Cosine Similarity

We analyze the learned input-output behavior of GLU-based neurons in large language models (LLMs). We propose a simple analysis method: For each neuron, we compute the cosine similarities between its input (reading) and output (writing) weight vectors. In this scheme, a strong negative cosine similarity indicates the neuron weakens the direction it detects in the residual stream, so we call this a weakening neuron. This allows us to gain a number of novel insights. First, we show that nine different LLMs have similar patterns: weakening neurons appear mostly in late layers whereas their counterparts, (conditional) strengthening neurons, are frequent in early-middle layers. Second, we find that weakening neurons display surprising behavior: even though there are few, they activate often and have a large influence on model behavior. Third, weakening neurons have a strong effect on model output when gate values are negative -- which is surprising since negative gate values are not expected to encode functionality.

Foundation models are increasingly adapted through fine-tuning, model editing, and alignment procedures while retaining previously acquired capabilities. Understanding the internal computations that support these adaptations is therefore becoming increasingly important for continual model evolution. Sparse autoencoders (SAEs) provide interpretable feature dictionaries for residual-stream activations and sublayer outputs, but it remains unclear how state features and update features interact to produce downstream residual features. In this work, we focus on MLP updates as a first test case. We construct a transition atlas of triples $s_k + u_j \rightarrow t_\ell$, where a residual-state feature and an MLP-update feature jointly predict a target residual feature, and validate candidate triples by ablating the decoded update feature. In a 20M-token Pythia-160M $L_7 \rightarrow L_8$ run, we find 38,125 strong ablation-effect transitions, but 88.0% have both state-target and update-target decoder cosine similarity below 0.7. As a preliminary cross-model check, a run of 20M-token Gemma-3-4B $L_{21} \rightarrow L_{22}$ causally validates only the top 30,000 ranked candidate triples by ablating the decoded update feature, and 53.6% of strong-effect triples have both state-target and update-target decoder cosine similarity below 0.7. The Gemma result is directionally consistent with Pythia, but weaker, since update-target cosine recovers many of the strongest Gemma effects and the run is not a full-atlas causal validation. Ultimately, our results suggest that feature flow atlases can serve as diagnostics of representation-update mechanisms and thereby inform tools for steering model updates. Future work will validate more complex patterns across layers, models, and SAE families.

We introduce AdamX, a first-order optimizer that incorporates cosine similarity as an adaptive mechanism for controlling update magnitudes. The proposed method is scalable, model-agnostic, and straightforward to integrate into existing training pipelines. We further introduce a variance rectification scheme that promotes smoother optimization during the early stages of training. Overall, we provide empirical evidence that AdamX achieves competitive convergence rates across a range of benchmark datasets and architectures. Performance is evaluated in terms of the number of epochs required to reach predefined performance thresholds under a fixed hyperparameter budget. Code and Experiments available at: https://github.com/FranciscoCaldas/adamX.

Large vision-language models (LVLMs) incur substantial inference costs due to their long and highly redundant visual-token sequences. Diversity-based pruning mitigates this cost by selecting token subsets based on pairwise cosine similarity. We find, however, that similarities between raw visual tokens are strongly concentrated in the positive range, limiting their ability to distinguish non-redundant tokens. A natural way to improve this resolution is to center token features before computing cosine similarity. Centering indeed reveals a substantially richer pairwise structure, yet unexpectedly degrades pruning performance when used alone. We show that this apparent contradiction arises because the raw geometry does more than represent pairwise diversity: it also implicitly favors globally distinctive tokens, which tend to contain semantically informative content. Centering better resolves subset diversity but loses this useful token-wise preference, revealing that diversity and distinctiveness are entangled in the raw geometry. Based on this analysis, we propose the \textbf{Cen}tered Geometry \textbf{Prune}r (Cen-Prune), which measures subset diversity using centered cosine similarity while retaining raw-space distinctiveness as a complementary token-wise preference. This lightweight, plug-and-play correction leaves the underlying selection mechanism unchanged and incurs negligible computational overhead. Extensive experiments across multiple image- and video-understanding benchmarks and LVLM architectures demonstrate that Cen-Prune provides robust improvements in overall performance across existing diversity-based pruners.

