> For the complete documentation index, see [llms.txt](https://docs.algenta.ai/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.algenta.ai/machine-learning/ml-svm.md).

# Trainable SVM classifier (simplified SMO)

Trainable soft-margin SVM classifier via simplified SMO with linear and RBF kernels.

`svm_solver` trains soft-margin SVM classifiers in pure Mojo — replacing `sklearn.svm.SVC` / `LinearSVC` fit + predict with zero Python imports. The solver is simplified sequential minimal optimization (Platt) with the random second-choice heuristic, driven by a seeded deterministic xorshift64 RNG. Labels must be in {−1.0, +1.0}; X is a flat row-major matrix. It complements `svm_kernel.mojo` (kernel functions only) but is fully self-contained.

## The modules

| Module       | What it does                                                                                                                                                               |
| ------------ | -------------------------------------------------------------------------------------------------------------------------------------------------------------------------- |
| `svm_solver` | Trains soft-margin SVM classifiers (linear or RBF kernel) via simplified SMO and predicts labels from the fitted model — fit, decision function, and single/batch predict. |

## Functions

| Function                                                         | What it does                                                                                                                                        |
| ---------------------------------------------------------------- | --------------------------------------------------------------------------------------------------------------------------------------------------- |
| `svm_fit_linear(X, y, n_features, C, tol, max_passes, seed)`     | Train a soft-margin linear SVM via simplified SMO; returns an `SVMModel` with dual alphas, bias `b`, and primal weights `w` (decision = `w·x + b`). |
| `svm_fit_rbf(X, y, n_features, C, tol, max_passes, gamma, seed)` | Train an RBF-kernel SVM (`K = exp(−γ‖a−b‖²)`); the model keeps support vectors and their `α·y` coefficients.                                        |
| `svm_decision(model, x)`                                         | Raw decision value `f(x) = Σ αᵢyᵢK(xᵢ,x) + b` — a single dot product for linear models.                                                             |
| `svm_predict(model, x)`                                          | Class label: +1.0 if the decision value ≥ 0, else −1.0.                                                                                             |
| `svm_predict_batch(model, X)`                                    | One label per complete row of a flat row-major query matrix; empty list for an empty model.                                                         |

## Determinism and cost

* **Deterministic per seed.** The only randomness is the seeded second-choice heuristic; refitting with the same seed reproduces `b`, `w`, and every alpha bit-for-bit (verified in the tests for RBF seed=7 and linear seed=42).
* Fitting is `O(passes · n² · d)` worst case and will exceed 1 ms for n in the thousands — acceptable because fitting is an offline / per-recalibration step. `svm_decision` / `svm_predict` are `O(d)` (linear) or `O(n_support · d)` (RBF) and stay sub-millisecond for any realistic model.
* Termination is `max_passes` consecutive no-change passes, with a hard cap of `100 · max_passes` total passes guarding pathological cases.
* Infeasible input (shape mismatch, `C ≤ 0`, `max_passes ≤ 0`, n = 0) yields an empty model whose decision is identically 0.

## Verification

Known-value tests pin the optimizer against hand-computed solutions:

* Two-point 1D problem (X = \[−1, +1], y = \[−1, +1], C = 10): SMO reaches the exact max-margin solution — alphas \[0.5, 0.5], b = 0, w = \[1.0], and `decision(2.0) = 2.0`, all to 1e-9. The dual constraint `Σ αᵢyᵢ = 0` holds to 1e-9.
* Linearly separable 2D set (true boundary `x0 + x1 = 4`): 100% train accuracy, minimum functional margin ≥ 0.9 (\~1 at the optimum), correct side-of-boundary signs, and 100% accuracy again under a different seed (99).
* XOR clusters with the RBF kernel (γ = 2, C = 10): ≥ 15/16 train accuracy, exact labels at all four corners, and support coefficients summing to 0 within 1e-6.
* Edge cases proven: n = 1 and all-equal labels leave all alphas at zero (no KKT-violating pair exists), and batch predictions match per-row predictions exactly.

**Next:** promote what you trained with [statistical model comparison](/machine-learning/ml-model-comparison.md), or browse the [engine catalog](/engine-runtime/catalog.md).


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