Simplicial Normalization (SimNorm)

• Simplicial normalization (SimNorm) is a framework that discretizes continuous geometric functionals by applying algebraic and topological methods on simplicial complexes.
• It employs integer linear programming and total unimodularity to achieve tractable, multiscale optimization in approximating discrete signals and structures.
• SimNorm underpins practical applications such as multiscale denoising, spectral analysis, and topological stability quantification in higher-order network data.

Simplicial normalization (SimNorm) refers to a collection of discrete, algebraic-topological frameworks for representing, analyzing, and optimizing structures and signals defined over simplicial complexes, with special emphasis on normalization, scaling, and approximation methods that preserve essential geometric, combinatorial, and statistical properties. These frameworks arise in geometric measure theory, algebraic statistics, spectral topology, higher-order network analysis, and related disciplines, and typically leverage normalization procedures on chain modules, boundary operators, associated Laplacians, or matrix/vector representations. Simplicial normalization is central for multiscale modeling, data denoising, model selection, and the quantification of topological stability in both theoretical and applied contexts.

1. Discrete Flat Norms and Simplicial Normalization

The multiscale simplicial flat norm (MSFN) (Ibrahim et al., 2011) is a foundational instance of simplicial normalization, providing a discrete analogue of the classical flat norm on currents. A current $T$ is modeled as an oriented $d$-chain (integer multiplicities) over a finite simplicial complex $K$, and deformable via a $(d+1)$-chain $S$ also supported on $K$. The objective is to find an $S$ minimizing the cost

$$F_S^\lambda(T) = \min_{s} \left\{ \sum_{i=1}^m w_i |x_i| + \lambda \sum_{j=1}^n v_j |s_j| \;\Big|\; x = t - [\partial_{d+1}] s \right\}$$

with $t$ the input chain, $w_i$ and $v_j$ simplex volumes, and $\lambda$ controlling multiscale weighting. The procedure discretizes deformation to be compatible with the combinatorial structure and topology of $K$, serving as a computational surrogate for continuous flat norm calculations.

SimNorm in this context is the restriction to decompositions and approximations entirely within the discrete skeleton of $K$, often with $\lambda=1$.

2. Integer Linear Programming, Total Unimodularity, and Complexity

Central to the practical use of SimNorm is casting decomposition problems as integer linear programs (ILPs), as in MSFN (Ibrahim et al., 2011). The decision problem—whether a chain can be decomposed within a given cost threshold—is NP-complete in general. However, topological properties of $K$ directly determine computational tractability. When the boundary matrix $[\partial_{d+1}]$ is totally unimodular (equivalently, the complex has no relative torsion), the linear programming relaxation yields integral solutions, enabling strongly polynomial-time algorithms.

A comparison table of properties relevant to Simplicial Normalization via the MSFN:

Property	General Simplicial Complex	No Relative Torsion / TU Matrix
Complexity	NP-complete	Polynomial-time
Optimization	Integer Linear Programming	LP relaxation exact
Deformation Type	Restricted by topology	Flexible, tractable

Absence of torsion holds, for example, for triangulations of orientable manifolds and Euclidean embeddings.

3. Simplicial Deformation Theorem and Approximation

A major theoretical result is the simplicial deformation theorem (Ibrahim et al., 2011), which guarantees that any $d$-current can be approximated, with explicit mass expansion bounds, by a simplicial $d$-current supported on $K$. The theorem provides quantitative control:

$$M(P) \leq (4\theta_K)^k M(T) + \Delta (4\theta_K)^{k+1} M(\partial T)$$

where $\theta_K$ encodes regularity parameters and $\Delta$ bounds diameters. As $K$ is refined (smaller simplices, controlled regularity), the approximation quality improves and the flat norm distance vanishes.

This result underlies the use of SimNorm in discretizing geometric functionals and for designing algorithms that approximate continuous objects by combinatorial surrogates.

