Embedding Models Measure in Peculiar Ways

Core Question§

Text embeddings are widely assumed to induce a meaningful geometry where cosine similarity tracks semantic equivalence. This paper interrogates that assumption specifically for physical measurements: quantities that possess a unique, objective ground-truth notion of equality and ratio distance (e.g., 1 kg == 1000 g, and 2 m is twice 1 m). The authors ask whether embedding spaces reproduce this objective structure.

Methodology§

For a quantity such as mass, each measurement can be canonically expressed in base SI units as a scalar magnitude plus a unit, e.g. $x = (m, u)$, normalized to $m_{\text{SI}}$. An objective similarity requires:

$$ \text{sim}_{\text{obj}}(x, y) = f\big(|m_x^{\text{SI}} - m_y^{\text{SI}}|\big) $$

with equal physical quantities mapping to identical embeddings (up to noise). The authors instead measure the learned similarity via cosine similarity of encoder representations:

$$ \text{sim}_{\text{emb}}(x, y) = \frac{E(x)^\top E(y)}{\lVert E(x)\rVert \, \lVert E(y)\rVert} $$

where $E$ is a sentence/embedding encoder over strings like "2 kg", "2000 g", "two kilograms". They compare across controlled sets of physically equivalent strings, ratio-related magnitudes, and unrelated-but-string-similar pairs, across mass, distance, time, and volume.

Key Findings§

• Physical measurement is only weakly modeled: embeddings of physically equal quantities are often far apart, while physically very different quantities can be close.
• Similarity appears strongly driven by superficial string similarity (shared tokens, characters, formatting) rather than encoded magnitude.
• Recalibration (e.g., isotonic/linear rescaling of similarity scores to better fit objective distances) does not substantially improve alignment, suggesting the deficiency is representational, not merely a monotone-rescaling artifact.

Implementation Sketch§

The experiment is essentially a probe over an encoder, scoring how well embedding geometry covaries with an objective metric.

import torch
import torch.nn.functional as F
from sentence_transformers import SentenceTransformer

model = SentenceTransformer("all-MiniLM-L6-v2")

def sim_pairs(a: list[str], b: list[str]):
    ea = model.encode(a, convert_to_tensor=True, normalize_embeddings=True)
    eb = model.encode(b, convert_to_tensor=True, normalize_embeddings=True)
    return (ea * eb).sum(-1)  # cosine sim per pair

# Physically equal but string-different pairs
A = ["2 kg", "1000 g", "one kilometre", "60 seconds"]
B = ["2000 g", "1 kilogram", "1000 m", "1 minute"]

sims = sim_pairs(A, B)
# Objective ground truth would be ~1.0 for all; measure gap empirically
print(sims)

# Recalibration: isotonic regression mapping sim -> objective ratio distance
from sklearn.isotonic import IsotonicRegression
ir = IsotonicRegression(out_of_bounds="clip")
sim_vals, obj_dist = [], []  # populate via full sweep
ir.fit(sim_vals, obj_dist)
print("Alignment after recalibration:", ir.score(sim_vals, obj_dist))

Why It Matters§

This is a cautionary result for anyone building retrieval, RAG, or reasoning pipelines that assume embedding distance is a proxy for physical or numeric truth. Superficial string overlap can produce misleading geometry for quantitative reasoning, and simple post-hoc score recalibration is insufficient. Robust numerical grounding likely requires architectural or training changes (e.g., explicit unit-aware encoders) rather than fixes at the similarity-scoring layer.