Additive-Multiplicative KANs

• Additive-multiplicative KANs are neural architectures that decompose multivariate functions using additive and multiplicative aggregation of univariate nonlinear transforms, based on the Kolmogorov–Arnold theorem.
• They achieve minimax optimal convergence in nonparametric regression with B-spline representations, outperforming traditional MLPs in structured settings.
• Their framework connects free probability, spectral analysis, combinatorial structures, and control theory, highlighting versatility in applications like network science and signal processing.

Additive-Multiplicative Kolmogorov–Arnold Networks (KANs) designate a class of neural architectures or mathematical models fundamentally characterized by the composition of univariate nonlinear functions via either additive or multiplicative aggregation, with deep connections to classical representation theorems, free probability, spectral analysis, ergodic theory, and combinatorial structures. They permeate theoretical and applied domains from random matrix theory and algebraic combinatorics to modern regression, non-Euclidean feature extraction, control theory, network science, and symbolic discovery.

1. Algebraic and Analytical Foundations

At the heart of additive-multiplicative KANs is the Kolmogorov–Arnold representation theorem, which guarantees that any continuous multivariate mapping $f: [0,1]^d \rightarrow \mathbb{R}$ admits a decomposition:

$$f(x_1,\ldots,x_d) = \sum_{q=1}^{Q} g_q\left(\sum_{j=1}^{d} \psi_{qj}(x_j)\right)$$

In hybrid extensions, the inner operations are allowed to be multiplicative:

$T_q(x) = \prod_{j=1}^{d} \psi_{qj}(x_j),$

$f(x) = \sum_{q=1}^{Q} g_q(T_q(x))$

This structure reflects an interplay—Editor's term: additive-multiplicative aggregation—where data undergoes either additive or multiplicative nonlinearity at each compositional stage. The two modes are deeply linked and, in certain cases, mathematically interconvertible (Liu et al., 24 Sep 2025, Ofir et al., 4 Jan 2024, Aryapoor, 2011).

2. Statistical Optimality and Nonparametric Regression

Recent theoretical analysis has established that both additive and hybrid additive-multiplicative KANs, leveraging B-spline representations for the univariate components $g_q$ and $\psi_{qj}$, attain the minimax-optimal convergence rate for nonparametric regression in Sobolev spaces of smoothness $r$:

$$\mathbb{E}[\|\widehat{f}_n - f\|^2_{L_2([0,1]^d)}] = O(n^{-2r/(2r+1)})$$

This rate persists for hybrid architectures, modulo a constant depending on input dimension and boundedness parameters, and is independent of $d$ due to the reduction of multivariate approximation to univariate spline estimation. The number of B-spline knots for each $\psi_{qj}$ should scale as $k \asymp n^{1/(2r+1)}$ to optimize bias-variance balance (Liu et al., 24 Sep 2025). Simulation studies consistently validate these rates and show superiority to standard MLP baselines in structured settings.

3. Algebraic Isomorphism and Functional Translation

The algebra of additive and multiplicative arithmetical functions is clarified by an explicit isomorphism: the logarithmic map translates the Dirichlet convolution product of multiplicative functions into addition, and vice versa via exponentiation (Aryapoor, 2011):

$Y(a) = u * \log(a)$

$Y^{-1}(b) = \exp(p * b)$

Such operator-level translation carries over to generalized KANs, enabling categorical duality between additive and multiplicative kernels and facilitating switching between summation-based and product-based modeling perspectives.

4. Spectral and Free-Probabilistic Frameworks

In the paper of spiked random matrices and their deformations, the "additive-multiplicative KANs" philosophy refers to analytic frameworks whereby subordination functions precisely determine both the spectral outlier locations and the asymptotic projection ("overlap") of spiked eigenvectors (Capitaine, 2011):

Additive case: $$p_0^{(j)} = H(\theta_j),\quad |P_{\ker(\theta_j I - A_N)} \xi^j|^2 \to H'(\theta_j)$$ Multiplicative case: $$p_0^{(j)} = Z(\theta_j),\quad |P_{\ker(\theta_j I - A_N)} \xi^j|^2 \to Z'(\theta_j)$$

Here, $H$ and $Z$ are inverses of analytic subordination maps arising from free convolution, governing both additive and multiplicative interactions.

