Title: Why Mean Pooling Works: Quantifying Second-Order Collapse in Text Embeddings ArXiv: 2604.27398 Authors: Tomomasa Hara, Hiroto Kurita, Masaaki Imaizumi, Kentaro Inui, Sho Yokoi Sections: 113 Estimated tokens: 26.2k ## Contents - 1 Introduction - 2 Related Work - 2.1 Text Embeddings via Mean Pooling - 2.2 Text Representations Preserving Higher-Order Statistics - 3 Preliminaries - 4 Second-Order Collapse by Mean Pooling - 4.1 When Does Collapse Arise? - When Collapse Arises - When Collapse Does Not Arise - 4.2 Quantify the Severity of Collapse - 4.2.1 Definition of SOCM - 4.2.2 Decomposing Distributional Distance - Decomposition - Motivation for W 2 2 W_{2}^{2} - 4.2.3 Validity of SOCM Form - Desirable Properties - (a) When Collapse Arises - (b) When Collapse Does Not Arise - (c) Monotonicity in d μ d_{\mu} - (d) Monotonicity in d Σ d_{\Sigma} - (e) Interaction between d μ d_{\mu} and d Σ d_{\Sigma} - Form of SOCM Satisfying the Properties - 5 Experiment - 5.1 Experimental Procedure and Setting - Overview - Normalization of Token Embedding Lists - Dataset - Model - 5.2 Results - Quantitative Analysis - Qualitative Analysis - 6 How Do Fine-Tuned Text Encoders Avoid Collapse? - 6.1 Theoretical Study - Assumption 1 (Setting) . - Definition 1 (Spread expansion/contraction under 𝑨 \bm{A} , 𝑾 v \bm{W}^{v} , and 𝑾 o \bm{W}^{o} ) . - Intuition . - Definition 2 (Ratio of token spread in input 𝑯 \bm{H} to the scale of the residual output 𝒀 \bm{Y} ) . - Intuition . - Definition 3 (Relative spread ratio of 𝑿 \bm{X} to 𝒀 \bm{Y} through transformation 𝒈 \bm{g} ) . - Intuition . - Theorem 1 (Token embeddings can become concentrated within each text) . - Intuition . - Sketch of proof. - Theorem 2 (Token embedding concentration leads to low SOCM) . - Intuition . - Sketch of proof. - 6.2 Empirical Study - Setting - Self-Attention and Projection ( λ \lambda ) - Residual Connection ( r r ) - Per-Token Transformation ( C C ) - Token Embedding Concentration - Why Concentration Occurs in Fine-Tuned Encoders - Connection to Prior Work - 7 Correlation with Downstream Task Performance - Setting - Result - Discussion - 8 Conclusion - Future Work - Limitations - Restriction to Second-order Statistics - Assumptions in the Proposed Metric - Toward Theoretical Understanding - Practical Utility of SOCM - Scope Limited to Text Embeddings - Focus on Mean Pooling - Ethical Consideration - Acknowledgments - References - Appendix A Gaussian Characterization - Advantages of the Gaussian Characterization - Information Captured by First- and Second-order Statistics - Appendix B Proof that SOCM Satisfies Desirable Properties - B.1 Proof of Property (a) - B.2 Proof of Property (b) - B.3 Proof of Property (c) - B.4 Proof of Property (d) - B.5 Proof of Property (e) - B.6 Generalization - Appendix C Proof of Normalization Property - Statement - Proof - Appendix D Implementation Details - D.1 Dataset Details - Preprocessing - Dataset URLs - Language - D.2 Model Details - Appendix E Additional Results for § 5 - Experimental Setting - Results - Appendix F Proof of Theorem 1 - Appendix G Proof of Theorem 2 - Appendix H Additional Results for § 6 - H.1 Results for Additional Models - H.2 Within-Text Token Embedding Concentration - Appendix I Analysis of d μ d_{\mu} and d Σ d_{\Sigma} - Observations - Connection to Token Embedding Concentration - Appendix J Trace Bound under Normalization - J.1 Setup and Notation - J.2 Assumptions - Assumption 1 - Assumption 2 - J.3 Proof - Relating normalized and unnormalized covariance matrices - Computing the trace of unnormalized covariance - Computing the squared norm of the mean - Deriving the expression for the trace of normalized covariance - Bounding the trace - Appendix K Computational Resources - Appendix L Use of AI Assistants ## Abstract Abstract For constructing text embeddings, mean pooling, which averages token embeddings, is the standard approach. This paper examines whether mean pooling actually works well in real models. First, we note that mean pooling can collapse information beyond the first-order statistics of the token embeddings, such as second-order statistics that capture their spatial structure, potentially mapping distinct token embedding distributions to similar text embeddings. Motivated by this concern, we propose a simple metric to quantify such a collapse induced by mean pooling. Then, using this metric, we empirically measure how often this collapse occurs in actual models and texts, and find that modern text encoders are robust to this collapse. In particular, contrastive fine-tuned text encoders tend to be less prone to the collapse than their pretrained backbone models. We also find that the robustness of these text encoders lies in the concentration of token embeddings within each text. In addition, we find that robustness to the collapse, as quantified by our proposed metric, correlates with downstream task performance. Overall, our findings offer a new perspective on why modern text encoders remain effective despite relying on seemingly coarse mean pooling. ## 1 Introduction Figure: Figure 1: Overview of this work. Top: Mean pooling can map distinct token embedding distributions to similar text embeddings. This is because mean pooling summarizes distributions using only their first-order statistics, collapsing higher-order statistics. Bottom: We empirically find that modern fine-tuned text encoders are robust to such a collapse (§ [5](#S5)). Each panel visualizes token and text embeddings for two texts, output by BERT Devlin2019-mb and GTE${}_{\text{base}}$ Li2023-oy via PCA projection without centering. This example was discovered using the metric in § [4](#S4). Refer to caption: 2604.27398v1/x1.png Text embeddings, which represent sentences, paragraphs, and documents as single vectors, are used across a wide range of NLP tasks Enevoldsen2025-hc, including information retrieval Thakur2021-ac and automatic evaluation Rei2020-pf. Also, text embeddings have become essential for modern retrieval-augmented generation Lewis2020-gr applications. Given their widespread use, understanding text embeddings remains an important open challenge, and prior work has approached this problem from the perspectives of geometry Xiao2023-vu and dimensionality Takeshita2025-eg. This