winnex-xfactor
Deterministic manifold embedding for native LLM inference.
The X-Factor is a training-free operator that embeds text into a pretrained
model's latent space — computed structurally from embed_tokens, never by
running the model.
What it is
For any model, the X-Factor is the orthonormal basis of the manifold where its token embeddings live:
Ē = E − mean(E) (center)
C = ĒᵀĒ / V (D×D covariance)
C = U Λ Uᵀ (eigh, Λ descending)
r = min{r : cumsum(Λ₁..r)/ΣΛ ≥ τ} (effective rank)
X = U[:, :r] (the X factor: D×r manifold basis)
P = X·Xᵀ (orthogonal projector)
The projector is idempotent (P²=P) and symmetric (Pᵀ=P) — the orthogonal projection onto the model's embedding manifold.
Why it is fast
The eigendecomposition uses power iteration with deflation:
O(D²·r·iters) vs O(D³) for a full Jacobi/eigh
Honest cost (corrected 2026-10-21). An earlier version of this README
claimed "~2000× faster at D=2048". That number does not reproduce, and the sign
is wrong at the dimension it names. Measured against numpy.eigh on the same
covariance:
| D | power iteration | numpy eigh | speedup |
|---|---|---|---|
| 128 | 0.25 s | 0.43 s | 1.7× |
| 384 | 7.4 s | 8.9 s | 1.2× |
| 2048 | 399 s | 34.7 s | 0.09× (11× slower) |
The cause is structural: max_candidates = D with up to 2000 iterations per
component. At D=2048 the effective rank is ~1656, so the cost is
1656 × 2000 × O(D²) against a single O(D³) eigh.
What the method actually wins is the low-rank regime — a corpus whose
effective rank is small relative to D. Measured on genuinely low-rank
embeddings (rank 40, in D=512 and D=1024): 8.3× and 5.6× faster than
eigh.
Recommendation. Use this operator when the manifold is known to be
low-rank and you want a deterministic, dependency-free implementation. For
D ≥ 2048 with a high effective rank, numpy.eigh — or a randomized SVD
(Halko–Martinsson–Tropp) — is an order of magnitude faster and should be
preferred.
The mathematical output is correct in every regime: the projector is idempotent (‖P²−P‖/‖P‖ = 3.3e-7), exactly symmetric, and the reported rank matches the true eigendecomposition. Only the performance claim was wrong.
The guarantee that does hold everywhere is determinism: fixed seed, fixed iteration count, same input → same basis.
Install
pip install winnex-xfactor
Requires Python ≥ 3.8 and NumPy. The C++ core ships pre-built in the wheel.
Quick start
import numpy as np
import winnex_xfactor
# 1. Compute the X-Factor from a model's embed_tokens (a sample is enough).
E = np.random.randn(5000, 2048).astype(np.float32)
P, rank, variance = winnex_xfactor.compute_xfactor(E, 5000, 2048, tau=0.95)
print(f"rank={rank}, variance captured={variance:.3f}")
# 2. Embed a spectral token into the model's manifold.
psi = np.ones(64, dtype=np.float32)
h = winnex_xfactor.embed_spectral(psi, P) # length 2048, on the manifold
API
compute_xfactor(embed_tokens, vocab, dim, tau=0.95) -> (P, rank, variance)XFactor(embed_tokens, vocab, dim, tau)— object with.project,.project_batch,.projector,.basis,.rank,.variance_captured,.dimexpand_spectral(psi, d, D)— zero-pad + L2 normalizeembed_spectral(psi, P)— full text→manifold projection
Benchmark
Run the benchmark on Kaggle — installs winnex-xfactor (and winnex-nano for
the safetensors loader) from PyPI and computes the X-Factor from real
embeddings of a Qwen model (Kaggle qwen-lm/qwen):
Result: effective rank 118 (D=128, τ=0.95, variance captured 0.954), compute 0.03s — power iteration with deflation (O(D²·r)) on real embeddings.
License
BSL 1.1 | pay@winnex.ai | (c) Winnex Brasil Soluções Empresariais LTDA-ME
1 maintainer
Metadata
1 maintainer
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