Pseudo-Convolution Structures

• Pseudo-Convolution Structures are flexible frameworks that generalize classical convolutions by relaxing algebraic constraints and encoding arbitrary topological, combinatorial, or analytic structures.
• They unify diverse operators across distribution theory, tensor-algebra, and semiring settings, providing a common ground for enhancing algorithmic efficiency and deep-learning methods.
• Practical applications demonstrate their power in deep neural networks, graph convolutions, and dynamic programming, achieving improved parameter sharing, computational speed, and analytic rigor.

A pseudo-convolution structure is a general mathematical or algorithmic framework that generalizes classical convolution operations by relaxing algebraic, structural, or analytic constraints. These constructions encompass a range of operators, from those in analysis and distribution theory (extending convolution and multiplication) to tensor-based models in machine learning and non-standard algebraic convolutions in algorithmics. Pseudo-convolutional frameworks acquire their unifying power by encoding arbitrary structure (topological, algebraic, or combinatorial) within the maps used and by permitting flexible parameterizations of interactions, thus subsuming standard convolution, graph and sequence convolutions, adaptive attention, and non-ring-based semiring convolutions.

1. Formal Definitions Across Domains

Distribution Theory and Pseudo-Differential Operators

Pseudo-convolution operators in analysis are defined as finite sums of the form

$Tg(x) = \sum_{j=1}^N f_j(x) \cdot (u_j * g)(x),$

where each $u_j$ is a compactly supported distribution and $f_j$ is a function. Alternatively, one may represent $T$ as a pseudo-differential operator with symbol $p(x,\xi)$:

$$Tg(x) = p(x,D)g(x) = (2\pi)^{-n} \int_{\mathbb{R}^n} e^{ix\cdot\xi}p(x,\xi)\widehat{g}(\xi)\,d\xi,$$

where $p(x,\xi) = \sum_j f_j(x)\widehat{u}_j(\xi)$ (Bonner, 2013).

Tensor-Algebraic Pseudo-Convolution in ML

A pseudo-convolution operator acting on structured embeddings generalizes matrix multiplication and classical convolution to arbitrary relations:

$$Y_{n,q} = \sum_{k=1}^K \sum_{m=1}^M \sum_{p=1}^P A_{k,m,n}X_{m,p}\Theta_{k,p,q},$$

or, in matrix notation,

$Y = \sum_{k=1}^K A_k^\top X \Theta_k,$

where the $A_k$ ("structure tensors") encode relations (shifts, adjacency) and the parameter tensors $\Theta_k$ specify inter-feature mixing. This template recovers standard CNNs, various graph convolutions, sequence models, and attention mechanisms as special cases (Andreoli, 2019).

(Min,+)-Pseudo-Convolution and Semiring Settings

A pseudo-convolution operation on sequences in the (min,+) or tropical semiring is formulated as:

$c_k = \min_{i+j=k} (a_i + b_j),$

eschewing classical ring structure in favor of semiring operations, which preclude invertibility and yield distinctive algorithmic properties (Gribanov et al., 2022).

2. Algebraic and Functional-Analytic Properties

Invertibility and Coercivity in Distribution Spaces

Pseudo-convolution operators generalize convolution by non-invertible distributions and analytic multipliers, acting on ultradifferentiable function spaces such as Beurling classes $E_w(X)$ and Denjoy–Carleman classes $C_L(X)$. A distribution $u \in E_w'(\mathbb{R}^n)$ is $w$-invertible if the convolution $T_u f = u * f$ surjects $E_w(\mathbb{R}^n)$. Coercivity—preservation of non-invertibility—is characterized by the property that for a linear map $T$, non-invertibility of $u$ implies non-invertibility of $Tu$. Elliptic real analytic pseudo-differential operators are coercive; relaxing analyticity (as in Gevrey or Denjoy–Carleman settings) leads to phenomena where coercivity fails, measurable by loss of non-invertibility in function classes with weaker regularity (Bonner, 2013).

Parameter Efficiency, Weight-Sharing, and Compositionality

In tensor-based pseudo-convolution frameworks, the parameter count scales as $K \cdot P \cdot Q$, with $K$ the number of basis relations, enabling parameter sharing and weight-tied operations across positions related structurally (e.g., neighbor nodes in graphs, translation on grids). The composition of pseudo-convolutions yields new pseudo-convolutions over product bases and parameter sets, attesting to the closure properties of these algebras (Andreoli, 2019).

3. Examples Across Application Domains

Classical Convolutions and Analysis

Standard convolution with a compactly supported distribution, multiplication operators, and elliptic pseudo-differential operators are immediate examples. In the analytic setting, these operators, when elliptic, both preserve and coerce invertibility between distribution spaces (Bonner, 2013).

Structured Data in Machine Learning

• Grid/CNN: For images and signals, the pseudo-convolution reduces to classic spatial convolution by identifying $A_k$ with shift (translation) matrices.
• Graph Convolution: By setting $\{A_k\}$ as normalized adjacency matrices or higher-order polynomials thereof, pseudo-convolution recovers message passing and spectral GCNs. Attention arises as a pseudo-convolution where $A$ is parameterized by data-dependent, learned affinity scores (Andreoli, 2019).
• Continuous Kernel Graph Convolution: The CKGConv model parameterizes kernels as continuous functions $\psi(\phi(u,v))$, where $\phi(u,v)$ encodes graph positional information (e.g., powers of the random-walk matrix). This framework unifies 1-hop MPNNs, polynomial spectral filters, diffusion processes, and permutation-invariant set models, and attains GD-WL power (Ma et al., 21 Apr 2024).