Omics datasets, particularly single-cell RNA sequencing data, are high-dimensional, sparse, noisy, and dominated by zero values, making faithful low-dimensional representation challenging. Existing dimensionality-reduction methods may distort local neighbourhoods, global organization, or the cohesion of meaningful populations, with similar limitations arising in genealogical data. We introduce Contrastive Manifold Approximation and Projection (CosMAP), a graph-based unsupervised dimensionality-reduction method for producing faithful and interpretable embeddings. CosMAP extends the graph-based framework of UMAP by combining cosine-similarity neighbourhoods with temperature-normalized contrastive affinities, which are optimized in the embedding space using an attractive--repulsive objective. It further employs a two-phase refinement strategy: an intermediate higher-dimensional representation is first learned and then used to reconstruct the neighbourhood graph and initialize the final low-dimensional embedding. We evaluate CosMAP on MNIST and USPS handwritten-digit datasets, mouse retina and cortex single-cell RNA-sequencing datasets, and a large genealogical kinship dataset derived from BALSAC-CARTaGENE. Compared with state-of-the-art dimensionality-reduction methods, CosMAP produces more coherent visual representations, improves neighbourhood preservation, and provides clearer global organization of digit classes, biological cell populations, and regional genealogical patterns. These results indicate that CosMAP offers a robust framework for exploratory analysis of complex, sparse, high-dimensional data. The implementation is publicly available at https://github.com/FenosoaRandrianjatovo/CosMAP-dr.

The direction of change --- whether a series will move up or down --- is often as important as its exact value in decisiondriven applications such as risk management and financial forecasting. However, most forecasting losses optimize either point magnitude or shape and frequency structure, and none explicitly targets the direction of change. In this paper, we find that MSE-trained forecasters fail on the direction of small moves. To address this, we propose CosDir, a simple yet effective direction-aware loss that aligns the difference vectors of the prediction and the target via cosine similarity. Being scale-invariant, CosDir keeps a directional gradient on small moves, re-injecting learning signal exactly where MSE neglects it. CosDir is a lightweight, plug-in term that attaches to any backbone without architectural modification. Since the best ratio for mixing the directional and magnitude terms differs across datasets, we further propose CosDir-UW, an extension that makes this ratio adaptive by learning it during training, matching a per-dataset tuned weight with no hyperparameter. We conduct over 100K experiments, demonstrating that our method consistently and significantly improves directional accuracy while preserving magnitude accuracy, and that it outperforms various loss functions. Code is available at: https://github.com/seunghan96/cosdir.

A growing number of applications, such as biometrics and retrieval-augmented generation (RAG), rely on cosine similarity scores computed between vector embeddings of text, images, or audio. These systems return similarity scores through their APIs for ranking and verification. However, such releases can leak information about individual records and enable membership inference attacks. While differential privacy (DP) provides a principled metric for quantifying attack risks, naïve application of DP mechanisms---such as adding i.i.d. Gaussian noise to vector entries---leads to excessive distortion (i.e., low utility) at a given privacy constraint that scales poorly with the number of released scores. We propose \textsc{ScoreShield}, a perturb-then-project mechanism that adds Gaussian noise calibrated to global sensitivity of the chosen score release regime and then projects the result onto the feasibility set of valid cosine objects. \textsc{ScoreShield} satisfies $(\varepsilon,δ)$-DP for releasing similarity score vectors and Gram matrices. We provide utility guarantees for the exact Frobenius metric projection used in the risk analysis, and prove convergence to feasibility for the practical averaged alternating-projection solver used for large-scale Gram releases. For full pairwise cosine Gram release under record-level replacement adjacency, the exact-projection bound improves the $n$-dependence of squared Frobenius risk from $Θ(n^3)$ for the naïve Gaussian baseline to $\mathcal{O}(n^2)$ for fixed privacy parameters, with sharper local bounds at low-rank Grams. We evaluate the mechanism across RAG, face recognition, semantic retrieval, image similarity, and recommender-system tasks.

