SFL-V1: Cross-Domain Metrics, Learning & Protocols

• SFL-V1 is a multifaceted concept that defines sub-Finsler metrics for V1, split federated learning protocols, heavy-ion flow observables, and off-path signaling methods.
• It introduces adaptive methodologies such as filter overlap-induced neural priors, client-specific server models ensuring cut-layer invariance, and UAV-optimized split learning.
• Empirical studies across neuroscience, distributed optimization, high-energy physics, and network management demonstrate its practical efficiency and robust performance.

SFL-V1 denotes a family of concepts in contemporary computational neuroscience, machine learning, high-energy nuclear physics, and network protocols. Despite the notational overlap, it is context-specific, designating: (1) the “Sub-Finsler/L₂-V1” metric architecture for the primary visual cortex (V1); (2) the “Split Federated Learning – Version 1” paradigm for distributed neural optimization; (3) the “Symmetrized Flow v₁” observable in heavy-ion collisions; and (4) a distributed off-path signaling protocol for service function localization in networks. Each instantiation exhibits distinctive theoretical foundations, algorithmic structure, and application domains.

1. Sub-Finsler/L₂-V1 Metric Model of Visual Cortex

SFL-V1, introduced by Montobbio, Citti, and Sarti, provides a metric model of V1’s functional geometry by representing the connectivity between simple cells via overlaps of their receptive profiles (RPs). Let $P$ denote the feature space parameterizing a bank of linear filters $\{\psi_p(x)\}_{p\in P}$, each $\psi_p$ of unit $L^2$-norm, which serve as RPs in $L^2(\mathbb{R}^2)$. The connectivity kernel is

$$K(p, q) = \Re \langle \psi_p, \psi_q \rangle_{L^2},$$

where $\Re$ indicates the real part. This induces the pointwise kernel distance:

$d^2(p, q) = 2(\eta - K(p, q)),$

and, globally, a metric $\tilde d(p, q)$ via minimal-length chains constrained to locally significant kernel overlap. For classical Gabor filters ($P = \mathbb{R}^2 \times S^1$), SFL-V1 recovers the sub-Riemannian geometry generated by vector fields $$X_1 = \cos\theta\,\partial_x + \sin\theta\,\partial_y$$, $X_2 = \partial_\theta$, thus reproducing the Citti–Sarti association field model. Critically, this construction generalizes to arbitrary filter banks, including those learned by unsupervised algorithms, as the metric depends solely on filter pairwise overlaps, not on any group structure on $P$.

Applications include modeling local association fields (collinear and cocircular connections) and simulating long-range horizontal connectivity by iterated kernel dynamics:

$$K_n^{p_0}(p) = \int_P S[K](p, q) K_{n-1}^{p_0}(q) \, d\mu(q),$$

with a normalized $S[K]$. This reproduces the anatomical spread of patchy axonal arborizations observed in V1 and can be directly utilized to define recurrent priors in convolutional neural networks without imposing group structure or introducing extra parameters. The SFL-V1 paradigm, therefore, induces neuro-geometric priors adaptively, fully determined by the chosen filter set (Montobbio et al., 2018).

2. Split Federated Learning: SFL-V1 Algorithm

SFL-V1 is one of two principal split federated learning (SFL) variants for distributed optimization. In SFL, a deep network is partitioned at a cut layer $L_c$, with the client maintaining the first $L_c$ layers and the server the remainder.

The key features of SFL-V1 are:

• The training server maintains separate server-side models $\theta^S_k$ for each client $k$.
• In each round: clients forward activations at $L_c$ to their own server-side model, receive gradient updates, complete the backward pass, and, after all local updates, synchronize both client and server weights across all clients by model averaging (FedAvg) (Dachille et al., 2024).
• Theoretical analysis shows that for any $L_c$, the convergence bound depends only on classical smoothness and variance parameters, but not on the cut position (Proposition 1), resulting in cut-layer invariance:

$$\frac{1}{T}\sum_{t=0}^{T-1}\eta^t\,\mathbb{E} [ \| \nabla f(\theta(t)) \|^2 ] \leq \text{(cut-layer-independent bound)}.$$

• Empirical results indicate SFL-V1 exhibits negligible accuracy (<3% variation) across cut points on multiple datasets and models (ResNet-18/50, CIFAR-10/100, TinyImageNet), both in IID and non-IID data regimes.

By contrast, SFL-V2 employs a shared server model and exhibits strong cut-layer sensitivity. In SFL-V1, decoupled server models guarantee that updates are fully client-specific until aggregation, making its dynamics and convergence equivalent to FedAvg on the full network (Dachille et al., 2024).

3. Split Federated Learning with UAV-Enabled ISCC

In wireless edge applications, SFL-V1 (SFLSCC) is deployed across a set of UAVs and an edge server, with joint optimization of split point, aggregation frequency, UAV positioning, and data volume. The system model incorporates stochastic sensing and communication links, models detailed computation and communication energy, and constrains optimization to guarantee uniform sensing quality and target accuracy.

