摘要：参数高效微调拥有数百种方法，通常按照参数数量和基准测试分数进行比较。我们将它们作为映射来比较。预训练检查点是权重空间中的一个基点，而权重空间方法则是从该点出发、由一个小型参数空间定义的光滑映射。这些映射构成一个切片范畴，其中冻结模型是始对象，而全量微调是终对象。映射的三个不变量承担了大部分分析工作。其在基点附近的像通过一阶像、维数、关于秩的最小割界，以及该点是光滑点还是锥顶，决定其表达能力。其纤维连同优化器的度量决定训练动力学。基变换会移动基点，并涵盖拆分初始化、量化、重启和合并。举例来说，我们证明：在给定的秩与奇偶性条件下，零初始化 LoRA，以及默认起点下的 Householder 反射适配，均从一个锥的锥顶开始；不存在任何精确初始化能让 LoRA 达到秩高于其自身秩两倍的更新；精确正交方法在精确算术下永远不会改变权重的奇异值，无论它们被合并多少次；在不使用 epsilon、权重衰减或梯度裁剪，并采用同一共享调度的 Adam 下，从标准零初始化出发的 LoRA+，其轨迹与普通 LoRA 完全一致，只是缩放系数乘以学习率比值；使用共享超参数的 Adam，在 LoRA 满秩因子连续对称群中仅保持带符号置换；以及激活函数决定缩放适配器何时能够折叠进前序权重。我们依据从公式中读取的七个坐标，对来自 138 篇论文的 382 种方法进行了分类，并给出 403 个模拟箭头和 485 个阻碍。每一项编号结果都给出了完整证明；对于我们重新推导的他人结果，均已注明归功来源。

Abstract:Parameter-efficient fine-tuning has hundreds of methods, usually compared by parameter count and benchmark score. We compare them as maps. A pretrained checkpoint is a base point in weight space, and a weight-space method is a smooth map from a small parameter space that starts at that point. These maps form a slice category in which the frozen model is initial and full fine-tuning terminal. Three invariants of the map do most of the work. Its image near the base point governs expressivity, through the first-order image, the dimension, a min-cut bound on rank and whether that point is smooth or an apex. Its fibres, with the optimizer's metric, govern training dynamics. Base change moves the base point and covers splitting initialisations, quantisation, restarts and merging. We prove, for example, that zero-initialised LoRA, and Householder reflection adaptation at its default start under stated rank and parity conditions, begin at the apex of a cone; that no exact initialisation lets LoRA reach an update of rank above twice its own; that exactly orthogonal methods never change a weight's singular values in exact arithmetic, however often they are merged; that under Adam with no epsilon, weight decay or clipping and one shared schedule, LoRA+ from the standard zero initialisation follows exactly the trajectory of plain LoRA with its scale multiplied by the learning-rate ratio; that Adam with shared hyperparameters respects only the signed permutations inside LoRA's continuous symmetry group of full-rank factors; and that the activation function decides when a scaling adapter folds into preceding weights. We classify 382 methods from 138 papers by seven coordinates read off their formulas, with 403 simulation arrows and 485 obstructions. Every numbered result is proved in full, and results due to others are credited where we re-derive them.