paper focuses on the aggregation method that constructs embeddings from token-level representations. As such, a standard aggregation method is mean pooling, which averages token embeddings. This simple aggregation has offered empirical advantages across embedding methods, from classical static token embeddings Wieting2015-fd; Shen2018-gg to modern contextualized embeddings from Transformer Vaswani2017-kx encoders Reimers2019-gc. Indeed, some state-of-the-art text encoders use mean pooling Lee2025-yl; Nussbaum2025-ys. While mean pooling is standard, does it actually work well in real models? Figure [1](#S1.F1) (top) illustrates a potential collapse mode: even when two texts yield distinct token embeddings in the embedding space ([$\bullet$$,\cdots,$$\bullet$] $\neq$ [$\bullet$$,\cdots,$$\bullet$]), mean pooled text embeddings may become indistinguishable ($\bigstar$ $\approx$ $\bigstar$). This is because the mean-pooled text embedding corresponds only to the first-order statistic (mean) of the token embedding distribution. Thus, mean pooling does not capture the spatial structure of token embeddings, as reflected in second- and higher-order statistics, such as the covariance matrix. Turning to text embeddings via mean pooling, several prior works have proposed alternative approaches to mean-pooled text embeddings that represent a text as a list of token embeddings prior to mean pooling, such as optimal transport between token embedding lists Kusner2015-ol; Zhao2019-mv; Yokoi2020-qw; Lee2022-lo (§ [2.2](#S2.SS2)). Nevertheless, mean-pooled text embeddings remain dominant in practice, as they are both computationally efficient and empirically effective. This potential collapse raises the question: do actual models suffer from it, or do they avoid it? To address this question, we first propose a framework to evaluate the robustness of a text encoder to such a collapse. Specifically, we introduce the degree of Second-Order Collapse by Mean pooling, a metric that quantifies the severity of the second-order statistics collapse by mean pooling (§ [4](#S4)). Next, using this proposed metric, we evaluate how well mean pooling works in actual models and texts. Our empirical results suggest that mean pooling works well in modern text encoders. In particular, we make the following findings: (i) We find that contrastive fine-tuned Transformer text encoders tend to be less prone to the collapse than their pretrained backbone models, suggesting that these fine-tuned encoders are robust to the collapse (§ [5](#S5), Figure [1](#S1.F1) bottom). (ii) We also find that this robustness lies in the concentration of token embeddings within each text, which arises through the Transformer layer (§ [6](#S6)). (iii) We further observe that the proposed metric correlates with downstream task performance, suggesting that robustness to the collapse can be one factor in the success of modern text encoders on downstream tasks (§ [7](#S7)). Overall, by rethinking mean pooling, which may appear to be a coarse aggregation method, this paper offers a new perspective on the effectiveness of modern text encoders. ## 2 Related Work This paper focuses on the coarseness of mean pooling in modern text embeddings, namely, its loss of higher-order statistics of token embeddings. Text encoders with mean pooling have become a standard paradigm (§ [2.1](#S2.SS1)). Meanwhile, as an alternative paradigm, methods that preserve higher-order statistics without pooling have also been proposed (§ [2.2](#S2.SS2)); yet modern text encoders achieve empirical performance competitive with these methods. We offer one perspective on why modern text encoders work well by rethinking mean pooling. ### 2.1 Text Embeddings via Mean Pooling In recent NLP, mean pooling over token embeddings from contrastive fine-tuned Transformer encoders has become a widely used paradigm for text embeddings Reimers2019-gc; Gao2021-ds; Wang2022-gw; Li2023-oy; Nussbaum2025-ys; Lee2025-yl. This popularity likely reflects its compatibility with contrastive fine-tuning Gao2021-ds and computational efficiency, including low memory use T2024-uo and compatibility with approximate nearest neighbor search Douze2025-wj. Beyond these engineering motivations, this paper offers a new perspective on mean pooling effectiveness, highlighting that the collapse of higher-order statistics is limited in modern text encoders. ### 2.2 Text Representations Preserving Higher-Order Statistics As an alternative paradigm to mean pooling, methods that handle text representations while preserving higher-order statistics have also been proposed. One approach is to work directly with the list of token embeddings before pooling. Distances of such lists have been computed using optimal transport, which measures the minimum cost of transporting one list to another Kusner2015-ol; Zhao2019-mv; Yokoi2020-qw; Lee2022-lo. Similarly, ColBERT Khattab2020-vk; Santhanam2022-ex measured the distance of token embedding lists by matching each query token to its most similar document token and aggregating the resulting score. BERTScore Zhang2019-fv also aggregated token-level similarities into an F-score-like metric. As another direction, Sen2Pro Shen2023-zb represents each text as a probability distribution over text embeddings by sampling multiple embeddings via MC dropout and data augmentation. GaussCSE Yoda2024-xu directly predicts first- and second-order statistics to represent each text as a Gaussian distribution. Meanwhile, modern text encoders with mean pooling are not only computationally lighter than these methods, but also achieve competitive empirical performance Lee2022-lo. This paper investigates why text encoders remain effective even when representing texts via simple mean pooling. ## 3 Preliminaries In this paper, we quantify how mean pooling, by using only the first-order statistics of the original token embeddings, collapses its second-order statistics (§ [4](#S4)). In this section, we formally introduce token embeddings and these statistics. We consider constructing text embeddings for two texts $t_{1}$ and $t_{2}$ using a model $\bm{f}$. Given $t_{i}$, $\bm{f}$ outputs a token embedding list $\bm{X}_{i}$: $$ $\displaystyle\bm{X}_{i}:=[\bm{x}_{i,1},\cdots,\bm{x}_{i,n_{i}}\,]=\bm{f}(t_{i})\in\mathbb{R}^{d\times n_{i}}.