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Ball vector machine

Algorithms AIMech:BallVectorMachine DRAFT

Aliases: BVM

An approximate SVM training procedure that uses fixed-radius enclosing-ball center updates in a transformed kernel feature space, with multiscale tolerance and radius-refinement variants that avoid CVM’s repeated core quadratic programs.

Identity Notes & Concept Scope This record denotes the Tsang-Kocsor-Kwok BVM procedure. Its fixed-radius core and optional refinement are distinguished explicitly; a feasible enclosing radius is not assumed to be the minimum radius. Compared all 442 pinned AIO IDs, labels and scoped synonyms with all 513 preceding native IDs, names and aliases. No normalized identity collision was found for either candidate. Whole-word fixed-string searches of both checkouts included ignored and hidden files, expanded names, CVM/BVM acronyms and primary-source identifiers. All 22 matches per checkout were inspected in 14 distinct text windows at 44 locations: prior queries and unresolved discovery notes, not existing records. All 93 semantic AIO rows and 35 native name/definition matches were read. Twelve annotated AIO CLASS nodes and outgoing edges and 17 contextual YAMLs were inspected or reused after unchanged-byte checks. This is a bounded local corpus assessment.

Claims & Grounded Evidence

7 Claims

Mechanism #fixed-radius-center-update

For a finite feature set, 0<epsilon<=1 and a supplied positive radius r>=R_star, initialize the center at a feature point. If a point z lies beyond (1+epsilon)*r, set beta=r/||c-phi(z)|| and c_new=phi(z)+beta*(c-phi(z)); otherwise terminate after checking all points. The update is the smallest center movement that places z on the radius-r boundary and is a convex combination. Any violating point suffices. Kernel coefficients can represent the center, and no inner quadratic program is required.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 3.1-3.2, printed pp912-914 / PDF pp2-4, Algorithms 2-3 and Equations 5-6

Limitation #iteration-guarantees-and-search-cost

The fixed-radius procedure requires at most O(1/epsilon²) violating-point updates under r>=R_star and the stated initialization. A multiscale version halves tolerance successively and warm-starts centers; its O(1/epsilon) iteration result is proved for r=R_star. That stronger rate is not established for every larger feasible radius. These are update bounds: finding a violator or certifying all-point coverage, kernel-evaluation costs and input storage must also be accounted for. A sampled stopping check is not an exhaustive certificate.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 3.1-3.2, printed pp913-914 / PDF pp3-4, Propositions 1 and 3; curator separation of search work

Mechanism #l2-transformed-kernel-and-feasible-radius

For binary L2 SVM training, use Ktilde_ij=y_i*y_j*(k_ij+1)+delta_ij/C. If k has constant diagonal kappa, the transformed diagonal is kappa_tilde=kappa+1+1/C, as follows from Equation 3, including its bias coordinate. The radius sqrt(kappa_tilde) is a feasible upper bound on the optimal MEB radius, since all transformed points have that norm. It need not be the minimum radius. For finite n and C>0, alpha^T*Ktilde*alpha>=1/(n*C)>0 on the simplex, hence R_star²<kappa_tilde. The nearby prose diagonal that omits +1 is not substituted for the displayed kernel.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 2 and 3.4, printed pp912,914-915 / PDF pp2,4-5, Equations 3 and 12; curator diagonal and finite-slack-term derivation
supports Source: Core Vector Machines: Fast SVM Training on Very Large Data Sets · Locator: Section 3.2.2, printed pp369-370 / PDF pp7-8, Equations 12-17 and following transformed-feature display

Limitation #fixed-radius-not-minimum-center

Reducing epsilon at a radius strictly larger than R_star does not in general recover the MEB center. For unit-norm points (1,0) and (-7/25,24/25), the fixed-r=1 update from the first point stops at (13/25,9/25), with distances 3/5 and 1. The true MEB center is (9/25,12/25), radius 4/5, and the center error remains 1/5 as tolerance tends to zero. This exact geometric illustration concerns Algorithm 2, not a historical SVM benchmark or the full radius-refinement procedure.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 3.1-3.2, printed pp912-914 / PDF pp2-4, Algorithms 2-3 and Equations 5-6
supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Section 3.2, printed pp913-914 / PDF pp3-4, Proposition 2; exact curator example in research/discovery/core-ball-vector-boundaries.py