5. Combinatorics, Signal Processing, and Algebraic Structures

In combinatorial and number-theoretic contexts, additive-multiplicative KANs manifest in results showing structural rigidity (e.g., uniqueness properties of multiplicative functions under additive cube constraints (Park, 2023)) and in combinatorial centrality phenomena (central sets in large integral domains simultaneously exhibit additive and multiplicative structure (Debnath et al., 11 May 2024)).

In signal processing, the interplay appears in the design and conversion of pattern-matching metrics—the LIP-multiplicative and LIP-additive Asplund's metrics are explicitly linked by an isomorphism $\varphi$:

$d_{As}^A = M(1 - e^{-d_s/M})$

This enables robust handling of both illumination (additive) and physical (multiplicative) invariances (Noyel, 2019).

6. Neural Network Architectures and Feature Representations

KANs advance neural network architecture by parameterizing activation functions as flexible, learnable univariate transforms on edges (using splines and weighted nonlinear components), enabling direct additive-multiplicative compositionality within deep models (Pourkamali-Anaraki, 16 Sep 2024, Zhang et al., 19 Jun 2024, Chen et al., 20 Oct 2024):

• In KANs, the forward pass at layer $\ell$ is:

$$x_j^{(\ell)} = \sum_i [ w_{b}^{j,i} \cdot \mathrm{SiLU}(x_i^{(\ell-1)}) + w_{s}^{j,i} \cdot \mathrm{spline}(x_i^{(\ell-1)}) ]$$

This edge-level additive-multiplicative nonlinearity allows for richer functional approximation and, when properly regularized, can outperform traditional MLPs—especially in settings where physical domain knowledge and interpretability are critical.

Empirical studies confirm the efficacy of KANs in feature extraction for graph neural networks, symbolic regression, building physics, and medical engineering, especially when the task structure is amenable to additive-multiplicative decompositions.

7. Connections to Matrix Theory and Control

The matrix theory of compounds is unified via Kronecker product and sum representations, relating multiplicative and additive compounds as:

$$A^{(k)} = L_{n,k} \cdot (A^{\otimes k}) \cdot M_{m,k}$$

$$A^{[k]} = L_{n,k} \cdot (A^{\oplus k}) \cdot M_{n,k}$$

These concise formulas allow explicit manipulation of high-order volume forms in control theory and can be generalized to networked systems—the so-called "KANs" in control analysis, which require both multiplicative and additive aggregation to formulate system stability and contractivity criteria (Ofir et al., 4 Jan 2024, Choudhury et al., 2023).

8. Synthesis and Prospects

Additive-multiplicative KANs unify analytic, algebraic, and computational models through compositionality, isomorphism, and structural decomposition. Their minimax optimality in regression tasks, explicit duality in function spaces, and their capacity to encode both physical invariances and complex interdependencies position them as a robust framework for interpretable, structured learning and analysis across diverse scientific domains.

Continued development aims to optimize parameterization (e.g., knot selection in splines for statistical learning), enhance computational efficiency, merge symbolic and data-driven discovery, and generalize combinatorial and algebraic classification for higher-order systems.

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Relevant Papers and Theories:

Area	Key Paper(s)	Main Conceptual Tool
Regression optimality, KAN structure	(Liu et al., 24 Sep 2025)	Minimax rate for additive/hybrid KANs, B-spline resolutions
Algebraic isomorphism (multiplicative/additive)	(Aryapoor, 2011)	Explicit log/exponential maps between functional classes
Free probability, spectral analysis	(Capitaine, 2011)	Subordination functions, eigenvector projections, additive-multiplicative framework
Compound matrices in networks/control	(Ofir et al., 4 Jan 2024, Choudhury et al., 2023)	Kronecker representations, master invariants, decompositions in additive-multiplicative settings
Graph feature extraction, neural architecture	(Zhang et al., 19 Jun 2024, Pourkamali-Anaraki, 16 Sep 2024)	Edge-centric learnable activations, interplay of parameter count and data availability
Symbolic regression, building physics	(Chen et al., 20 Oct 2024)	Additive-multiplicative formula discovery, decision frameworks, modular updates
Combinatorial / central set rigidity	(Park, 2023, Debnath et al., 11 May 2024)	Uniqueness phenomena, centrality and partition results in large domains

This taxonomy underpins the interplay and structural power of additive-multiplicative KANs throughout contemporary mathematics, statistics, and machine learning.