Vertical Recurrence: Activation Methods

• Vertical Recurrence (Activation-Based Methods) is a unified framework that uses iterative, activation-driven processes to propagate structure and information across systems.
• The approach distinguishes between dynamic regimes by analyzing recurrence patterns, enhancing neural sequence generation, and optimizing quantum chemical computations.
• It bridges theory and practice by linking combinatorial and matrix recurrence structures with scalable applications in model pruning and activation function optimization.

Vertical Recurrence (Activation-Based Methods) encompasses a spectrum of techniques in dynamical systems analysis, neural computation, quantum chemistry, network pruning, and theoretical mathematics, unified by the principle of structured recurrence—often operating along a temporal or hierarchical axis—and by the decisive role of activation states or functions in shaping the system’s evolution. Vertical recurrence frequently refers to iterative, hierarchical, or time-like propagation of activity, error, or structure, with "activation-based" signifying that activation events, magnitudes, or functions themselves drive or modulate these processes. This article provides a comprehensive survey of the concept, its theoretical foundations, methodologies, applications, and implications, as derived from a range of seminal works.

1. Recurrence-Based Methods in Dynamical Systems

The paper of recurrence in continuous dynamical systems, especially for distinguishing chaotic from periodic behavior, is a prominent application of activation-based vertical recurrence (1011.5172). The recurrence plot (RP) is foundational: for a state sequence $\{\mathbf{x}_i\}$, the recurrence matrix is

$$R_{i,j}(\varepsilon) = \Theta(\varepsilon - \|\mathbf{x}_i - \mathbf{x}_j\|),$$

where $\Theta$ is the Heaviside function, and $\varepsilon$ a threshold parameter. Diagonal structures in the RP can reveal periodic orbits, whereas more scattered patterns are associated with chaos.

Extending RPs, the activation-based perspective forms a recurrence network (RN) by interpreting $R_{i,j}$ as an adjacency matrix (with self-loops removed: $A_{i,j} = R_{i,j} - \delta_{i,j}$). Network-theoretic metrics, notably global clustering coefficient ($\mathcal{C}$) and average path length ($\mathcal{L}$), capture how activations (recurrences) cluster and propagate. High $\mathcal{C}$ and $\mathcal{L}$ often reflect periodicity, while their reduction signals chaotic spreading of system trajectories.

Crucially, these approaches enable discrimination of dynamical regimes from short time series—particularly valuable for experimental settings where the underlying system equations are not directly accessible.

2. Activation-Based Recurrence in Neural Networks

Sequential activity in neural systems can be produced via vertical recurrence mechanisms modulated by activation-based learning (1603.04687). In partially structured recurrent neural networks, sequence generation emerges through a small fraction of connections being learned (Partial In-Network Training, PINning), with the rest remaining random. The network’s dynamics,

$$\frac{dx_i}{dt} = -x_i + \sum_j J_{ij} \varphi(x_j) + h_i,$$

show that recurrent connections and external inputs jointly mediate the propagation of activation, forming transient "bumps" of activity that represent sequences or memories.

Activation-based modification of synapses—directly targeting internal activity patterns—contrasts with architectures that induce recurrence only via fixed-point attractors or purely feedforward paths. This form of vertical recurrence, shaped by activation history and plasticity, has been implicated in working memory, decision-making, and temporal pattern generation.

3. Recurrence Relations and Vertical Recurrence in Quantum Chemistry

In quantum chemistry, vertical recurrence relations (VRRs) are fundamental to the efficient computation of many-electron integrals over Gaussian basis functions (1704.08051). VRRs provide explicit formulas for iteratively "raising" angular momentum on a given center—analogous to recurrently expanding the activation state in a stacked or deeply recursive network:

$$\left[ \cdots a_i^+ \cdots \right]^m = (Z_i A_i - D_i Y_0) \left[ \cdots a_i \cdots \right]^m + \sum_{j=1}^n a_j \left( \cdots \right).$$

This recursion enables the systematic construction of higher-order integrals from fundamental "activated" states (s-type integrals).

A key connection to activation-based methods is the selective computation: only those integrals (activated paths in the recursive graph) exceeding significance thresholds (evaluated via upper bounds) are propagated. This principle—computing only "activated" branches—minimizes computational waste and underpins modern scalable algorithms in electronic structure.

