Embedding spaces define notions of semantic similarity and distance. We study whether those embeddings reflect physical measurements of mass, distance, time and volume, which admit a unique, objective notion of semantic equivalence and distance. We find that physical measurement is only weakly modeled in the embedding space, and that instead quite peculiar measurement patterns can be observed. Further analysis indicates that embedding representations of physical measurements are strongly influenced by superficial string similarity, and recalibration of similarity does not substantially improve the alignment.
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.
In this paper, we show that high-performing embedding models organize their embedding spaces in a consistent way. We evaluate 25 contemporary embedding models on five MTEB tasks spanning four diverse task categories (retrieval, bitext mining, pair classification, and summarization) in both English and multilingual settings, and reveal that nearest-neighbor overlap and magnitude differences in independent component analysis (ICA) between paired text instances strongly correlate (even up to 0.97) with performance on the given task. Ultimately, we show that embedding tasks display varying degrees of linearity and reliance on retention of local information. Our results further the understanding of embeddings, their relation to model performance, and shed light on possible future training objectives and optimizing conditional embeddings.
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.