分析式持续学习中 ETF 目标不可行，Equinorm 后处理改善长尾分类
An Impossibility Result for ETF Target Geometry, and an Equinorm Remedy for Imbalanced Analytic Continual Learning
想在持续学习里用神经塌缩那套几何的可以看看：作者证明 ETF 目标加分析式更新其实没用，但对权重做 Equinorm 后处理在长尾场景提升明显，100:1 不平衡下尾部类准确率直接涨 32 个点。
该论文研究将 ProNC 的 Simplex ETF 目标与 ACIL、GACL 等分析式持续学习方法融合的可能性，证明这种融合在数学上等价于普通 one-hot ACIL。论文还指出 Simplex ETF 在类别数变化时并不嵌套，直接渐进扩展 ETF 会带来相当于经典 ACIL 头 4 倍的灾难性遗忘。作者转而将 ETF 的两个性质拆开，对分类器权重做 Equinorm 后处理：在 CIFAR-100/ViT 上 100:1 不平衡设置下，尾部类准确率提升 32.0 个点，平均准确率提升 16.0 个点，且在 5:1 到 100:1 范围内跨两个骨干网络、两个数据集保持符号一致。代价是接近均衡（约 3:1 以下）的流上 CIFAR-100/ViT 平均准确率下降 5.4 个点。
An Impossibility Result for ETF Target Geometry, and an Equinorm Remedy for Imbalanced Analytic Continual Learning
Analytic continual learning (ACL) methods such as ACIL and GACL fit a ridge-regression classifier head on frozen features via the Sherman-Morrison-Woodbury identity, achieving exemplar-free, exact updates. A separate line of work, Progressive Neural Collapse (ProNC), shows standard continual classifiers benefit from a progressively expanding Simplex Equiangular Tight Frame (ETF) target. It is natural to ask whether fusing the two ideas, exact analytic updates and ETF targets, would yield a better closed-form continual learner. We show this is not the case as the fusion turns out to be identical to plain one-hot ACIL. Further, Simplex ETFs are not nested across class counts, so the literal "progressively expand the ETF" construction stops yielding an ETF as new tasks arrive, and it introduces catastrophic forgetting ($4\times$ that of a classic ACIL head). We therefore separate an ETF's two defining properties, equiangularity and equinormality, and apply each on its own to the classifier weights rather than the targets, where the invariance above does not hold. Equinorm post-processing of the fitted weights recovers both tail-class and overall average accuracy across the long-tailed regime from 5:1 to 100:1 imbalance (e.g. +32.0 tail-accuracy points and +16.0 average-accuracy points at 100:1 on CIFAR-100/ViT), transferring in sign across two backbones and two datasets, at a mild cost of 5.4 average-accuracy points on CIFAR-100/ViT on near-balanced streams (below $\approx 3{:}1$). Equiangular geometry, by contrast, is exactly inert on the target side but actively harmful on the weight side, usually losing to plain one-hot. Hence, the generally useful part of Neural Collapse geometry in Analytic Continual Learning is equinorm post-processing of the weights, becoming more useful as class imbalance grows; and equiangularity is at best a no-op and induces catastrophic forgetting at worst.