$ (1) $$ Here, $\bm{x}_{i,j}\in\mathbb{R}^{d}$ is the $j$-th token embedding in $t_{i}$, $d$ is the embedding dimension, and $n_{i}$ is the number of tokens in $t_{i}$. Mean pooling averages token embeddings to construct a text embedding $\bm{\mu}(\bm{X}_{i})\in\mathbb{R}^{d}$: $$ $\bm{\mu}(\bm{X}_{i}):=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\bm{x}_{i,j}.$ (2) $$ This text embedding $\bm{\mu}(\bm{X}_{i})$ corresponds to the first-order statistic of $\bm{X}_{i}$ when viewed as an empirical distribution in the embedding space. Meanwhile, the token embedding distribution also has a second-order statistic $\bm{\Sigma}(\bm{X}_{i})\in\mathbb{R}^{d\times d}$: $$ $\displaystyle\bm{\Sigma}(\bm{X}_{i})$ (3) $\displaystyle:=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}(\bm{x}_{i,j}-\bm{\mu}(\bm{X}_{i}))(\bm{x}_{i,j}-\bm{\mu}(\bm{X}_{i}))^{\top}.$ $$ This second-order statistic captures the spatial structure of token embeddings. For simplicity, we focus on it as the lowest-order statistic not retained by mean pooling. Based on these definitions, we quantify the collapse induced by mean pooling (§ [4](#S4)) and examine how often it arises in practice (§ [5](#S5)). ## 4 Second-Order Collapse by Mean Pooling In this section, we provide both intuitive and formal characterizations of collapse by mean pooling, based on first- and second-order statistics. We first provide an intuitive understanding of when this collapse occurs by examining the similarity of first- and second-order statistics (§ [4.1](#S4.SS1)). Based on this intuition, we then introduce a metric to quantify the severity of this collapse (§ [4.2](#S4.SS2)). ### 4.1 When Does Collapse Arise? Figure: Figure 2: When mean pooling collapse arises (red) and does not (green). Mean pooling collapse occurs when the first-order statistics are similar while the second-order statistics differ. Each panel depicts a conceptual distribution in a two-dimensional embedding space, characterized by its first- and second-order statistics. Refer to caption: 2604.27398v1/x2.png As discussed in § [1](#S1), mean pooling may collapse two distinct token embedding distributions into similar text embeddings. We characterize whether this collapse arises based on the similarity of their first-order statistics and that of their second-order statistics, as shown in Figure [2](#S4.F2). ##### When Collapse Arises The collapse arises when first-order statistics are similar but second-order ones differ, i.e., $\bm{\mu}(\bm{X}_{1})\approx\bm{\mu}(\bm{X}_{2})\land\bm{\Sigma}(\bm{X}_{1})\neq\bm{\Sigma}(\bm{X}_{2})$. In this case, distributions that are distinct up to the second-order statistics may collapse into similar representations after mean pooling. ##### When Collapse Does Not Arise We also consider when the collapse does not arise. One case is when first-order statistics differ, i.e., $\bm{\mu}(\bm{X}_{1})\neq\bm{\mu}(\bm{X}_{2})$. Here, the distributions are separated by their means unless the variance is excessively large. Another case is when both first- and second-order statistics are similar, i.e., $\bm{\mu}(\bm{X}_{1})\approx\bm{\mu}(\bm{X}_{2})$ and $\bm{\Sigma}(\bm{X}_{1})\approx\bm{\Sigma}(\bm{X}_{2})$. Here, the distributions are similar, as reflected in their means. ### 4.2 Quantify the Severity of Collapse Based on these intuitions, we introduce the degree of Second-Order Collapse by Mean pooling (hereafter referred to as SOCM), a metric that quantifies the collapse of second-order statistics by mean pooling. As discussed in § [4.1](#S4.SS1), whether the collapse by mean pooling arises depends on the similarity of first-order statistics and the similarity of second-order statistics. Thus, we define SOCM using the distance of the first-order statistics and the distance of the second-order statistics (§ [4.2.1](#S4.SS2.SSS1)). Specifically, we decompose the distance of token embedding distributions into first- and second-order components and construct SOCM from them (§ [4.2.2](#S4.SS2.SSS2)). We also design SOCM to satisfy desirable properties such as boundary conditions and monotonicity (§ [4.2.3](#S4.SS2.SSS3)). #### 4.2.1 Definition of SOCM Given two token embedding lists $\bm{X}_{1}$ and $\bm{X}_{2}$, SOCM quantifies the severity of the collapse when applying mean pooling to them. Specifically, SOCM is defined as: $$ $\mathrm{SOCM}(d_{\mu},d_{\Sigma}):=(1-d_{\mu})d_{\Sigma}.$ (4) $$ Here, $d_{\mu}$ is the distance of the first-order statistics $\bm{\mu}(\bm{X}_{1})$ and $\bm{\mu}(\bm{X}_{2})$, and $d_{\Sigma}$ is the distance of the second-order statistics $\bm{\Sigma}(\bm{X}_{1})$ and $\bm{\Sigma}(\bm{X}_{2})$. $d_{\mu}$ is the scaled squared Euclidean distance: $$ $d_{\mu}:=\|\bm{\mu}(\bm{X}_{1})-\bm{\mu}(\bm{X}_{2})\|_{2}^{2}/4.$ (5) $$ We assume unit-norm means, $\|\bm{\mu}(\bm{X}_{i})\|_{2}=1$, as is common in the use cases of text embeddings Enevoldsen2025-hc. Under this normalization, $d_{\mu}\in[0,1]$(^1^11A factor $1/4$ rescales $d_{\mu}$ to $[0,1]$.) and corresponds to a common distance for text embeddings Enevoldsen2025-hc. $d_{\Sigma}$ is defined as the scaled Bures Wasserstein distance Dowson1982-sq: $$ $\displaystyle d_{\Sigma}:=\mathrm{tr}\bigl(\bm{\Sigma}(\bm{X}_{1})+\bm{\Sigma}(\bm{X}_{2})$ $\displaystyle-2\bigl(\bm{\Sigma}(\bm{X}_{1})^{1/2}\bm{\Sigma}(\bm{X}_{2})\bm{\Sigma}(\bm{X}_{1})^{1/2}\bigr)^{1/2}\bigr)/4.$ (6) $$ We assume $\mathrm{tr}(\bm{\Sigma}(\bm{X}_{i}))\leq 2$ to put $d_{\Sigma}$ in the same range as $d_{\mu}$. This assumption corresponds to the scenario described in § [4.1](#S4.SS1), where the variance of token embeddings does not become excessively large. Under this assumption, $d_{\Sigma}\in[0,1]$(^2^22As with $d_{\mu}$, the factor $1/4$ rescales $d_{\Sigma}$ to $[0,1]$ under the trace bound.), matching the range of $d_{\mu}$. Since $d_{\mu},d_{\Sigma}\in[0,1]$ as above, Eq. ([4](#S4.E4)) implies $\mathrm{SOCM}\in[0,1]$. This SOCM value indicates a more severe collapse for larger values. As discussed in § [4.1](#S4.SS1), the collapse arises when first-order statistics are similar ($d_{\mu}$ is small), but second-order statistics differ ($d_{\Sigma}$ is large). SOCM reflects this intuition via $(1-d_{\mu})d_{\Sigma}$. In the following subsections, we discuss the validity of this SOCM design. #### 4.2.2 Decomposing Distributional Distance We adopt $d_{\mu}$ and $d_{\Sigma}$ as the first- and second-order distances, respectively. This choice follows from decomposing the distance of token embedding distributions into first- and second-order components. ##### Decomposition $d_{\mu}$ and $d_{\Sigma}$ correspond to the first- and second-order components of the $L_{2}$-Wasserstein distance $W_{2}^{2}$ of token embedding distributions Dowson1982-sq: $$ $\displaystyle W_{2}^{2}\bigl(\mathcal{N}(\bm{\mu}(\bm{X}_{1}),\bm{\Sigma}(\bm{X}_{1})),\mathcal{N}(\bm{\mu}(\bm{X}_{2}),\bm{\Sigma}(\bm{X}_{2}))\bigr)/4$ $\displaystyle=d_{\mu}+d_{\Sigma}.