity by amplifying deviations in degree distributions.
• Perron-biased amplification: Exploits the spectral radius (via the Perron eigenfunction) to transfer spectral lower bounds at scale.

These two forms undergird the main regularization and spectral transfer principles.

Equality Regularization Principle

A foundational theorem established is that, in any admissible class, for any non-matching Sidorenko graph $H$ (i.e., $2e(H) > v(H)$), the only graphons achieving equality in the Sidorenko inequality are those which are regular. More precisely:

• If $H$ is $\mathcal{C}$-Sidorenko and $t(H, W) = p(W)^{e(H)}$, then $W$ is $p(W)$-regular for any $W^{\otimes k}$0 in $W^{\otimes k}$1.

This result is optimal: for matchings, the statement does not hold since every graphon trivially achieves equality.

A corollary is that, for non-matching graphs, relative forcing and regular-forcing are equivalent notions: Any obstruction to forcing appears among regular graphons.

Spectral Transfer Principle

A complementary result in the range $W^{\otimes k}$2 establishes an equivalence between the ordinary Sidorenko property and a strict spectral lower bound:

• **$W^{\otimes k}$3 is $W^{\otimes k}$4-Sidorenko if and only if for every non-zero $W^{\otimes k}$5,

$W^{\otimes k}$6

where $W^{\otimes k}$7 is the operator norm (Perron spectral radius).**

For constant graphons, this recovers the Sidorenko bound; for non-constant $W^{\otimes k}$8, the inequality becomes strictly stronger whenever $W^{\otimes k}$9. The proof employs the weak spectral regularization theorem, demonstrating that tensor powers and suitable restrictions allow detection of the spectral radius on the appropriate scale, ultimately yielding a universal transfer principle over admissible classes.

This spectral formulation generalizes and sharpens earlier results, directly connecting to recent advances for finite graphs and their homomorphism counts in terms of adjacency spectral data (Li et al., 26 May 2026).

Near-Equality Regimes and Quantitative Stability

Strong quantitative forms of the above results are derived. Notably:

• If the Sidorenko ratio is close to equality, the degree-biased amplification yields tight bounds on the relative entropy between the degree distribution and the base measure, quantifying proximity to regularity.
• In the spectral regime, a tight $W$0-bound between the degree function and its mean is established in terms of the Sidorenko ratio, further quantifying stability and structural regularization for near-equality cases.

Application to Doubly Nonnegative Kernels and Sidorenko-Good Graphs

The framework is applied to the class of doubly nonnegative (DNN) graphons and, more broadly, to bounded DNN kernels. For such positivity-preserving classes:

• Spectral equivalence is proven for all Sidorenko-good graphs when $W$1.
• Sidorenko-good forcing is shown to be equivalent to regular-KNRS forcing for non-matching Sidorenko-good graphs.

This connects the Sidorenko-good framework with the regular locally dense setting prevalent in contemporary KNRS-oriented literature and recasts classical positivity-based inequalities within the admissible-class methodology.

Technical Innovations and Proof Techniques

The methodology is characterized by the systematic use of tensor power constructions and principal restrictions, supported by probabilistic and operator-theoretic tools:

• Use of degree-biased and spectral (Perron-biased) tilting measures to amplify structural deviations.
• Application of law of large numbers, entropy, and Pinsker-type inequalities to obtain quantitative stability.
• Spectral regularization derived via level-set techniques on Perron eigenfunctions in high-dimensional tensor spaces.
• Measure-theoretic arguments ensuring that sets in tensor powers concentrate mass or energy appropriately to force contradictions under non-regularity or sub-optimal spectral behavior.

The analytic approach dispenses with finite approximation schemes, working directly at the graphon/operator level—a point of conceptual and technical advancement, especially for infinite-dimensional positivity classes.

Implications and Future Developments

Structural Consequences

The identification of tensor and principal restriction closure as the minimal structure required for amplification arguments establishes a flexible platform for analyzing Sidorenko-type inequalities, equality, and forcing properties across a broad array of graphon classes. This formalization should enable more systematic study of extremal subgraph densities and regularization in dense limit objects.

The equivalence between Sidorenko, spectral-Sidorenko, and regular-forcing properties in admissible classes—especially in positivity-preserving settings—has theoretical impact for operator inequality generalizations and regularity method developments.

Prospects for Algebraic Operations and Exponent-Monotone Algebra

As articulated in the conclusion, the tensor amplification paradigm suggests a broader algebraic program: to classify and exploit algebraic operations (beyond tensor powers) that preserve or sharpen Sidorenko exponents, potentially yielding new notions of admissibility and error amplification. This has ramifications for sparse graph limits, higher symmetry host constructions, and potentially for non-dense regimes as well.

Potential Impact on Forcing and Existence Questions

The techniques sharply delimit the locus of any remaining obstruction to regular forcing of Sidorenko-type inequalities—showing that any such obstruction must manifest within the regular world. This narrows the search for counterexamples or proofs in open cases and provides a clean dichotomy guiding future research.

Conclusion

This paper establishes a rigorous tensor-power framework for Sidorenko-type inequalities, proving powerful regularization and transfer principles in structurally elegant admissible graphon classes. The equality and spectral strengthening results, along with their quantitative forms and applications to doubly nonnegative kernels, markedly extend the conceptual and technical scope of Sidorenko theory. The algebraic approach grounded in tensor operations opens further prospects for deepening the understanding of extremal subgraph densities, regularity, and spectral properties in dense graph limits, with potential for broader influence on algebraic and operator-theoretic combinatorics.