Centered Kernel Alignment

Learning symmetries through data augmentation offers an alternative to architectural equivariance, but does not guarantee that a network respects those symmetries. Measuring learned equivariance is especially difficult for intermediate representations, whose group actions are not specified. In a finite discrete setting, we show that equivariance requires representations of original and transformed inputs to determine one another exactly, yielding a test based on normalized mutual information (NMI). For high-dimensional activations, we use centered kernel alignment (CKA) as a scalable test, with an exact guarantee for orthogonal feature transformations. Across architectures and tasks, augmentation usually increases transformation consistency, showing that learned equivariance can be probed directly from representations without recovering the hidden transformation rules that relate them.

Decoder hidden states from different layers of large language models (LLMs) often encode complementary information. However, existing speculative decoding methods typically rely on a small set of target-model layers chosen using fixed, position-based heuristics. Selecting an optimal subset of layers is inherently a combinatorial problem, making exhaustive evaluation infeasible even for moderately deep models. In this work, we present a principled framework for multi-layer hidden-state selection for speculative decoding based on two complementary criteria: informativeness, measured through matrix-entropy analysis, and diversity, quantified using centered kernel alignment (CKA). We investigate layer-wise Shannon entropy, aggregate entropy under hidden-state concatenation, and pairwise CKA dynamics across depth, revealing distinct geometric and informational compartments within decoder representations. Motivated by these findings, we propose entropy and CKA-based layer selection algorithms that favor hidden-state sets that are both highly informative and minimally redundant. As part of this framework, we introduce token-entropy filtering, which removes hidden states associated with low predictive-entropy tokens, thereby improving the quality of layer-selection statistics. In EAGLE3-style speculative decoding, our selected sets outperform fixed baseline choices, including the default $[2, N//2, N-3]$ rule, while providing a clearer explanation of why some defaults can remain competitive. Our results provide a practical recipe for multi-layer hidden-state selection for speculative decoding, insights into the geometry and information in LLM decoder states, and a foundation for extending similar analysis to related applications.

Robust perception in intelligent vehicles demands 3D object detectors that remain dependable under domain shifts, such as changes in time of day, location, or weather. However, due to costly annotation and rare shifts, some environments lack sufficient data to train a standalone detector. Federated learning offers a privacy-preserving framework for collaborative model training, enabling clients to benefit from shared learning across diverse environments. Yet, this framework traditionally relies on a single global consensus model, which struggles to perform across heterogeneous local data distributions. Local conditions are better captured by adapting a subset of the model, but many personalization approaches rely on predefined layer partitions or fixed personalization ratios, thereby limiting adaptation to client-specific divergence. To reduce this rigidity, we propose FedCKA, a Centered Kernel Alignment (CKA)-based strategy that dynamically handles the personalization-globalization trade-off. Specifically, FedCKA computes layer-wise feature similarities between local client models and the global consensus model during training. By converting layer-wise similarity scores into client-specific aggregation masks, FedCKA selectively shares representation-consistent layers. Evaluation on a unified multi-domain benchmark based on nuScenes shows that FedCKA outperforms established federated baselines, including FedBN, FedRep, and FedSelect, improving average NDS by 7 percentage points over the strongest baseline. The findings offer both a comparative benchmark and a promising direction for robust federated 3D perception across shifts in location, weather, and illumination. Code is available at https://github.com/j-verhoog/FedCKA.

Do different LLM architectures encode high-level concepts in structurally compatible ways? We systematically characterize a geometric-functional universality dissociation: across multiple concept domains and architectural families, moderate geometric convergence coexists with near-perfect functional transfer. Using contrastive-difference CKA (CKA_Delta), a training-free diagnostic that computes kernel alignment on per-sample contrastive differences, we isolate concept-specific convergence from generic similarity -- achieving significant discrimination where standard CKA cannot. The dissociation replicates across all six concept domains we test (five with p <= 0.017 geometric discrimination and safety as a converging-functional trend, p = 0.08), including two non-instruction concepts (code-vs-NL, reasoning-vs-recall) validated without system prompts; a single 70B--70B pair provides an observational note that universality may strengthen with scale, requiring replication with additional >=70B models. We position CKA_Delta as a practical regime classifier and architectural outlier detector (Gemma: d = 1.08, AUC = 0.79) rather than an absolute transfer-accuracy predictor, providing a training-free diagnostic for cross-architecture concept monitoring.

Demand for low-precision inference, including NVFP4-based approaches, has grown as large language models are increasingly deployed in latency and cost constrained production environments. Quantization-aware distillation (QAD) helps recover accuracy lost under low bit quantization by training a quantized student to match the output distribution of a frozen higher precision teacher via a KL-divergence loss. In this work, we first provide a representation level diagnosis of QAD: output matching alone can mask internal degradation, because many intermediate activation geometries can yield similar teacher-aligned logits. Using CKA, we show that KL-only QAD can reduce layerwise representational similarity relative to the BF16 teacher, with especially severe drift in RL-post-trained models. This drift correlates with downstream bottlenecks on reasoning and coding tasks, suggesting that low bit recovery requires preserving internal geometry rather than matching outputs alone. Motivated by this finding, we propose CKA-QAD, a CKA-guided representational alignment method for NVFP4 QAD and low bit LLM accuracy recovery. The method adds a lightweight regularizer that preserves internal representational geometry during distillation by aligning layerwise Gram matrices through CKA. Across Nemotron 3 Nano and Qwen3-4B-Thinking-2507, CKA-QAD substantially improves representational alignment and improves downstream reasoning and coding accuracy with modest training overhead. Our findings position CKA-guided representational alignment as a practical complement to output matching for quantized LLM recovery.