Common chunking strategies in Retrieval-Augmented Generation (RAG) systems often create redundant chunks. These redundant chunks make the vector database bigger and slow down retrieval. A common fix is cosine-similarity thresholding. This method reduces each chunk to a single vector, then compares vectors using a similarity score. But a single vector can lose the fine-grained, token-level detail needed to tell a true duplicate apart from a chunk that just shares the same topic. We propose Cross-Attention Calibrated Deduplication (CACD). CACD checks each new chunk against an in-memory pool of chunks already kept, using a cross-encoder instead of a single pooled vector. This keeps token-level detail all the way to the final comparison. CACD combines three parts: the cross-encoder comparison itself, a New Information Score (NIS) that measures how much of a chunk is not explained by a candidate already kept, and a majority vote across several candidates rather than a single best match. NIS is calculated from the attention entropy of the cross-encoder. We tested CACD against five existing filtering methods, nine chunking strategies, and 18 configurations, all on the full SQuAD 1.1 validation set. In our experiments, CACD removes 9.75% of chunks on average. This drop rate is close to other semantic-level methods, and much higher than exact-match filters, which barely remove anything. In these experiments, CACD also processes each configuration in 51.0 seconds on average, about 27% faster than the strongest baseline, NERExact (69.6s), and about 7x faster than cosine-similarity filtering (356.7s). These results come from a single dataset, so we present them as an early comparison, not a general claim. Code for the baseline evaluation and for CACD is available at https://github.com/lehuyphuong/rag_bench and https://github.com/lehuyphuong/cacd_dedup.

Scaled dot product attention conflates directional alignment and vector magnitude, limiting its effectiveness as a similarity metric in Transformer models. We introduce L1 augmented attention, a simple and computationally parallelizable modification that subtracts a learned, head specific L1 distance between queries and keys from the dot product score. This hybrid similarity captures complementary geometric information. Dot product rewards directional alignment, while L1 penalizes coordinate deviations. To reduce the cost of L1 computation, we project queries and keys into low dimensional subspaces whose parameters specialize to preserve informative L1 structure. Evaluated on WikiText 2 using a compact transformer, L1 augmented attention achieves up to a 14.5% reduction in perplexity over the original transformer baseline and outperforms an RBF L2 kernel. Analysis of norm variance and learned L1 weights reveals distinct geometric roles across layers and strong head level specialization. These results demonstrate that enriching attention with L1 geometry provides a principled and effective improvement to similarity computation in modern language models, with practical benefits for both accuracy and parallel efficiency.

We study whether \sigreg -- LeJEPA's anti-collapse objective -- can reshape representations during standard autoregressive language-model pretraining, and when the resulting geometry helps \kv-cache quantization. We train 110M-parameter models on 10B FineWeb tokens and report three findings. \textbf{(1)} At $λ{=}0.01$, \sigreg reduces hidden-state pairwise-cosine anisotropy by $38\%$ across three paired seeds. Perplexity increases by less than $0.35\%$ in every pair, with no consistent zero-shot loss. \textbf{(2)} This change does not propagate from hidden states to the \kv cache. Applying \sigreg directly to K and V during continued training, however, reduces mean cache anisotropy by $94\%$ across four checkpoints. A matched continuation without the \kv term leaves cache geometry nearly unchanged, and the frozen-trunk retrofits we tested do not reproduce the effect. \textbf{(3)} Under untransformed symmetric group-free quantization, direct \kv regularization is the only training condition that prefers per-channel scaling in all three seeds, and under that same 3-bit per-channel scheme the baseline incurs $4.3$--$7.9\times$ the directly regularized model's \dnll. Under the full simulated KIVI-style configuration (mixed arrangement, zero-points, grouped scales), however, all models reach near-parity, including when storage overhead is approximately matched. In this 110M regime, the training intervention helps when quantizer scales are coarse; the advantage vanishes under the tested combination of token-local grouping, mixed \kv scaling, and zero-points. To our knowledge this is the first training-time \emph{distributional} regularization of standard \kv-cache geometry evaluated against post-hoc cache quantization.