Convergence is rigorously derived under smoothness, variance, and heterogeneity assumptions, leading to explicit bounds on rounds to target precision (Hou et al., 2 Apr 2025). A four-block coordinate descent method is used for energy minimization, optimizing (i) aggregation frequency, (ii) minibatch size, (iii) split layer, and (iv) UAV positioning, each with closed-form or low-complexity solutions.

Notable empirical results:

• SFLSCC (SFL-V1) achieves up to 40% lower energy and 40% faster convergence than baseline methods.
• The energy-accuracy objective is robust to environment conditions (dense urban/high-rise), outperforming standard federated and split learning baselines.
• Shallower splits and frequent client aggregation minimize UAV-side energy and raise convergence rates (Hou et al., 2 Apr 2025).

4. Communication-Pipelined SFL-V1 in Foundation Model Fine-Tuning

For foundation model (FM) fine-tuning in UAV networks, SFL-V1 incorporates additional communication and scheduling innovations. The model is split at a tunable layer $u$; UAVs hold and update client-side LoRA parameters, while the BS fine-tunes the server-side submodel.

Key characteristics:

• Sequential Gradient Transmission (GT): Downlink resources are allocated to one client at a time (as opposed to parallel), minimizing per-round latency in networks where communication dominates computation.
• CPSFL Enhancements: Incorporates (a) client-lag-based scheduling (priority to more lagging clients per iteration), and (b) intra-round asynchrony (server transmits gradients to next client immediately upon completion, without idle waiting) (Zhou et al., 19 Nov 2025).
• Optimization Framework: Balances weighted objectives of round latency and worst-case energy by selecting split point, bandwidth allocation, and server compute rate per round, all driven by historical UAV trajectory data.
• Attention-based DRL Policy: The base station implements a PPO agent with attention over variable-length UAV trajectories for adaptive control.

Simulations show that DRL-based CPSFL achieves ~30% latency reduction compared to ablations and approaches the best split-fixed solution, particularly benefiting under client/channel heterogeneity (Zhou et al., 19 Nov 2025).

5. SFL-V1 in Heavy-Ion Physics: Symmetrized Flow Component

In nuclear physics, SFL-V1 denotes the “symmetrized flow” observable $v_1^S(y)$ in azimuthal correlations of particle emissions from heavy-ion collisions:

$v_1^S(y) = \frac{v_1(y) + v_1(-y)}{2},$

where $v_1(y)$ is the first Fourier coefficient of the particle azimuthal distribution relative to the reaction plane. This component isolates the mirror-symmetric (global) part of the directed flow, filtering out rapidity-odd contributions from initial state fluctuations.

Hydrodynamic modeling with MIT bag equation of state and PIC numerics predicts that, at LHC energies, the global $v_1$ reverses sign relative to RHIC, reflecting macroscopic rotation and pressure gradients in the quark-gluon plasma (Csernai et al., 2011). The symmetrized $v_1^S$ is less susceptible to event-by-event rapidity fluctuations and enables robust extraction of collective flow features, serving as a sensitive probe of initial angular momentum and QGP properties.

6. SFL-V1 as Service Function Localization Protocol

In computer networking, SFL-V1 is synonymous with the Off-path Signaling Protocol (OSP) for distributed service function localization (Femminella et al., 2016). OSP structures the network into SA (Signaling Application) and ST (Signaling Transport) layers and combines:

• Background gossip for peer discovery, maintaining a Peer Table (PeT) with hop/RTT information.
• On-path packet interception with controlled off-path signaling flood within a configurable radius $r$ to discover and aggregate information from network nodes hosting a given service function (SF).
• TLV-based message flow for registration, responses, queries, errors, and data collection, engineered with FSMs for state management.
• Sub-second discovery times and bandwidth overheads several factors below prior GIST-based proposals.

OSP’s path-coupled yet off-path signaling, integrating discovery and status monitoring, has demonstrated high scalability and efficacy in large topologies under realistic experimental conditions (Femminella et al., 2016). Limitations include coarse hop-based metrics, potential for excessive flooding in dense deployments, and lack of security/authentication—future work is suggested in these directions.

7. Summary and Distinctions

Context	SFL-V1 Meaning	Core Mechanism
Visual Cortex Geometry	Sub-Finsler/L₂-V1 metric	Filter overlap, induced metric & kernel
Federated Machine Learning	Split Fed Learning V1	Per-client server models, cut-invariant
UAV/Edge Machine Learning	SFLSCC (SFL-V1)	Joint split/aggregation/placement
Fine-Tuning (UAV/FMs)	SFL-V1 w/ CPSFL	Pipelined, prioritized comm. & DRL
Heavy-Ion Physics	Symmetrized $v_1$	Flow observable, even-rapidity part
Network Protocols	Off-path Signaling (OSP)	Gossip+Flood for SF discovery

Despite their divergent fields, all instances share the principle of partitioned, distributed, or structurally-induced interaction: whether found in functional brain geometry, distributed optimization, collective physical flows, or networked system management. Each SFL-V1 implementation is accompanied by precise theoretical formulations, algorithmic mechanisms, and empirical evidence demonstrating efficiency or invariance appropriate to its domain (Montobbio et al., 2018, Dachille et al., 2024, Hou et al., 2 Apr 2025, Zhou et al., 19 Nov 2025, Csernai et al., 2011, Femminella et al., 2016).