$ (7) $$ For simplicity, we characterize each token embedding list $\bm{X}_{i}$ as a Gaussian $\mathcal{N}(\bm{\mu}(\bm{X}_{i}),\bm{\Sigma}(\bm{X}_{i}))$. This means that higher-order moments are not explicitly distinguished, though the Gaussian characterization is motivated by computational tractability and stability in Wasserstein-based evaluation (see Appendix [A](#A1) for details). ##### Motivation for W 2 2 W_{2}^{2} This decomposition of the Wasserstein distance is a natural choice for quantifying the severity of the collapse by mean pooling. The Wasserstein distance is a widely used metric for measuring the distance between distributions, considering not only first- but also higher-order statistics Villani2009-go. Indeed, optimal transport, which measures semantic similarity over token embedding lists (discussed in § [2.2](#S2.SS2)), is based on this Wasserstein distance Peyre2018-yj. Therefore, $d_{\mu}$ and $d_{\Sigma}$, which decompose the Wasserstein distance, are natural components for constructing SOCM. #### 4.2.3 Validity of SOCM Form We adopt Eq. ([4](#S4.E4)) as SOCM. To verify its validity, we enumerate desirable properties for quantifying the severity of the collapse and show that this form satisfies all of them. ##### Desirable Properties To quantify the collapse severity, we define five desirable properties: - (a) When Collapse Arises: $\displaystyle{d_{\mu}=0\land d_{\Sigma}=1\Leftrightarrow\mathrm{SOCM}=1}$. - (b) When Collapse Does Not Arise: $d_{\mu}=1\lor d_{\Sigma}=0\Leftrightarrow\mathrm{SOCM}=0$. - (c) Monotonicity in $d_{\mu}$: $\displaystyle{\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}\leq 0}$. - (d) Monotonicity in $d_{\Sigma}$: $\displaystyle{\frac{\partial\,\mathrm{SOCM}}{\partial d_{\Sigma}}\geq 0}$. - (e) Interaction of $d_{\mu}$ and $d_{\Sigma}$: $\displaystyle{\frac{\partial^{2}\,\mathrm{SOCM}}{\partial d_{\mu}\,\partial d_{\Sigma}}\leq 0}$. ##### (a) When Collapse Arises Property (a) requires that $\mathrm{SOCM}=1$ if and only if the collapse arises in the extreme case: $d_{\mu}=0\land d_{\Sigma}=1$. As discussed in § [4.1](#S4.SS1), collapse arises when first-order statistics are similar but second-order statistics differ. This extreme case corresponds to identical first-order statistics ($d_{\mu}=0$) and maximally different second-order statistics ($d_{\Sigma}=1$). ##### (b) When Collapse Does Not Arise Property (b) requires that $\mathrm{SOCM}=0$ if and only if the collapse does not arise: $d_{\mu}=1\lor d_{\Sigma}=0$. As discussed in § [4.1](#S4.SS1), collapse does not arise when first-order statistics differ or when both first- and second-order statistics are similar. $d_{\mu}=1\lor d_{\Sigma}=0$ indicates these cases. Specifically, $d_{\mu}=1$ indicates maximally different first-order statistics. Additionally, $d_{\Sigma}=0$ indicates identical second-order statistics, so similarity is determined by $d_{\mu}$. ##### (c) Monotonicity in d μ d_{\mu} Property (c) requires that SOCM is monotonically non-increasing in $d_{\mu}$. This reflects that the collapse is less severe as the first-order statistics become more distant. ##### (d) Monotonicity in d Σ d_{\Sigma} Property (d) requires that SOCM is monotonically non-decreasing in $d_{\Sigma}$. This reflects that the collapse becomes more severe as the second-order statistics become more distant. ##### (e) Interaction between d μ d_{\mu} and d Σ d_{\Sigma} Property (e) requires that the impact of $d_{\Sigma}$ on SOCM decreases as $d_{\mu}$ increases. While property (d) requires SOCM to increase with $d_{\Sigma}$, this increase should be smaller for large $d_{\mu}$. When first-order statistics are already distant (large $d_{\mu}$), they capture distributional differences, so second-order differences matter less. The mixed derivative enforces this: as $d_{\mu}$ increases, SOCM becomes less sensitive to $d_{\Sigma}$. ##### Form of SOCM Satisfying the Properties Figure: Figure 3: SOCM values for each combination of $(d_{\mu},d_{\Sigma})$. The figure confirms that SOCM satisfies all properties (a)–(e). The four corners correspond to the scenarios in Figure [2](#S4.F2). SOCM is higher when mean pooling collapse occurs (top-left) and lower when it does not (the other three cases). Refer to caption: 2604.27398v1/x3.png The adopted form in Eq. ([4](#S4.E4)) satisfies all desirable properties (a)–(e) (see Appendix [B](#A2) for the proof). Figure [3](#S4.F3) plots values of SOCM over $d_{\mu}$ and $d_{\Sigma}$, confirming properties (a)–(e). The four corners of the plot correspond to the scenarios in Figure [2](#S4.F2). SOCM is high when collapse arises and low otherwise. ## 5 Experiment Using SOCM, we empirically measured how often the collapse induced by mean pooling arose across actual models and texts. We found that contrastive fine-tuned Transformer text encoders tend to be less prone to such collapse than their pretrained backbone models. ### 5.1 Experimental Procedure and Setting ##### Overview To examine how often the collapse arises, we prepared a text pair dataset $D=\{(t_{1},t_{2})\}$ and a model $\bm{f}$. Given a text $t_{i}$, we obtain a token embedding list $\bm{X}_{i}$ from the model $\bm{f}$. We computed SOCM for each token embedding list pair $(\bm{X}_{1},\bm{X}_{2})$. ##### Normalization of Token Embedding Lists We normalized each token embedding list. The definition of SOCM in § [4.2.1](#S4.SS2.SSS1) assumes this normalization, $\|\bm{\mu}(\bm{X})\|_{2}=1$. This assumption aligns with standard practice in text embedding applications, where a unit-normalized embedding $\bm{\mu}(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}$ is used Enevoldsen2025-hc. To satisfy this assumption, given a token embedding list $\bm{X}_{i}=[\bm{x}_{i,1},\ldots,\bm{x}_{i,n_{i}}]\in\mathbb{R}^{d\times n_{i}}$ from model $\bm{f}$, we use the normalized version $\bm{X}_{i}^{\text{norm}}$ defined as: $$ $\bm{X}_{i}^{\text{norm}}=\left[\frac{\bm{x}_{i,1}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}},\ldots,\frac{\bm{x}_{i,n_{i}}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}\right]\in\mathbb{R}^{d\times n_{i}}.