4. Structured Pruning via Activation-Based Recurrence

Structured network compression now leverages activation-based, vertically recurrent strategies for model pruning (2201.09881). In Iterative Activation-based Pruning (IAP), the pruning criterion is the mean activation value of filters:

$$\bar{A}_{ij} = \frac{1}{b} \sum_{k=1}^{h^i} \sum_{l=1}^{w^i} a_{k,l},$$

where activations $a_{k,l}$ are measured across batches. Filters with persistently low activation are iteratively pruned, with thresholds updated adaptively in AIAP. Unlike traditional weight-based pruning, this activation-centric approach reflects a filter’s operational contribution as determined by repeated (vertical) activation in forward passes.

Empirical results demonstrate superior compression ratios with negligible accuracy loss, especially on edge-oriented architectures. The process highlights vertical recurrence not only in the selection of effective subnetworks but also in maintaining functionality as layers are iteratively compressed.

5. Vertically-Recurrent Matrices and Algebraic Structures

The mathematical abstraction of vertical recurrence finds expression in vertically-recurrent matrices—structured arrays whose entries are defined by a weighted generalization of the "hockey stick" recurrence from combinatorics (2206.02758). Given a sequence $\Lambda = \{\lambda_n\}$, entries $a_{n,k}$ obey

$$a_{n,k} = \sum_{\ell = k-1}^{n-1} \lambda_{n-1-\ell} a_{\ell, k-1}.$$

Vertically-recurrent matrices encompass Pascal matrices as a special case and admit lower-triangular (Toeplitz-block) decompositions, linking them to admissible matrices and circuit analogs such as ladder networks.

The paper of their powers, minimal polynomials, and combinatorial identities underscores the structural depth of vertical recurrence—that is, how repeated, activation-like propagation along the matrix’s vertical axis shapes global properties relevant both to pure mathematics and applied network analysis.

6. Activation Functions, Information-Theoretic Optimization, and Vertical Recurrence

The selection and optimization of neural activation functions has been theoretically grounded in information entropy (2405.12954). The Entropy-based Activation Function Optimization (EAFO) framework posits that the entropy of the activation-induced output distribution,

$$H(y(x)) = -\int p(y(x)) y'(x) \log(p(y(x)) y'(x))\,dx,$$

serves as a guiding functional. Corrections to existing activations (e.g., ReLU) are derived by minimizing this entropy—yielding improved forms such as the Correction Regularized ReLU (CRReLU):

$f(x) = \max(0, x) + \varepsilon x e^{-x^2/2}.$

Vertical recurrence here is realized by iteratively refining activation functions—moving stepwise away from the "worst-case" (max-entropy) baseline toward lower-entropy, higher-performing functions. This dynamic adaptation parallels the recursive modernizations seen in pruning, sequence memory, and algebraic matrix constructions.

7. Broader Implications and Practical Applications

Across disciplines, vertical recurrence anchored in activation-based principles enables robust classification of dynamical regimes from time series (1011.5172), efficient construction of many-body integrals (1704.08051), scalable pruning of deep neural networks (2201.09881), and optimization of network nonlinearity (2405.12954). Critical properties include:

• Scalability: Vertical recurrence supports algorithms that scale efficiently with system complexity by focusing computational resources on activated (significant) states or components.
• Data Efficiency: Activation-based vertical strategies, as in recurrence network analysis, enable reliable inference from limited or noisy observations.
• Structural Compression: Iterative, vertically recurrent pruning exploits activation profiles to retain performance in compact representations, crucial for deployment on resource-constrained devices.
• Generalization: Information-theoretic optimization of activation functions via vertical recurrence mechanisms can yield functions with superior empirical generalization, across both vision and language tasks.
• Unified Mathematical Frameworks: Vertically-recurrent matrices and their decompositions offer unifying structures spanning combinatorics, linear algebra, and signal processing, echoing recurrent propagation of activations in complex networks.

Vertical recurrence (activation-based methods) thus constitutes an integrated framework for understanding, analyzing, and improving both synthetic and natural dynamical systems, with core ideas now permeating theoretical, computational, and applied research at the interface of nonlinear dynamics, neural computation, and network science.