Contrastive embedding models trained with scale-invariant losses are typically paired with distance metrics like cosine similarity, effectively ignoring embedding magnitudes. However, surprisingly, empirical studies reveal that despite this, these "discarded" norms seem to correlate with semantic properties such as concept specificity, token frequency, and human uncertainty. In this work, we provide a formal theoretical framework explaining this phenomenon. By analyzing the optimization dynamics, we derive an analytic formula demonstrating that embedding length naturally encodes this information as a byproduct of the training process. We also show how this gives rise to signals that can serve as "free" calibration tools in specific models and retrieval tasks, providing a grounded explanation for a previously heuristic observation.

$\mathrm{Cl}(3,0)$ interatomic potentials, despite their algebraic elegance, predict force magnitudes accurately but force directions poorly. Across ten rMD17 molecules, every $L \leq 1$ baseline in our twelve-model study attains aggregate force-cosine similarity below $0.25$. The cause is structural. The geometric product of two vectors in $\mathbb{R}^3$ realises only the $L=0$ and $L=1$ components of its irreducible representation content, leaving the symmetric-traceless rank-2 component absent from the per-edge bilinear that drives each message-passing layer. We address this with CliffordSTF, which couples the Clifford multivector to closed-form symmetric-traceless tensor tracks at ranks two and three through bilinear cross-track contractions, using a single learned bilinear and no Clebsch--Gordan tables, Wigner-$D$ matrices, or e3nn calls. On rMD17, CliffordSTF raises aggregate force-cosine similarity from $0.055$ (base Clifford) to $0.551$, an order-of-magnitude relative directional gain, alongside improved magnitude accuracy (force MAE $15.8\%$ lower; energy MAE $10.9\%$ lower). It outperforms all CG-free or body-ordered baselines in our study (all $\leq 0.17$). On catalysis benchmarks, CliffordSTF achieves the best out-of-distribution S2EF energy MAE on OC22 in our experiments, and the best in-distribution energy MAE among $L \geq 2$ methods on OC22 IS2RE. An eleven-variant ablation shows the two tracks are complementary: neither alone matches the combined model.

The standard way to compare two text embeddings is cosine similarity. Scattered studies report that a different metric does better, but never pin down the geometric condition that decides when, or why. We settle both with a comprehensive empirical study: nineteen parameter-free similarity metrics on nineteen encoders, from compact sentence transformers up to seven-billion-parameter large language models, across seven datasets. The answer is geometric. When an encoder spreads its variance evenly across directions, cosine is the best parameter-free choice and no other metric helps by a usable margin. When the variance concentrates into a few dominant directions, a property known as anisotropy, rank-based and L1-type metrics beat cosine by a clear margin. The absolute gain is modest, but because cosine starts low on these encoders it is a sizable relative improvement, around twenty percent on average and largest where cosine is weakest. What decides this is the geometry of the embedding space, not how the model was trained: where the two disagree, the metric follows the geometry. One number, the fraction of variance held by the single most dominant dimension, predicts how much the alternatives help across all nineteen encoders, with a rank correlation of 0.86 and a linear correlation of 0.95. To test this as the cause rather than a correlate, we project out the dominant directions: cosine recovers and the advantage of the other metrics nearly vanishes, but only on the encoders that were anisotropic to begin with. The effect is directional, not magnitude based, since it survives normalizing every vector to unit length. Among parameter-free metrics, then, cosine is the right tool wherever an encoder is well spread, which includes the fine-tuned embedders commonly deployed for retrieval, and we give a one-number diagnostic for when it is not.