$ (8) $$ $\bm{\mu}(\bm{X}_{i}^{\text{norm}})=\bm{\mu}(\bm{X}_{i})/\|\bm{\mu}(\bm{X}_{i})\|_{2}$ holds under this normalization (see Appendix [C](#A3)). We then computed SOCM for each pair $(\bm{X}_{1}^{\text{norm}},\bm{X}_{2}^{\text{norm}})$. ##### Dataset We constructed a dataset of text pairs from Wikipedia Gao2021-ds, which contains 1 million texts. Specifically, we randomly sampled 1,000 texts and generated 499,500 text pairs by comparing them pairwise. Experiments on another dataset, MS MARCO Nguyen2018-ql, led to similar conclusions (Appendix [E](#A5)). ##### Model We examined popular Transformer text encoders that use mean pooling. Specifically, we used the following models, each obtained by contrastive fine-tuning the respective backbone model: Unsupervised SimCSE Gao2021-ds with mean pooling, E5${}_{\text{base}}$ Wang2022-gw, and GTE${}_{\text{base}}$ Li2023-oy (BERT Devlin2019-mb); all-MiniLM-L12-v2 Reimers2019-gc, E5${}_{\text{small}}$ Wang2022-gw, and GTE${}_{\text{small}}$ Li2023-oy (MiniLM Wang2020-mg); all-mpnet-base-v2 Reimers2019-gc (MPNet Song2020-ls); and nomic-embed-text-v1.5 Nussbaum2025-ys (nomic-bert-2048 Nussbaum2025-ys). We also applied SOCM to the respective backbone models. ### 5.2 Results ##### Quantitative Analysis **Table 1: Average SOCM values for each model on text pairs from Wikipedia. For text encoders derived from backbone models, values in parentheses show the change in SOCM. Bold values indicate a reduction.** | Model | Avg. SOCM $\downarrow$ | | --- | --- | | BERT | $0.396$ | | $\rightarrow\text{Unsup-SimCSE-mean}$ | $\bm{0.193}$ $(-0.203)$ | | $\rightarrow\text{E5}_{\text{base}}$ | $\bm{0.029}$ $(-0.367)$ | | $\rightarrow\text{GTE}_{\text{base}}$ | $\bm{0.018}$ $(-0.378)$ | | MiniLM | $0.242$ | | $\rightarrow\text{all-MiniLM-L12-v2}$ | $0.313$ $\hskip 3.41432pt(+0.071)$ | | $\rightarrow\text{E5}_{\text{small}}$ | $\bm{0.099}$ $(-0.143)$ | | $\rightarrow\text{GTE}_{\text{small}}$ | $\bm{0.055}$ $(-0.187)$ | | MPNet | $0.117$ | | $\rightarrow\text{all-mpnet-base-v2}$ | $\bm{0.100}$ $(-0.017)$ | | nomic-bert-2048 | $0.139$ | | $\rightarrow\text{nomic-embed-text-v1.5}$ | $\bm{0.122}$ $(-0.017)$ | Table [1](#S5.T1) shows the average SOCM for each model on Wikipedia. Overall, contrastive fine-tuned text encoders tended to have lower SOCM than their backbone models. This suggests that contrastive fine-tuned text encoders are less prone to collapse induced by mean pooling. ##### Qualitative Analysis We further validated our findings by visualizing embedding spaces. The bottom of Figure [1](#S1.F1) shows 2D PCA projections (without centering) for BERT and GTE${}_{\text{base}}$ on two semantically unrelated texts: “Virginia Woolf set many scenes of her novel “Night and Day” (1919) in Russell Square” and “Ghiz was born in Charlottetown, Prince Edward Island, to Atallah Joseph Ghiz, a Lebanese corner store owner, and Marguerite F. Ghiz (née McKarris).”. For this example, BERT exhibits high $\mathrm{SOCM}=0.618$ while GTE${}_{\text{base}}$ shows low $\mathrm{SOCM}=0.024$.(^3^33SOCM values are computed in the original 768-dimensional space before dimensionality reduction for visualization.) For BERT, the token embedding distributions differed in spread but had similar means. In contrast, for GTE${}_{\text{base}}$, the means of the two token embedding distributions were separated. These visualizations are consistent with the quantitative analysis. In summary, these results indicate that contrastive fine-tuned text encoders are more robust to the collapse induced by mean pooling. This suggests that mean pooling, despite appearing coarse, works well in contrastive fine-tuned text encoders. ## 6 How Do Fine-Tuned Text Encoders Avoid Collapse? In this section, we investigate how fine-tuned text encoders avoid collapse induced by mean pooling. We argue that this robustness lies in the concentration of token embeddings within each text, as shown in the case of GTE${}_{\text{base}}$ in Figure [1](#S1.F1) (bottom). Specifically, we theoretically show that token embeddings within each text can become concentrated under certain conditions in a simplified Transformer formulation, and that such concentration reduces SOCM (§ [6.1](#S6.SS1)). Then, we empirically examine whether fine-tuned text encoders behave consistently with this theoretical account (§ [6.2](#S6.SS2)). ### 6.1 Theoretical Study In the following, using a simplified Transformer formulation, we formally show how token embeddings become concentrated within each text under certain conditions, thereby lowering SOCM. To analyze how Transformer-layer components contribute to token embedding concentration within a text, we introduce a simplified formulation based on a single-head self-attention block. This formulation abstracts the layer into three components: self-attention with projection, residual connection, and a per-token transformation. ###### Assumption 1 (Setting) . Let $\bm{H}=[\bm{h}_{1},\cdots,\bm{h}_{n}]\in\mathbb{R}^{d\times n}$ denote the input token embeddings for a text. The self-attention and projection components produce the attention-branch output $\bm{Z}$: $$ $\bm{Z}=\bm{W}^{o}\bm{W}^{v}\bm{H}\bm{A}^{\top}\in\mathbb{R}^{d\times n},$ (9) $$ where $\bm{A}\in\mathbb{R}^{n\times n}$ is the attention weight matrix after softmax, $d_{v}$ is the value dimension, and $\bm{W}^{v}\in\mathbb{R}^{d_{v}\times d}$ and $\bm{W}^{o}\in\mathbb{R}^{d\times d_{v}}$ are the value and output projection matrices, respectively.(^4^44For simplicity, we omit bias terms from $\bm{W}^{v}$ and $\bm{W}^{o}$.) The residual connection adds $\bm{Z}$ to the input $\bm{H}$, yielding the post-residual embeddings $\bm{Y}$: $$ $\bm{Y}=\bm{Z}+\bm{H}\in\mathbb{R}^{d\times n}.