SOAP Algorithm: Overview & Applications

Updated 2 October 2025

SOAP Algorithm is a multifaceted framework that encompasses XML messaging, scheduling policies, deep learning optimizers, clinical note structuring, astrophysical data analysis, and secure authentication.
It integrates techniques such as XML signature binding to prevent rewriting attacks, eigenbasis preconditioning in optimizers, and geometric variational methods for gait optimization.
Its practical significance is highlighted by enhanced system security, efficient response time calculations, reduced training iterations in language models, and improved clinical and astrophysical analyses.

SOAP (Simple Object Access Protocol) spans a diverse set of meanings in computing, mathematics, optimization, web services, natural language processing, astrophysics, and other fields. The term “SOAP algorithm” does not refer to a single specific algorithm, but rather to a cluster of formalisms, analytical frameworks, middleware implementations, computational protocols, and optimization methods that share the SOAP acronym—sometimes as historical accident, sometimes as wordplay, and sometimes as deliberate design. The following sections systematically cover the most technically significant SOAP algorithms from the recent academic literature, contextualizing each within their specific research area and formal mechanism.

1. SOAP in Web Services and Security

SOAP as XML-Based Web Services Messaging

SOAP is fundamentally an XML-based protocol for structured information exchange, especially in Service Oriented Architecture (SOA) environments (Karimi, 2011). It encapsulates both data payload and security tokens using the envelope–header–body structure, and supports rich security extensions, notably WS-Security and WS-Policy. WS-Security allows for message integrity, confidentiality, authentication, and secure session management through XML digital signatures, XML encryption, and token-based mechanisms (such as SAML assertions):

Message integrity is achieved via signing: $S = \operatorname{Sign}(\operatorname{Hash}(M))$ .
Confidentiality uses XML encryption on designated payload sections.
Authentication and trust leverage identity providers and token formats (SAML, WS-Trust).
Secure sessions use protocol extensions (WS-SecureConversation) to establish and reuse context.

The stack is tightly integrated with WSDL (service description) and UDDI (discovery and publication), and is often enforced via compliance with interoperability profiles (WS-I Basic Profile) and domain-specific rulesets (Schaaff et al., 2011).

Security Risks: XML Rewriting Attacks and Countermeasures

A critical weakness arises from the semantics of XML Digital Signatures: the signature binds only the referenced digest of an XML element, not its location in the tree, enabling “XML rewriting attacks” (0812.4181). In such attacks, an adversary can relocate the signed element within the SOAP structure, or wrap it with illegitimate containers, without invalidating the signature. This is formally expressed as:

Let $E$ be a signed element; the signature checks $D = \operatorname{Digest}(\operatorname{canonicalize}(E))$ , with no verification of the enclosing structure.
Attackers exploit the fact that as long as $\operatorname{canonicalize}(E)$ is unchanged, $D$ remains valid regardless of $E$ 's position.

Proposed solutions and their limitations are summarized below:

Solution	Mechanism	Limitation
SOAP Account	Records structural info, signs structure	Sensitive to header manipulations and intermediaries, not robust to replay attacks
WS-Policy	Documents constraints on structure	Syntax-centric; cannot specify semantic/relational invariants for all attack variants
WSE Policy Advisor	Rule-based policy checker	No formal guarantees; may miss subtle or reordered attacks
Formal Methods	Model protocol in (e.g.) $\pi$ -calculus	Typically ignore message manipulation unless key is known; weak on insider scenarios

Authors recommend strengthening the signature by binding the depth and parent identity of signed elements, and enforcing unique wsu:Id attributes, so that the signature validates the full context: $E \rightarrow (\text{Parent}, \text{content})$ .

Defenses Against XML Signature Wrapping

Further, SESoap (Kouchaksaraei et al., 2013) proposes cryptographically signing the entire SOAP envelope except the <Signature> element. This approach ensures that any change in the message—movement, addition, deletion, or wrapping of any element—breaks signature validation. This counters all known classes of XML signature wrapping attack (Simple Ancestry Context, Optional Element Context, Sibling Value Context, Sibling Order Context). Performance experiments show SESoap achieves approximately three times the signing throughput of XPath-based verification.