Mechanism #optional-radius-refinement

The source’s Algorithm 4 augments every feature by a zero coordinate and starts the center with an extra coordinate r. After an approximate radius-r enclosure, let u be the final extra center coordinate. If u²>(2*epsilon+epsilon²)*r², replace r by sqrt((1+epsilon)²*r²-u²) and repeat; otherwise stop. The paper’s experiments explicitly use only one iteration of Algorithm 4. This refinement is distinct from assuming the initial feasible radius is already optimal.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Section 3.3, printed p914 / PDF p4, Algorithm 4; Section 4 footnote 4, printed p915 / PDF p5

Limitation #asymptotic-and-prediction-scope

The paper motivates near equality of the feasible and optimal radii using high-dimensional Gaussian data and a Gaussian kernel. Its limiting argument additionally needs the class-count imbalance divided by n to vanish; merely having two reasonably populated classes does not ensure that. The center-distance bounds retain their dependence on r/R_star, and coefficient or score proximity does not by itself guarantee identical predicted labels or test accuracy. No finite-sample equality R_star=sqrt(kappa_tilde) is asserted.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 3.2 and 3.4, printed pp913-915 / PDF pp3-5, Propositions 2-3 and Equations 9-12; curator finite-radius qualification

Scope #procedure-identity-and-evaluation

BVM denotes the enclosing-ball approach to the stated SVM formulation, with multiscale and radius-refinement variants described in the same paper. It is separate from CVM’s repeated core MEB fits, generic SVM, support-vector domain estimation and sphere-based cluster labeling. The ten-dataset experiments use particular kernels, C values, tolerances, versions, hardware, time cutoffs and multiclass availability; they do not establish a universal speed or accuracy ranking.

supports Source: Simpler Core Vector Machines with Enclosing Balls · Locator: Sections 2-5, printed pp912-918 / PDF pp2-8; Tables 1-4 and Figures 2-4

⚠️ Curated Limitations & Negative Results

Kernel validity and normalization, regularization, numerical tolerances, violation-search policy and data geometry affect guarantees and costs. A geometric approximation is not a calibrated uncertainty estimate or a generalization guarantee.
All 30 CVM and eight BVM primary PDF pages were visually read; the CVM critique’s 11 pages were also read for that record. The reply remains abstract-only context and its software diagnosis was not independently verified.
Exact rational formula and boundary checks were executed. No historical binary, full optimizer, trained model or benchmark was executed. Original software access links failed; current implementation behavior is not asserted.
This maintained native record remains DRAFT, with a local related AIO mapping. Generalized solvers, data-description variants and other discovered source leads remain unresolved.

Artificial Intelligence Ontology (AIO) Grounding

Mapped AIO Classes

AIO Target Class	Mapping Relation	Mapping Rationale
`AIO:SupportVectorMachine` Support Vector Machine	related_match	The algorithm trains a particular support-vector formulation, making AIO:SupportVectorMachine a related model-family target. The pinned target is a Network with layer relations and supervised classification/regression scope; it is not identical to this training algorithm. Related matching asserts neither equivalence nor subclassing. BinaryClassification, Classification, SupervisedLearning and UnsupervisedLearning are tasks, not the specific solver; encoding labels in a feature map does not make binary training unsupervised. KernelLayer is a component, TrainingStrategy is described as preprocessing, and DataPreparation or SelectionAndSamplingBias do not identify the procedure. Native SVDD fits a support boundary, SVC adds component labeling, kernel PCA extracts variance directions, and margin clustering infers labels. Shared kernels or enclosing spheres do not establish identity.

Evidence Sources & Primary Literature

2 Sources

Source ID	Publication / Document	Type / Access	Reference & Checksum
`tsang2007`	Simpler Core Vector Machines with Enclosing Balls Ivor W. Tsang, Andras Kocsor, James T. Kwok	primary paper full_text	sha256:3bb017075a61…
`tsang2005`	Core Vector Machines: Fast SVM Training on Very Large Data Sets Ivor W. Tsang, James T. Kwok, Pak-Ming Cheung	primary paper full_text	sha256:8ba1804ac233…

Curation Provenance

Curator: Codex

Method: ai_assisted

Model: GPT-6

Created: 2026-10-12T09:31:37Z

Updated: 2026-10-12T09:40:04Z

Source File: data/curated/BallVectorMachine.yaml