$ (10) $$ Finally, a per-token transformation $\bm{g}$, which abstracts post-attention operations such as LayerNorm and FFN, is applied to each $\bm{y}_{i}$ to produce the final token embeddings $\bm{X}$: $$ $\bm{x}_{i}=\bm{g}(\bm{y}_{i}),\quad\bm{X}=[\bm{x}_{1},\cdots,\bm{x}_{n}]\in\mathbb{R}^{d\times n}.$ (11) $$ We assume $\bm{h}_{1},\cdots,\bm{h}_{n}\overset{\mathrm{i.i.d.}}{\sim}\mathcal{N}(\bm{\eta},c\bm{I}_{d})\quad(\bm{\eta}\neq\bm{0},c>0)$ and treat $\bm{A}$ as fixed for analytical tractability. We write $\mathbb{E}_{\bm{H}}[\cdot]$ for the expectation over the randomness of $\bm{H}=[\bm{h}_{1},\ldots,\bm{h}_{n}]$ induced by $\bm{h}_{1},\cdots,\bm{h}_{n}\overset{\mathrm{i.i.d.}}{\sim}\mathcal{N}(\bm{\eta},c\bm{I}_{d})$. We also define the spread of any matrix $\bm{M}=[\bm{m}_{1},\cdots,\bm{m}_{n}]$ to quantify the degree of token embedding concentration: $$ $S(\bm{M})\coloneqq\frac{1}{n}\sum_{j=1}^{n}\|\bm{m}_{j}-\bm{\mu}(\bm{M})\|_{2}^{2}.$ (12) $$ Based on this formulation, token embedding concentration can arise under the following three conditions: (i) the self-attention and projection components reduce the spread of the attention output $\bm{Z}$, (ii) the residual connection preserves this concentration in the residual output $\bm{Y}$, and (iii) the per-token transformation $\bm{g}$ does not excessively increase the spread of the final token embeddings $\bm{X}$. We formalize these conditions as follows. ###### Definition 1 (Spread expansion/contraction under $\bm{A}$ , $\bm{W}^{v}$ , and $\bm{W}^{o}$ ) . Let $\bm{1}\in\mathbb{R}^{n}$ be the all-ones vector and $\bm{P}=\bm{I}_{n}-\frac{1}{n}\bm{1}\bm{1}^{\top}\in\mathbb{R}^{n\times n}$. Define $$ $\lambda:=\|\bm{W}^{o}\bm{W}^{v}\|_{\mathrm{op}}^{2}\frac{\|\bm{P}\bm{A}\|_{F}^{2}}{n-1},$ (13) $$ where $\|\cdot\|_{\mathrm{op}}$ denotes the operator norm and $\|\cdot\|_{F}$ the Frobenius norm. ###### Intuition . This $\lambda$ relates the spread of the attention-branch output $\bm{Z}$ to that of the input $\bm{H}$; a smaller $\lambda$ indicates that $\bm{Z}$ is more concentrated. ###### Definition 2 (Ratio of token spread in input $\bm{H}$ to the scale of the residual output $\bm{Y}$ ) . Define $$ $r:=\frac{\mathbb{E}_{\bm{H}}\left[S(\bm{H})\right]}{\mathbb{E}_{\bm{H}}\left[\|\bm{\mu}(\bm{Y})\|_{2}^{2}\right]}.$ (14) $$ ###### Intuition . A smaller value of $r$ indicates that the influence of the spread in the input $\bm{H}$ is smaller in the residual output $\bm{Y}=\bm{Z}+\bm{H}$. ###### Definition 3 (Relative spread ratio of $\bm{X}$ to $\bm{Y}$ through transformation $\bm{g}$ ) . Let $C>0$ denote the smallest constant such that $$ $\frac{\mathbb{E}_{\bm{H}}\left[S(\bm{X})\right]}{\mathbb{E}_{\bm{H}}\left[\|\bm{\mu}(\bm{X})\|_{2}^{2}\right]}\leq C\frac{\mathbb{E}_{\bm{H}}\left[S(\bm{Y})\right]}{\mathbb{E}_{\bm{H}}\left[\|\bm{\mu}(\bm{Y})\|_{2}^{2}\right]}.$ (15) $$ ###### Intuition . A smaller value of $C$ indicates that the per-token transformation $\bm{g}$ does not significantly increase the relative spread from $\bm{Y}$ to $\bm{X}$. ###### Theorem 1 (Token embeddings can become concentrated within each text) . Using Definitions [1](#Thmdefinition1)–[3](#Thmdefinition3), if $\lambda<1$, then the normalized spread of the final token embeddings satisfies $$ $\frac{\mathbb{E}_{\bm{H}}\left[S(\bm{X})\right]}{\mathbb{E}_{\bm{H}}\left[\|\bm{\mu}(\bm{X})\|_{2}^{2}\right]}=O(rC)\quad(r,C\to 0).$ (16) $$ ###### Intuition . When $\lambda<1$, if $r$ and $C$ are also small, $\mathbb{E}_{\bm{H}}[S(\bm{X})]/\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{X})\|_{2}^{2}]$ becomes small, meaning that token embeddings concentrate. ###### Sketch of proof. $\bm{Z}$ is concentrated after passing through the self-attention and projection (captured by $\lambda$); the influence of the spread of $\bm{H}$ on the residual output $\bm{Y}=\bm{Z}+\bm{H}$ becomes relatively small (captured by $r$); and $\bm{g}$ does not disperse $\bm{Y}$ (captured by $C$), so $\bm{X}$ remains concentrated. A detailed proof is provided in Appendix [F](#A6). ∎ Having established conditions for token embedding concentration, we now connect this concentration result to SOCM. ###### Theorem 2 (Token embedding concentration leads to low SOCM) . Let $\bm{f}$ be a model that outputs token embeddings $\bm{X}_{i}=[\bm{x}_{i,1},\cdots,\bm{x}_{i,n_{i}}]=\bm{f}(t_{i})$ for input texts $t_{1}$ and $t_{2}$. Suppose that, for $i=1,2$, $S(\bm{X}_{i})/\|\bm{\mu}(\bm{X}_{i})\|_{2}^{2}<\varepsilon$. Then, the SOCM value for $(\bm{X}_{1}^{\mathrm{norm}},\bm{X}_{2}^{\mathrm{norm}})$ (Eq. ([8](#S5.E8))) satisfies $\mathrm{SOCM}=O(\varepsilon)\quad(\varepsilon\to 0)$. ###### Intuition . That is, when $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ is small, meaning that token embeddings concentrate within each text, SOCM also becomes small. ###### Sketch of proof. When $\bm{X}$ is concentrated, the token embeddings are nearly identical within each text, so $d_{\Sigma}$ is small, and hence SOCM is small. A detailed proof is provided in Appendix [G](#A7). ∎ ###### Assumption 1 (Setting) . ###### Definition 1 (Spread expansion/contraction under 𝑨 \bm{A} , 𝑾 v \bm{W}^{v} , and 𝑾 o \bm{W}^{o} ) . ###### Intuition . ###### Definition 2 (Ratio of token spread in input 𝑯 \bm{H} to the scale of the residual output 𝒀 \bm{Y} ) . ###### Intuition . ###### Definition 3 (Relative spread ratio of 𝑿 \bm{X} to 𝒀 \bm{Y} through transformation 𝒈 \bm{g} ) . ###### Intuition . ###### Theorem 1 (Token embeddings can become concentrated within each text) . ###### Intuition . ###### Sketch of proof. ###### Theorem 2 (Token embedding concentration leads to low SOCM) . ###### Intuition . ###### Sketch of proof. ### 6.2 Empirical Study Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x4.png In the theoretical study, we showed that if the self-attention and projection term satisfies $\lambda<1$, then the normalized spread of the final token embeddings is $O(rC)$, where $r$ and $C$ characterize the residual connection and the per-token transformation, respectively. Thus, when $r$ and $C$ are small, token embeddings concentrate, leading to lower SOCM. In the following, we empirically examine each of these quantities in actual models. ##### Setting We used the same 1,000 Wikipedia texts as in § [5](#S5). While § [6.1](#S6.SS1) considers a simplified setting, we computed the corresponding quantities