Long-horizon agents depend on context management: systems compress, summarize, and evict old tokens so tasks can continue beyond finite windows. That is safe only when dropped information is no longer needed or has been internalized. Plans are the stress case: they are written early, used for many steps, and first to be evicted. We introduce replay pairing, a diagnostic that runs the same trajectory with and without the plan in history and measures hidden-state cosine distance. On Llama-3.1-70B, plan signal spikes to 0.453 one step after the plan, then falls 4.1x in a single action-observation step; HotpotQA falls 12.4x. This is evidence that standard LLM agents do not carry plans forward as persistent state, and instead depend on the plan remaining in context. A layer-L32 probe detects this decay as a diagnostic, not as proof that it reads plan content itself. Reasoning models add a measurement confound: their <think> traces re-derive plan content, so standard stripping leaves plan evidence in the stripped condition. We name this the reasoning-trace confound and fix it with strict stripping, which removes prior <think> blocks from the stripped run only. It recovers +163% of the step+1 signal in-sample and +153% held out, while not meaningfully changing non-reasoning Llama (+4.8%). On DeepSeek-R1-Distill-Llama-70B, a Llama-trained probe transfers at AUROC 0.748 (p=6e-4), while R1-specific probes reach 1.000, suggesting R1 encodes plan signal in a different hidden-state direction. Finally, a compression stress test shows the practical cost: naive plan eviction cuts ALFWorld success by 34.7pp, while probe-gated re-surfacing does not recover it. The contribution is a measurement and stress-test framework showing that agent-critical information can be context-resident rather than persistent. Context management is load bearing, but plan protection alone is not enough.

Linear probes are widely used in interpretability research and often compared by cosine similarity. The Mahalanobis cosine similarity (MCS) between two directions, which reweights the inner product by test data covariance, is a natural task-aware refinement. Ying et al. (2026) report that a probe's MCS to a reference probe trained on the out-of-distribution (OOD) data near-perfectly linearly predicts the probe's OOD AUROC (R^2 = 0.98). Here, we extend this empirical finding across models, layers, and concept domains, and prove this general phenomenon in closed form: For balanced classes whose projections are Gaussian, OOD AUROC and MCS to the reference probe are linear because both are sigmoid-shaped functions of the probe's signal-to-noise ratio (SNR) on the test data. The theory also predicts when this linearity fails, which we verify empirically. MCS offers a theoretically grounded and empirically effective alternative to Euclidean cosine similarity for comparing linear probes.

We investigate whether task-vector arithmetic, successful for cross-speaker emotional intensity control in modular text-to-speech (TTS), transfers to large-scale TTS systems built on language-model backbones with in-context learning (LM-TTS). Through a systematic elimination study over four progressively narrower operands on Qwen3-TTS-12Hz-1.7B - model weights via LoRA fine-tuning, continuous codec embeddings, discrete codec tokens, and the speaker embedding (x-vector) produced by an ECAPA-TDNN encoder jointly trained with the synthesis backbone - we localize the dominant carrier of emotional prosody to the x-vector. Building on this finding, we propose a training-free method based on centroid arithmetic in x-vector space: an emotion direction $τ= \mathbb{E}_i[x(s_i,\text{emo})] -\mathbb{E}_i[x(s_i,\text{neutral})]$ applied to an unseen target speaker as $x_{\text{new}} = x(\text{target},\text{neutral}) + α\cdotτ$. Using ESD (English) as the $τ$ source and emoUERJ (Brazilian Portuguese) as a cross-lingual ground-truth target, we observe average gains of $+0.29$ in emotion2vec cosine over the ICL baseline on English held-out speakers and $+0.09$ on Brazilian Portuguese held-out speakers, while largely preserving identity (WavLM SECS $\gtrsim 0.88$ for the multi-speaker $τ$ variant) and intelligibility (WER $\approx 0$ in PT-BR). These results offer initial evidence that the dominant carrier of emotional prosody in this class of models is localizable, by elimination, to the co-trained speaker embedding, where training-free centroid arithmetic remains effective even under cross-lingual transfer.

