cs.LGSep 9, 2026

Exact Degeneracy Under Balanced k-Shot Sampling:Consequences for Small-Sample Discriminant Analysis on LLM Embeddings

Authors: Lingxiao Qu

Abstract

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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Sep 9, 2026cs.LG

A Kernel-Based Modular Discriminant Analysis Framework for Small-Sample Learning

The small-sample-size (SSS) problem remains a fundamental challenge in machine learning when labeled data are scarce due to cost, accessibility, or ethical constraints. While numerous approaches have been proposed, existing methods often struggle to maintain stable and discriminative representations under high-dimensional and limited-data conditions. Kernelized Linear Principal Component Discriminant Analysis (KLPCDA), a recently proposed modular framework, integrates variance preservation, inter-class separability, and intra-class compactness within a unified kernel space. Although its formulation has shown promising initial results, a systematic understanding of how its components interact across diverse SSS scenarios remains lacking. In this paper, we present a systematic cross-domain study of KLPCDA to characterize the interaction mechanisms among its core objectives. We analyze the behavior of its seven variants across multiple real-world SSS tasks, including hyperspectral image classification, mechanical fault diagnosis, medical diagnosis, and face recognition. Through extensive experiments and ablation studies, we investigate how different objective combinations influence performance under varying conditions such as noise, class imbalance, and high dimensionality. Our analysis reveals consistent patterns in the interaction of the three core objectives variance, between-class, and within-class terms, providing a unified and interpretable understanding of their roles in stabilizing representations and enhancing discrimination in SSS settings. Based on these findings, we further derive practical guidelines for selecting appropriate KLPCDA variants under different data characteristics. Experimental results demonstrate that KLPCDA achieves strong and robust performance across domains, while maintaining low computational complexity suitable for resource-constrained environments.
Lingxiao Qu, Yan Pei
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