Algorithmic Pseudo-Convolutions

The (min,+)-convolution generalizes the dynamic programming step in separable problems. When the kernel sequence admits additional structure (linear, monotone, convex, concave, piecewise linear, polynomial), subquadratic algorithms are attainable, e.g., O(n) for linear/convex cases via double-ended queues or sliding window minima, O(n^{4/3} log^2 n) for concave cases via block decompositions and augmented segment trees, etc. These routines are then instrumental in efficient solvers for the separable nonlinear knapsack, shortest/closest vector in lattices, and related optimization problems (Gribanov et al., 2022).

4. Unified Frameworks and Theoretical Implications

A core insight of pseudo-convolutional formalism is the unification of operations across domains by encoding structural constraints in the basis tensors or kernels, while parameterizing adaptive mixing via independently learned or preset weights. This abstraction (i) generalizes equivariance and locality to arbitrary structure, (ii) allows analytic control over invertibility and surjectivity in functional spaces, and (iii) enables new algorithmic speedups by exploiting non-ring or non-invertible algebraic contexts.

In the analysis setting, the algebra of pseudo-convolution operators is closed under composition and admits parametrices (analytic inverses) when the symbols are elliptic and analytic. Failure of analyticity leads to measurable "loss" of invertibility dependent on the regularity of the symbol and the function space (e.g., Denjoy–Carleman vs. Beurling class) (Bonner, 2013).

In deep learning contexts, pseudo-convolution templates capture and elucidate the parameter sharing, adaptivity, and compositionality properties of convolutional, graph, and attention modules, providing a transparent mechanism to interpolate between fixed-structure (CNN, GCN, ChebNet) and learned-structure (attention, transformer) interactions (Andreoli, 2019).

Theoretical work on continuous kernel graph convolution (CKGConv) demonstrates that, by parameterizing the kernel over pseudo-coordinates (e.g., relative random walk positional encodings), one recovers and strictly generalizes classical graph convolutions and set operations, and reaches the expressivity limits defined by generalized distance Weisfeiler–Lehman (GD–WL) tests, matching Transformers in distinguishing power (Ma et al., 21 Apr 2024).

Algorithmic frameworks benefit by recasting dynamic programming or combinatorial problems in pseudo-convolutional forms, where structural properties of the kernel sequence (e.g., convexity, piecewise linearity, or polynomiality) directly inform the design and complexity of underlying solvers (Gribanov et al., 2022).

5. Practical Implementations and Empirical Outcomes

Deep Learning

Empirical results indicate that CKGConv-based networks consistently achieve top-tier performance across graph property prediction and node/graph classification benchmarks, matching or outperforming both classical GNNs and transformer-style architectures in various tasks. Typical choices include batches of CKGConv layers with norm and feedforward blocks, depthwise-separable operations for parameter savings, and kernel MLPs with normalization and residual connections (Ma et al., 21 Apr 2024).

Algorithmics

By leveraging pseudo-convolutional algorithms, separable nonlinear knapsack and group-based shortest vector problems can be solved in near-linear or sub-quadratic time, with complexity bounds dictated by the structural properties of the kernel $f(i)$. Explicit algorithm pseudocode and block-based segment tree constructions are provided for different functional forms, as well as illustrative worked examples contrasting O(nm) naive convolutions to optimal O(n) procedures in structured regimes (Gribanov et al., 2022).

Applications in Analysis and PDE

Pseudo-convolution structures enable explicit criteria for PDE solvability modulo convolution operators, invariant properties under small analytic perturbations, and novel insights in micro-local and distributional algebra analysis, including propagation of non-invertibility and identification of zero-divisors in ultradifferentiable or Beurling-type distribution algebras (Bonner, 2013).

6. Extensions, Limitations, and Generalizations

The range of pseudo-convolutional constructions is critically determined by the algebraic, analytic, or combinatorial structure imposed:

• Ellipticity and analyticity are necessary for coercivity and invertibility preservation in pseudo-differential settings.
• Structural regularity (convexity, monotonicity) admits algorithmic acceleration.
• Learnability of structure (as in attention) transforms the static prior into an adaptive operator.
• Loss of coercivity or invertibility occurs predictably as analytic or structural assumptions are relaxed, measurable via quantification in Denjoy–Carleman or Beurling scale regularities.

A plausible implication is that ongoing generalization of pseudo-convolution frameworks will further unify operator-theoretic, algorithmic, and deep-learning-based approaches to high-dimensional, structured data processing and functional analysis, especially as more complex or learned structures replace fixed discrete symmetries.

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References:

• (Bonner, 2013) Operators that coerce the surjectivity of convolution
• (Andreoli, 2019) Convolution, attention and structure embedding
• (Ma et al., 21 Apr 2024) CKGConv: General Graph Convolution with Continuous Kernels
• (Gribanov et al., 2022) Structured $(\min,+)$-Convolution And Its Applications For The Shortest Vector, Closest Vector, and Separable Nonlinear Knapsack Problems