We introduce Adaptive Calibration (AC), a novel calibration strategy for facial recognition that maps cosine similarity between normalized embeddings to well-calibrated probabilities. By incorporating local context into calibration, Adaptive Calibration corrects for a fundamental mismatch in cosine similarity, whereby the same distance can correspond to different match probabilities in different embedding regions. Our approach improves both overall performance and results in a fairer calibration without requiring demographic metadata. Our approach often improves worst-group ranking and probability quality across a variety of pretrained models and standard benchmarks, with gains in low-FPR verification performance depending on the setting. AC provides a practical solution for equitable facial recognition, without requiring demographic group annotations, and while improving overall performance. Our method provides continuous, region-specific calibration, and we examine per-group performance to assess "leveling down" where fairness comes at the cost of degraded performance for some groups.

Clark Hash is a small method for storing neural embeddings in less space. It normalizes each database vector, applies a deterministic sparse signed Johnson-Lindenstrauss projection, clips the result, and stores a fixed-width scalar-quantized code. Queries stay in floating point and are scored against the stored sketches. In the default 384-dimensional sentence-embedding setting, Clark Hash stores a cosine-search vector in 48 bytes instead of 1536 bytes for dense f32 storage. This is 32x smaller. The method does not need a training pass, learned codebooks, rotations, or corpus statistics before new vectors can be stored. We describe the codec, the Rust implementation, and a multilingual sentence-similarity evaluation on 9,304 labeled pairs from 29 subsets. With a multilingual MiniLM encoder, the 48-byte sketches reached 0.910 and 0.946 macro Pearson correlation with dense cosine scores on STS17 and STS22. Clark Hash is not a new Johnson-Lindenstrauss theorem and it is not a replacement for approximate nearest-neighbor indexes. It is a simple stateless codec for compact embedding storage.

We introduce XFP, a dynamic weight quantizer for LLM inference that inverts the conventional workflow: the operator specifies reconstruction quality floors on per-channel cosine similarity (one strict floor for attention and shared experts, one lazy floor for routed-expert MoE); XFP determines codebook size, outlier budget, and packing per layer automatically -- no Hessian, no calibration data, no manual bit-width selection. Each weight matrix is decomposed into a sparse fp16 outlier residual and a dense sub-byte index tensor into a per-group learned codebook. Two storage modes share one auto-select frontend and one fused decode kernel: V2 (per-channel Lloyd) and V2a (shared library of L=32 codebooks per layer). On Qwen3.5-122B-A10B under V2, XFP reaches 138 tok/s single-stream decode on workstation hardware (RTX PRO 6000 Blackwell, TP=2) at 94.49% GSM8K strict-match (3 seeds, n=3957), and is 49% faster than Marlin INT4 at TP=1. For models that do not fit in the target memory envelope, we present the H-Process: a quality-driven iteration over the two cosine thresholds that finds the operating point at which the model just fits while still producing sensible output. Three constraints define its search space: the operator-set thresholds, an OOM boundary at quantize-on-load, and a garbage boundary in generation (cosine similarity steers; benches verify). On Qwen3.5-397B-A17B (512 routed experts/layer), the H-Process fits the full expert population into 2x96 GB at ~3.4 effective bits and delivers 100.9 tok/s long-output decode at 66.72% GSM8K strict-match on the full 1319-problem set (single seed at submission; multi-seed evaluation in progress), exceeding INT4 with routed-expert pruning on memory, throughput, and accuracy simultaneously.

Large language models (LLMs) have revolutionized natural language processing. Understanding their internal mechanisms is crucial for developing more interpretable and optimized architectures. Mechanistic interpretability has led to the development of various methods for assessing layer relevance, with cosine similarity being a widely used tool in the field. On this work, we demonstrate that cosine similarity is a poor proxy for the actual performance degradation caused by layer removal. Our theoretical analysis shows that a layer can exhibit an arbitrarily low cosine similarity score while still being crucial to the model's performance. On the other hand, empirical evidence from a range of LLMs confirms that the correlation between cosine similarity and actual performance degradation is often weak or moderate, leading to misleading interpretations of a transformer's internal mechanisms. We propose a more robust metric for assessing layer relevance: the actual drop in model accuracy resulting from the removal of a layer. Even though it is a computationally costly metric, this approach offers a more accurate picture of layer importance, allowing for more informed pruning strategies and lightweight models. Our findings have significant implications for the development of interpretable LLMs and highlight the need to move beyond cosine similarity in assessing layer relevance.