2. SOAP in Scheduling: Schedule Ordered by Age-Based Priority

The SOAP (“Schedule Ordered by Age-based Priority”) framework defines and solves an exceptionally broad class of single-server (M/G/1) queueing policies, unifying nearly all known scheduling rules under the following abstraction (Scully et al., 2017):

Each job has a static descriptor $d$ and a dynamic age $a$ (service received).
A rank function $r: \mathcal{D} \times \mathbb{R}_+ \to \mathcal{R}$ maps the job's information into a totally ordered set $\mathcal{R}$ .
At any instant, the server works on the job with minimal rank.

This encompasses FCFS ( $r(a) = -a$ ), FB ( $r(a) = a$ ), SRPT ( $r(x, a) = x-a$ ), class/priority-based policies, checkpoint/recycling policies, and even nonmonotonic constructs such as the Gittins index. The framework’s technical core rests on:

Pessimism Principle: Analyze each "tagged job" by assuming its worst future rank $r_{\text{worst}}[d,x](a) = \sup_{a \leq b < x} r(d, b)$ , treating all delay as occurring before reaching this worst priority.
Vacation Transformation: Model interference from older jobs as a virtual busy period with “vacations” (inactive intervals when the preemption rule discards jobs exceeding the tagged job's rank).

Analytically, for arrival rate $\lambda$ , the mean response time is derived via the busy period of an M/G/1 queue with respect to the new $r$ -work and old $r$ -work (see paper’s Theorem 5 for full formula), and the Laplace–Stieltjes transform $T_{d,x}(s)$ factorizes into waiting and residence time components. This approach allows, for the first time, mean response time calculations for policies with arbitrarily complex (even nonmonotonic) age-dependent priorities.

3. SOAP as a Modern Optimization Algorithm

Gradient Preconditioning via Shampoo, Adafactor, and Adam

Recent work introduces SOAP as an algorithm in deep learning optimization that builds on the Shampoo optimizer, which uses second-order preconditioning via Kronecker-factored approximations (Vyas et al., 17 Sep 2024, Lu et al., 26 Sep 2025). SOAP (“ShampoO with Adam in the Preconditioner’s eigenbasis”) is motivated by the following formal connection:

Shampoo exponent- $\frac{1}{2}$ with exact statistics is formally equivalent to running a diagonal-preconditioned Adam (or Adafactor) update in the eigenbasis defined by the left and right Kronecker factors $L$ and $R$ (i.e., the eigendecomposed per-layer covariance matrices).
For a weight matrix $W \in \mathbb{R}^{m \times n}$ and gradient $G$ , construct $L = GG^{\top}$ , $R = G^{\top}G$ , with eigendecompositions $L = Q_L \Lambda_L Q_L^{\top}$ , $R = Q_R \Lambda_R Q_R^{\top}$ .
Rotate the gradient: $G' = Q_L^{\top} G Q_R$ , perform an Adam update on $G'$ , then rotate back: $N = Q_L (\operatorname{Adam}(G')) Q_R^{\top}$ .

The whitening matrix $\Sigma = \mathbb{E}[gg^{\top}]$ is (ideally) approximated as $(\mathbb{E}[G^{\top}G] \otimes \mathbb{E}[GG^{\top}])/ \operatorname{Tr}(\mathbb{E}[GG^{\top}])$ , and thus both Shampoo and SOAP implement a block-wise second-order preconditioner.

Empirical results in language modeling (360M/660M models) show SOAP reduces training iterations and wall clock time by over 40% and 35% respectively compared to AdamW, with approximately 20% further improvement over Shampoo (Vyas et al., 17 Sep 2024). Notably, empirical studies indicate that while SOAP and Shampoo exhibit faster convergence, their final achieved loss is comparable to AdamW and to each other (Lu et al., 26 Sep 2025).

Performance is robust to lower preconditioning frequency $f$ (eigen-decomposition interval), with SOAP degrading more gracefully than Shampoo.