Diffusion geometry is a manifold learning framework that uses random walks defined by Markov transition matrices to characterize the geometry of a dataset at multiple scales. We use diffusion geometry for neural representations, incorporating tools from multi-view learning into this field for the first time. Our key technical observation is that a broad class of similarity measures based on representational similarity matrices (RSMs) admits a closed-form equivalent formulation in terms of row-stochastic Markov matrices, opening the door to manipulations from diffusion geometry. As a first application, we develop multi-scale variants of Centered Kernel Alignment and Distance Correlation, which utilise the $t^{th}$ power of the underlying transition matrix to probe the data geometry at adjustable diffusion scales. Going further, we introduce variants of these measures which fuse the Markov matrices of several layers via alternating diffusion into a single operator that captures the network's joint sample geometry, allowing similarity to be computed across multiple layers and shifting the comparison from layer-to-layer to network-to-network. We perform extensive numerical experiments, evaluating our measures on the Representational Similarity (ReSi) benchmark comprising 14 architectures trained on 7 datasets across three different domains. Our methods achieve SoTA results in accuracy and output correlation for both language and vision tasks across different models. We furthermore show SoTA performance on an additional benchmark evaluating on out-of-distribution data.

Mean-pooled cosine similarity is the default metric for comparing neural representations across languages, modalities, and tasks. We establish that this metric is not length-invariant: under the anisotropy that characterizes modern transformer representations, mean-pooled cosine grows monotonically in sequence length, independent of representational content. Empirically, on HumanEvalPack across four code LLMs, the length ratio alone explains $R^2 = 0.52$--$0.75$ of cross-language "Python proximity," while AST depth and shared-token fraction add less than 3% of explained variance beyond length. Substituting Centered Kernel Alignment (CKA) reduces explained variance by 83% and reverses the sign of the length coefficient ($β_{\mathrm{len}}: +0.86 \to -0.37$). The same pattern holds in Mistral-7B on parallel WMT pairs ($R^2 = 0.23$ EN-FR, $R^2 = 0.33$ EN-DE for cosine; $R^2 < 0.01$ for CKA). In CLIP ViT-B/32, mean-pooling reduces the length effect relative to EOS-pooling ($R^2: 0.21 \to {<}0.01$), as predicted by the theory's dependence on anisotropy. We argue that length-invariant metrics such as CKA should be the default for cross-representation comparisons, and that recent claims of cross-lingual representational convergence built on mean-pooled cosine warrant re-examination.

Training of multimodal foundation models is currently restricted to centralized data centers containing massive, aligned datasets (e.g., image-text pairs). However, in realistic federated environments, data is often unpaired and fragmented across disjoint nodes; one node may hold sensor data, while another holds textual logs. These datasets are strictly private and share no common samples. Current federated learning (FL) methods fail in this regime, as they assume local clients possess aligned pairs or require sharing raw feature embeddings, which violates data sovereignty. We propose a novel framework to train a global multimodal transformer across decentralized nodes with disjoint modalities. We introduce a small public anchor set to align disjoint private manifolds. Using Gram matrices calculated from these public anchors, we enforce semantic alignment across modalities through centered kernel alignment without ever transmitting private samples, offering a mathematically superior privacy guarantee compared to prototype sharing. Further, we introduce a subspace-stabilized fine-tuning method to handle FL with huge transformer models. We strictly decouple domain-specific magnitude shifts from semantic direction, ensuring that nodes with varying sensor characteristics align geometrically to the global consensus. Lastly, we propose precision weighted averaging, where efficiently obtained uncertainty estimates are used to downweight uncertain nodes. This paper establishes the mathematical backbone for federated unpaired foundation models, enabling a global model to learn a unified representation of the world from fragmented, disjoint, and private data silos without requiring centralized storage or paired samples.

Centered kernel alignment (CKA) is a popular metric for comparing representations, determining equivalence of networks, and conducting neuroscience research. However, CKA does not account for the underlying manifold and relies on numerous heuristics that cause it to behave differently at different scales of data. In this work, we propose Manifold-approximated Kernel Alignment (MKA), which incorporates manifold geometry (i.e., local neighborhood) into the alignment task. We derive a theoretical framework for MKA. Our construction removes representation-dependent degree weighting, yields a closed-form alignment score, and gives each row a probabilistic interpretation as a local Markov transition distribution. We further establish kernel- and score-level stability with respect to the neighborhood size. We perform empirical evaluations on synthetic datasets and real-world examples to characterize and compare MKA to its contemporaries. In controlled synthetic experiments, MKA is consistently less sensitive to hyperparameters and captures the selected topology-preserving relationships. On the large-scale benchmark, MKA achieves its strongest aggregate performance in vision, is competitive on graphs, and is weaker than CKA in several NLP evaluations. Our findings suggest that manifold-aware kernel alignment provides a robust foundation for measuring representations and is potentially useful when local neighborhood geometry is important, with possible applications in representation learning.