from the actual models: $\bm{H}$ for the input hidden states to the layer, $\bm{Y}$ for the residual output combining the input hidden states with the output of the actual attention and projection operations, and $\bm{X}$ for the output hidden states, together with the attention weights $\bm{A}$ and projection matrices $\bm{W}^{o}$ and $\bm{W}^{v}$. In the following, we compare BERT and GTE${}_{\text{base}}$ (see Appendix [H](#A8) for results on other models). ##### Self-Attention and Projection ( λ \lambda ) We verify whether $\lambda<1$ holds in practice. Figure [4(a)](#S6.F4.sf1) shows the layer-wise average of $\lambda$ for BERT and GTE${}_{\text{base}}$. In both models, $\lambda<1$ held across all layers. ##### Residual Connection ( r r ) We examine the empirical counterpart of $r$, which compares the spread of the input token embeddings with the scale of the residual output. Figure [4(b)](#S6.F4.sf2) shows the layer-wise average of $r\coloneqq\frac{S(\bm{H})}{\|\bm{\mu}(\bm{Y})\|_{2}^{2}}$ for BERT and GTE${}_{\text{base}}$. In GTE${}_{\text{base}}$, $r$ was lower than in BERT in some later layers and decreased toward zero in the final layer. ##### Per-Token Transformation ( C C ) We examine the empirical counterpart of $C$, which measures the degree to which the per-token transformation $\bm{g}$ disperses token embeddings. Figure [4(c)](#S6.F4.sf3) shows the layer-wise average of $C\coloneqq\frac{S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}}{S(\bm{Y})/\|\bm{\mu}(\bm{Y})\|_{2}^{2}}$ for BERT and GTE${}_{\text{base}}$. $C$ was broadly similar in scale across both models, though GTE${}_{\text{base}}$ showed smaller values in some later layers. ##### Token Embedding Concentration We examine the empirical counterpart of the concentration measure $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$. Figure [4(d)](#S6.F4.sf4) shows the layer-wise average of $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ for BERT and GTE${}_{\text{base}}$. This value was lower in GTE${}_{\text{base}}$ than in BERT, especially in the later layers. This is consistent with Theorem [1](#Thmtheorem1), given the observed values of $\lambda$, $r$, and $C$. Also, the concentration in the final layer is consistent with the low SOCM in GTE${}_{\text{base}}$, as implied by Theorem [2](#Thmtheorem2). Looking more closely at the final layer, both $r$ and $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ were closer to zero in GTE${}_{\text{base}}$ than in BERT. Here, a smaller $r$ indicates a smaller influence of the spread in $\bm{H}$ on $\bm{Y}$. By contrast, $\lambda$ was below 1 in both GTE${}_{\text{base}}$ and BERT, suggesting that the attention and projection branch tends to concentrate token embeddings in both models. These observations suggest that the attention and projection components may tend to concentrate token embeddings both before and after fine-tuning, while the influence of input spread in the residual connection may be smaller after fine-tuning, resulting in more concentrated output token embeddings. ##### Why Concentration Occurs in Fine-Tuned Encoders One possible explanation for concentration in fine-tuned encoders is that contrastive learning supervises mean-pooled embeddings. In this setting, contrastive learning does not directly supervise individual token embeddings, but optimizes them through the resulting text embedding. Because contrastive learning encourages this embedding to remain discriminative Wang2020-ru, representations are more useful when discriminative properties are reflected stably in the mean. One way this can happen is for token embeddings within a text to become more concentrated around their mean, so that discriminative properties are more directly reflected in it. From this perspective, in fine-tuned models, the attention component may promote concentration by aggregating information from other tokens, and this concentration may also be preserved in the residual connection. ##### Connection to Prior Work These results are consistent with prior work on the geometry of token embeddings in fine-tuned text encoders (Xiao2023-vu). Xiao2023-vu reported that contrastive fine-tuning leads to anisotropic token embeddings within each text, particularly in the later layers. Our findings connect this observation with robustness against collapse induced by mean pooling. ## 7 Correlation with Downstream Task Performance In this section, we examine whether robustness to collapse by mean pooling, as quantified by SOCM, correlates with downstream task performance. ##### Setting As a downstream task, we used MTEB (eng, v2) Enevoldsen2025-hc, a standard text embedding evaluation benchmark consisting of 41 tasks. For each model used in § [5](#S5), we compared its MTEB score with the SOCM in Table [1](#S5.T1). ##### Result Figure: Figure 5: Scatter plot of average SOCM and MTEB (eng, v2) score. Each point represents a model. Marker types indicate the backbones. Models with a BERT backbone are annotated with their names. Refer to caption: 2604.27398v1/x8.png Figure [5](#S7.F5) shows a scatter plot of SOCM and MTEB score for each model. We observed a negative correlation (Spearman’s $\rho=-0.678$, $p=0.015$), indicating that models with lower SOCM tended to achieve higher downstream task performance. This suggests that robustness to collapse may be one factor behind the success of modern text encoders on downstream tasks. ##### Discussion **Table 2: Spearman’s $\rho$ between MTEB (eng, v2) scores and each of SOCM and $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$.