Raw cosine in the 768-dimensional output space of the Contrastive Style Descriptor (CSD) is now widely read as an absolute, calibrated style-fidelity score for text-to-image and style-imitation evaluation. We introduce the discrimination gap, a corpus-internal, prototype-free and threshold-free diagnostic that tests whether contrastive style cosines admit an absolute same-versus-different interpretation on a candidate artist corpus. On a 1799-artwork, 91-artist public-domain corpus, raw CSD cosine yields negative point-estimate gaps for $23/91$ artists at the pairwise level ($2/91$ robust under bootstrap) and for $15/91$ in the aggregated-pool scoring regime style-fidelity evaluations typically use. CSLS readout on the frozen backbone reduces the aggregated negative-gap count to $4/91$; combined with positional-embedding interpolation to $336$ pixels it raises unsupervised pair-verification AUC from $0.883$ to $0.905$ across $25$ artist-disjoint splits. We refer to this diagnostic-driven readout protocol on the frozen backbone (CSLS as default, pos-interp $336$ as the stronger optional setting) as CSD+, not a new encoder.A cross-backbone check on CLIP-ViT-L/14, SigLIP-large and DINOv2-Large reproduces the same shared-tradition failure pattern, providing evidence that the residual reflects a shared limitation of the four backbones we tested rather than a CSD-specific artefact. Practical implication: before reporting CSD cosine as an absolute style-fidelity score, run the diagnostic on the candidate corpus; CSLS is the minimal correction when it fails.

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.

Reinforcement learning fine-tuning has become the dominant approach for aligning diffusion models with human preferences. However, assessing images is intrinsically a multi-dimensional task, and multiple evaluation criteria need to be optimized simultaneously. Existing practice deal with multiple rewards by training one specialist model per reward, optimizing a weighted-sum reward $R(x)=\sum_k w_k R_k(x)$, or sequentially fine-tuning with a hand-crafted stage schedule. These approaches either fail to produce a unified model that can be jointly trained on all rewards or necessitates heavy manually tuned sequential training. We find that the failure stems from using a naive weighted-sum reward aggregation. This approach suffers from a sample-level mismatch because most rollouts are specialist samples, highly informative for certain reward dimensions but irrelevant for others; consequently, weighted summation dilutes their supervision. To address this issue, we propose MARBLE (Multi-Aspect Reward BaLancE), a gradient-space optimization framework that maintains independent advantage estimators for each reward, computes per-reward policy gradients, and harmonizes them into a single update direction without manually-tuned reward weighting, by solving a Quadratic Programming problem. We further propose an amortized formulation that exploits the affine structure of the loss used in DiffusionNFT, to reduce the per-step cost from K+1 backward passes to near single-reward baseline cost, together with EMA smoothing on the balancing coefficients to stabilize updates against transient single-batch fluctuations. On SD3.5 Medium with five rewards, MARBLE improves all five reward dimensions simultaneously, turns the worst-aligned reward's gradient cosine from negative under weighted summation in 80% of mini-batches to consistently positive, and runs at 0.97X the training speed of baseline training.

Sparse autoencoders (SAEs) have become central to mechanistic interpretability, decomposing transformer activations into monosemantic features. Yet existing analyses characterise features almost exclusively through top-activating token lists or decoder weight vectors, leaving the higher-order co-occurrence structure shared across features largely unexamined. We introduce a graph-structured representation in which each SAE feature is modelled as a token co-occurrence graph: nodes are the tokens most frequent near strong activations, and edges connect pairs that co-occur within local context windows. A custom WL-style, frequency-binned graph kernel then provides a similarity measure over this structural space. Applied as a proof of concept to features from a large SAE trained on GPT-2 Small and probed with a synthetic mixed-domain corpus, our clustering recovers heuristic motif families (punctuation-heavy patterns, language and script clusters, and code-like templates) that are not recovered by clustering on decoder cosine similarity. A token-histogram baseline achieves higher overall purity, so the contribution of the graph view is complementary rather than dominant: it surfaces structural relationships that token-frequency and decoder-weight views alone do not capture. Cluster assignments are stable across graph-construction hyperparameters and random seeds.

