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Life sciences · Preprint
arXiv · September 9, 2026
Early or partial results. Treat as a signal, not a conclusion.
This preprint identifies and proves an exact degeneracy in Kernelized Linear Principal Component Discriminant Analysis (KLPCDA) under balanced k-shot sampling, where two of seven variants have equal signal eigenvalues, rendering their selection criterion indifferent. Empirical validation on frozen LLM embeddings and decoder-model activations shows that logistic regression still outperforms KLPCDA on most datasets, and that performance gaps close substantially (>80%) when sample size grows from k≤10 to k=30–50, suggesting an estimation-efficiency effect rather than a fundamental limitation of the embedding space.
Mathematical analysis with empirical validation. Few-shot text classification tasks on high-dimensional LLM embeddings and decoder model activations; high-dimensional regime where sample size n is much smaller than dimensionality d.. Intervention: Balanced k-shot sampling inducing exact degeneracy; in-formula tie-break repair applied to two repairable KLPCDA variants.. Compared with: Logistic regression, SetFit, LoRA, and in-context learning baselines..
Within-class scatter operator under balanced k-shot sampling is exactly a scaled orthogonal projector, inducing exact degeneracy in KLPCDA variants. Two of KLPCDA's seven variants have every signal eigenvalue exactly equal, making eigenvector selection criterion provably indifferent rather than ill-conditioned. A third KLPCDA variant has a provably void objective under balanced sampling.
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This is a preprint describing a mathematical degeneracy in small-sample discriminant analysis on LLM embeddings, with empirical validation on few-shot text classification; the work is novel and methodologically sound but unreviewed, addresses a technical limitation rather than a clinical or practice outcome, and does not yet establish superiority of a proposed solution.
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Balanced k-shot sampling draws exactly k labeled examples per class. We show that it induces an exact, provable degeneracy in a family of small-sample discriminant estimators. Under balanced sampling, the within-class scatter operator of Kernelized Linear Principal Component Discriminant Analysis (KLPCDA) is not merely rank-deficient but exactly a scaled orthogonal projector. We derive the consequences in closed form: two of KLPCDA's seven variants have every signal eigenvalue exactly equal, so their eigenvector selection criterion is provably indifferent rather than ill-conditioned, and a third has a provably void objective. This follows from the estimators' construction, not any dataset; we confirm it on frozen sentence embeddings and, separately, on residual-stream activations from a decoder-only generative model. An in-formula tie-break repairs the two repairable variants, with recovery gated by class count: the residual subspace constraint costs 5x more on few-class than many-class datasets (p=0.000001). We then evaluate the repaired framework on few-shot text classification on frozen LLM embeddings (n much smaller than d, up to 4096), across four datasets, three embedding sizes, and three trained baselines (SetFit, LoRA, in-context learning). A properly cross-validated logistic-regression probe still beats every KLPCDA variant on three of four datasets, at every embedding size; guidance carried from pixel, vibration-signal, and gene-expression data does not directly generalize to this feature space. Three independent geometric separability metrics fail to explain why one high-dimensional decoder-based embedding model underperforms smaller bidirectional encoders, ruling out anisotropy; the gap is substantially an estimation-efficiency effect, not a permanent ceiling, closing by more than 80% when the support set grows from k<=10 to k=30-50 (p=0.00195, both many-class datasets).
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