4. SOAP in Multimodal Clinical and Biomedical NLP

In medical documentation, SOAP references the standardized structuring of clinical notes into Subjective, Objective, Assessment, and Plan sections. Recent machine learning research has targeted automated classification and generation of these sections:

Automatic SOAP Classification (Kwon et al., 2022): Employs weak supervision (rule-based header extraction and propagation) to generate large labeled datasets for training neural classifiers (BioBERT/BioSentVec + Bi-LSTM-CRF), and implements a transfer learning framework to enable inter-hospital adaptation. Transfer boosts F1 from $<14$ (MIMIC, when trained on unrelated notes) to $62$ or $90$ (target-like notes), demonstrating both performance and the challenge of generalizing across variable EHR formats.
Clinical Note Generation (Skin-SOAP) (Kamal et al., 7 Aug 2025): Utilizes weak supervision, retrieval-augmented generation, and parameter-efficient fine-tuning (e.g., QLoRA on Vision‑LLaMA), to synthesize dermatology SOAP notes from clinical images and minimal feature text. Novel evaluation metrics (MedConceptEval and Clinical Coherence Score) measure semantic alignment with clinical descriptors and input features, reflecting domain-specific reliability and coherence. Skin-SOAP matches or exceeds large LLMs (GPT‑4o, Claude) in clinical relevance metrics.

5. SOAP in Cosmological Simulation Analysis

SOAP also functions as the acronym for robust astrophysical data analysis packages:

Spherical Overdensity and Aperture Processor (McGibbon et al., 30 Jul 2025): A Python package tailored for large cosmological simulations, calculating over 250 galaxy and halo properties from subhalo catalogs (e.g., from HBT-HERONS, SubFind, VELOCIraptor). It supports varied halo definitions (spherical overdensity, fixed/aperture regions), is parallelized via mpi4py, and integrates with swiftsimio for seamless unit handling and HDF5 output. The primary mathematical core is determination of the SO radius $r$ by solving

$\rho(<r) = \frac{M(<r)}{\frac{4}{3}\pi r^3} = \Delta \rho_\text{crit}$

with $M(<r)$ as enclosed mass, $\Delta$ as the overdensity factor, and $\rho_\text{crit}$ as the cosmic critical density.

SOAP (“Social Authentication Protocol”) is a cryptographically formalized protocol for messaging application authentication, binding the user's messaging key to one or more external digital identities (managed by identity providers like Microsoft or GitLab) (Linker et al., 5 Feb 2024). The binding uses OpenID Connect with the session's safety number ( $k$ ) cryptographically blinded ( $h_s(k, s)$ ) and included in the OIDC nonce. The protocol guarantees "sender correspondence"—the message stream cryptographically and organizationally maps to unique identities—thus raising the attack bar: successful compromise requires breaching both the messaging application and all IdP-managed identities.

A formal security proof is given using the Tamarin prover, confirming that successful authentication is only possible if both pseudonyms (messaging key and digital ID) belong to the same agent and remain uncompromised. The protocol is implemented in a modified Signal client and as a web-based React app, with direct usability and practical deployability benefits.

7. SOAP-Bubble Optimization for Locomotion

Soap-bubble Optimization of Gaits (SOAP) is a geometric variational algorithm for optimizing cyclic motions ("gaits") in kinematic locomotion systems such as Purcell’s swimmer (Ramasamy et al., 2016). The optimizer represents candidate gaits as closed curves in shape space and evolves them according to a dynamical rule:

$\dot{\lambda} = \nabla_{\lambda}g - \frac{g}{\ell} \nabla_{\lambda}\ell + \nabla_{\lambda}\sigma$

where:

$g \approx \iint_{a} -D\mathcal{A}$ is the net displacement ("pressure," from the Lie bracket curvature).
$\ell$ is the metric-weighted pathlength ("surface tension," penalizing cost).
$\sigma$ is a reparameterization term for even waypoint distribution.

This yields balanced gaits that maximize displacement for given cost, directly analogous to a soap bubble where internal pressure is balanced by surface tension. The method generalizes across locomotion systems and captures both maximum displacement and maximum efficiency flows.

This account demonstrates that the "SOAP algorithm" label encompasses a spectrum of methodological paradigms, with meanings ranging from XML-based security and web interoperability to scheduling analysis, neural optimization, medical NLP, astrophysical data handling, messaging protocols, and geometric mechanics. Each context provides its own definition, mathematical formalism, and state-of-the-art implementation, reflecting the term’s highly polysemous role in contemporary computational and mathematical research.