While representation and similarity learning have improved the sample efficiency of Reinforcement Learning (RL), they are rarely used to shape policy updates directly in the action space. To bridge this gap, a geometry-aware RL algorithm that explicitly incorporates value-based similarity into the policy update, State-Action Value Geometry Optimization (SAVGO), is proposed. In detail, SAVGO learns a joint state-action embedding space in which pairs with similar action-value estimates exhibit high cosine similarity, while dissimilar pairs are mapped to distinct directions. This learned geometry enables the generation of a similarity kernel over candidate actions sampled at each update, allowing policy improvement to be guided directly toward higher-value regions beyond local gradient-based updates. As a result, representation learning, value estimation, and policy optimization are unified within a single geometry-consistent objective, while preserving the scalability of off-policy actor-critic training. The proposed method is evaluated on standard MuJoCo continuous-control benchmarks, demonstrating improvements over strong baselines on challenging high-dimensional tasks. Ablation studies are done to analyze the contributions of value-geometry learning and similarity-based policy updates.

We show that the geometric relations between semantic features in large language models' hidden states closely mirror human psychological associations. We construct feature vectors corresponding to 360 words and project them on 32 semantic axes (e.g. beautiful-ugly, soft-hard), and find that these projections correlate highly with human ratings of those words on the respective semantic scales. Second, we find that the cosine similarities between the semantic axes themselves are highly predictive of the correlations between these scales in the survey. Third, we show that substantial variance across the 32 semantic axes lies on a low-dimensional subspace, reproducing patterns typical of human semantic associations. Finally, we demonstrate that steering a word on one semantic axis causes spillover effects on the model's rating of that word on other semantic scales proportionate to the cosine similarity between those semantic axes. These findings suggest that features should be understood not only in isolation but through their geometric relations and the meaningful subspaces they form.

In this work, we developed and tested 3 techniques for vector quantization (VQ) based model weight compression. To mitigate codebook collapse and enable end-to-end training, we adopted cosine similarity-based assignment. Building on ideas from attention-based formulations in Differentiable K-Means (DKM), we further improved this approach by using cosine similarity for assignment combined with top-1 sampling and a straight-through estimator, thereby eliminating the need for weighted-average reconstruction. Finally, we investigated the use of differentiable neural architecture search (NAS) to adaptively select layer-wise quantization configurations, further optimizing the compression process. Although our method does not consistently outperform existing approaches across all quantization levels, it provides useful insights into the design trade-offs and behaviors of VQ-based model compression methods.

Symmetric objects are common in daily life and industry, yet their inherent orientation ambiguities that impede the training of deep learning networks for pose estimation are rarely discussed in the literature. To cope with these ambiguities, existing solutions typically require the design of specific loss functions and network architectures or resort to symmetry-invariant evaluation metrics. In contrast, we focus on the numeric representation of the rotation itself, modifying trigonometric identities with the degrees of symmetry derived from the objects' shapes. We use our representation, SARR, to obtain canonic (symmetry-resolved) poses for the symmetric objects in two popular 6D pose estimation datasets, T-LESS and ITODD, where SARR is unique and continuous w.r.t. the visual appearance. This allows us to use a standard CNN for 3D orientation estimation whose performance is evaluated with the symmetry-sensitive cosine distance $\text{AR}_{\text{C}}$. Our networks outperform the state of the art using $\text{AR}_{\text{C}}$ and achieve satisfactory performance when using conventional symmetry-invariant measures. Our method does not require any 3D models but only depth, or, as part of an additional experiment, texture-less RGB/grayscale images as input. We also show that networks trained on SARR outperform the same networks trained on rotation matrices, Euler angles, quaternions, standard trigonometrics or the recently popular 6d representation -- even in inference scenarios where no prior knowledge of the objects' symmetry properties is available. Code and a visualization toolkit are available at https://github.com/akriegler/SARR .