Finite Probes Suffice: Identifiability and Universality for Weight-Space Learning

Neural representations are not unique objects. Even when two systems realize the same downstream computation, their hidden coordinates may differ by reparameterization. A probe family intended to reveal structure already present in a representation should therefore be stable under the relevant representation symmetries rather than be tied to a particular basis. We prove that, under standard nondegeneracy assumptions, behaviorally equivalent LLMs have hidden representations related by an invertible affine transformation. The resulting symmetry principle singles out a unique hierarchy of shallow coordinate-stable probes, with linear probes as its degree-1 member. We also show that a natural object for cross-model probe transfer is a shared probe-visible quotient--the representation modulo directions invisible to the probe family--rather than the full hidden state. Experiments on synthetic and real-world tasks support both predictions, showing where degree-2 probes help beyond linear ones and how quotient-based transfer enables coverage-aware monitor portability across model families. These results point toward a broader geometric representation theory of neural probing, with coverage-aware monitor transfer as a concrete operational consequence.

The explosive growth of open-source model repositories has created a Model Jungle, where checkpoints are frequently shared without adequate documentation or metadata. While weight-space learning offers a pathway to identify and analyze these models directly from their parameters, processing full-scale weights is computationally prohibitive. Probing-based methods have emerged as a lightweight alternative, extracting permutation-equivariant representations via learnable probe vectors. However, existing probing methods are limited by a single-view design: they capture first-order structures but fail to encode the rich, higher-order correlation patterns inherent in row-column interactions. To bridge this gap, we introduce MVProbe, a multi-perspective probing framework that synthesizes first-order signals with interaction-aware (Gram-based) views. Our approach is theoretically grounded; we analyze the scaling laws of different probing orders to derive a principled standardization and fusion strategy that ensures balanced contributions from all branches. On the Model Jungle benchmark, MVProbe consistently outperforms the state-of-the-art ProbeX across diverse architectures, including discriminative backbones (ResNet, SupViT, MAE, DINO) and large-scale generative LoRA adapters (Stable Diffusion LoRA).

Learned feature representations in neural networks often contain structure beyond that directly used by the final task output. We study this structure using $\textit{passive probes}$ that apply fixed, untrained, property-independent projections to representations as they evolve during training. We motivate this approach through the task of prediction on $S^2$ where equivalent vector and Hermitian parameterizations reveal an additional loss-invariant trace coordinate. This motivates a general construction in which fixed random projections serve as observers of learned features. Because the observer is loss-invariant and independent of the property being studied, changes in accessibility reflect changes in the representation relative to the fixed observer rather than adaptation of the observer itself. We show that ensembles of passive probes can directly reflect task-relevant information such as target alignment. Under our constructions, the accessibility of eventual difficulty evolves differently across tasks. It increases during training in the regression tasks of surface-normal estimation and image inpainting but remains near its initial level in image classification. Comparisons with learned linear probes further show that recoverability and passive accessibility can evolve differently during training. Together, these results show how passive probes can separately characterize changes in representation geometry and the accessibility of eventual task difficulty.