4. Operations Preserving Normality in Simplicial Models

In the statistical context (Bernstein et al., 2015), matrices arising from the design of hierarchical log-linear models are studied for normality—every lattice point in the cone generated by columns is realized by a nonnegative integer combination. Operations on simplicial complexes such as vertex deletion, edge contraction, gluing along faces, and taking links (often implemented via projection operators $P_a = I - \frac{a a^{\top}}{\|a\|^2}$) are shown to preserve normality of the corresponding configuration matrices. Normality is crucial for the existence of Markov bases and for well-behaved integer programming models.

Classification of complexes on up to six vertices provides a benchmark for which configurations admit SimNorm-compatible models (normal/ compressed matrices).

5. Normalized Laplacians and Simplicial Diffusion

Spectral approaches on simplicial complexes (Schaub et al., 2018, Millán et al., 2021) generalize normalized Laplacians to higher dimensions, enabling random walks and diffusion models in the space of $n$-simplices. In these settings, normalization ensures that differential and combinatorial properties, not scale or degree heterogeneity, govern analytic outcomes.

A normalized Hodge $1$-Laplacian:

$$\mathcal{L}_1 = D_2 B_1^{\top} D_1^{-1} B_1 + B_2 D_3 B_2^{\top} D_2^{-1}$$

enables diffusion dynamics to reflect cycle space topology rather than raw connectivity. Lifting operators allow one to disentangle orientation and magnitude, so that the projected actions correspond to proper stochastic processes on the edge-space.

In synchronization models, normalized Laplacians encode both geometry and topology; spectral dimension $d_s$ determines the regime. Simplicial normalization of coupling and boundary terms is required to probe meaningful phase transitions across scales.

6. Stability, Perturbation, and Quantification in Simplicial Homology

Recent work (Guglielmi et al., 2023) considers SimNorm procedures for quantifying structural stability of simplicial complexes: measuring the minimal weighted perturbation required to alter the homology (Betti number) of the complex. The approach introduces normalized (weighted) boundary operators, generalized Laplacians

$$\bar{L}_k = \bar{B}_k^{\top} \bar{B}_k + \bar{B}_{k+1} \bar{B}_{k+1}^{\top}$$

and recasts the stability question as a spectral matrix nearness problem, solvable via bilevel constrained gradient flows. Minimization of

$$F(\epsilon, E) = \frac{1}{2}[\lambda_{+}(\epsilon, E)]^2 + \frac{\alpha}{2} \max(0, 1 - \frac{\mu_2(\epsilon, E)}{\mu})^2$$

detects perturbations sufficient to create new topological features, facilitating vulnerability analysis in networked data.

The normalization step is critical: invariance under rescaling ensures the topological features are intrinsic to the structure, not artifacts of weighting.

7. Applications and Implications

Simplicial normalization underpins a range of theoretical and practical applications:

• Multiscale denoising and feature extraction: MSFN enables simultaneous control across scales and yields tractable models for shape decomposition, segmentation, and measurement in arbitrary dimensions (Ibrahim et al., 2011).
• Hierarchical statistical models: Normality and compressedness classifications inform selection of tractable, interpretable models for contingency tables and algebraic statistics (Bernstein et al., 2015).
• Spectral network analysis: Normalized Laplacians provide the infrastructure for PageRank generalization, cycle-space centrality, and embedding models in trajectory and co-purchasing data (Schaub et al., 2018).
• Topological stability and robustness: The matrix nearness framework delivers exact estimates of “weak spots” in infrastructure and synthetic networks, informing resilience and vulnerability analyses (Guglielmi et al., 2023).

Simplicial normalization thus provides rigorous, scalable strategies for modeling higher-order relational, geometric, and topological phenomena within discrete data structures. Theoretical advances such as total unimodularity criteria, explicit deformation bounds, and bilevel optimization enable robust, quantifiable analysis across diverse application domains.