** | | $\rho$ | | --- | --- | | SOCM | $\mathbf{-0.678}$ | | $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ | $-0.622$ | We also compared SOCM with $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$, the token concentration measure introduced in § [6](#S6), in terms of their correlations with downstream task performance. Table [2](#S7.T2) shows that SOCM is more negatively correlated with downstream task performance. One possible reason is that $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ does not capture inter-text separation. As a result, this measure can be small even when semantically dissimilar texts have similar mean-pooled text embeddings. By contrast, SOCM also decreases as $d_{\mu}$ increases, thereby capturing this separation. This suggests that, by reflecting the separation of negative pairs encouraged by contrastive fine-tuning Wang2020-ru, SOCM may better capture improvements in downstream task performance brought by fine-tuning. Further investigation into how contrastive fine-tuning improves downstream task performance remains an important direction for future work. ## 8 Conclusion This paper offered a new perspective on why modern text encoders remain effective despite adopting seemingly coarse mean pooling. First, we argued that mean pooling can discard second-order statistics, causing distinct token embedding distributions to collapse into similar text embeddings. We then proposed a metric to quantify this collapse (§ [4](#S4)). Using this metric, we empirically found that contrastive fine-tuned text encoders are less prone to collapse than their backbone models in practice (§ [5](#S5)). We also found that the robustness of fine-tuned text encoders against this collapse lies in the concentration of token embeddings within each text (§ [6](#S6)). We further observed that this metric correlates with downstream task performance (§ [7](#S7)). ##### Future Work A fuller mathematical account of why contrastive fine-tuning reduces SOCM and improves downstream performance remains an important direction. SOCM may also be useful for encoder development; one direction is to use it as a regularization term during training. Another direction is to extend this analysis to LLM-based generation, for example, by examining context compression by mean pooling Feldman2025-va. ## Limitations This work has several limitations. ##### Restriction to Second-order Statistics This paper approximates the information in token embedding lists using only first- and second-order moments. Specifically, we approximate token embedding lists with Gaussian distributions $\mathcal{N}(\bm{\mu}(\bm{X}_{i}),\bm{\Sigma}(\bm{X}_{i}))$, which are fully characterized by their first- and second-order moments. However, the actual distribution of token embedding lists may also contain information from third-order and higher moments. Analyzing third-order and higher moments in token embedding lists remains an interesting direction for future work. ##### Assumptions in the Proposed Metric In defining the proposed metric SOCM, we impose the assumptions $\|\bm{\mu}(\bm{X}_{i})\|_{2}=1$ and $\mathrm{tr}(\bm{\Sigma}(\bm{X}_{i}))\leq 2$. As shown in Appendix [J](#A10), these assumptions hold under the normalization procedure in § [5](#S5) and certain conditions on model architecture (LayerNorm with shared parameters across dimensions and sufficient similarity of token embeddings within texts). However, the proposed metric may not work effectively in cases where these assumptions are violated, for instance, when the second-order moment $\mathrm{tr}(\bm{\Sigma}(\bm{X}_{i}))$ becomes extremely large. Extending the generality of the metric to broader settings remains future work. ##### Toward Theoretical Understanding In § [6](#S6), we partially addressed the theoretical understanding of the observed robustness. Specifically, we showed theoretically that token embedding concentration leads to lower SOCM, and empirically verified that fine-tuned text encoders satisfy the conditions under which such concentration occurs. However, why contrastive fine-tuning induces token embedding concentration in the first place remains an open question. A deeper theoretical investigation of this mechanism is an interesting direction for future work. ##### Practical Utility of SOCM In this paper, we use SOCM as an analysis tool for characterizing robustness to collapse by mean pooling across models and texts. An interesting direction for future work is to explore practical uses of SOCM beyond analysis, such as incorporating it into training objectives as a regularization term. ##### Scope Limited to Text Embeddings This paper focuses on text embeddings and does not address LLM-based applications, such as generation or reasoning. Broadening the scope to these LLM-based applications, for example, by investigating how context compression by mean pooling Feldman2025-va influences generation or reasoning, remains an interesting direction for future work. ##### Focus on Mean Pooling This paper focuses on mean pooling as the target aggregation method. Mean pooling remains the dominant pooling strategy in modern text encoders. However, richer pooling methods, such as SIF weighting Arora2017-pn, have also been proposed as alternatives to simple averaging. Investigating such methods is an interesting direction for future work. ## Ethical Consideration This study analyzes embeddings using models released under the MIT License (MiniLM, $\text{E5}_{\text{small}}$, $\text{E5}_{\text{base}}$, $\text{GTE}_{\text{small}}$, $\text{GTE}_{\text{base}}$, Unsupervised SimCSE, MPNet) and Apache License 2.0 (all-MiniLM-L12-v2, all-mpnet-base-v2, nomic-bert-2048, nomic-embed-text-v1.5). Our analysis, which examines the embedding representations output by these models for given input texts, falls within their intended use cases. We use datasets (Wikipedia, MS MARCO, MTEB (eng, v2)) released under Apache License 2.0. These datasets are used as provided without additional preprocessing to remove social biases, personal information, or offensive content, and thus may reflect various biases present in the original data. ## Acknowledgments This work was supported by AMED Grant Number JP26wm0625405, JSPS KAKENHI Grant Number JP22H05106, the JST FOREST Program Grant Number JPMJFR2331, and JST BOOST Grant Number JPMJBS2421. We would like to thank the members of the Tohoku NLP Group, as well as the organizers and participants of IBIS 2025 and NLP 2026, for their insightful and encouraging feedback. ## References ## Appendix A Gaussian Characterization As noted in § [4](#S4), our method evaluates the target distributions by approximating them with Gaussian distributions. This is an intentional design choice motivated by computational tractability and stability in Wasserstein-based evaluation. ##### Advantages of the Gaussian Characterization First, when using the Wasserstein distance, the Gaussian characterization offers substantial advantages despite ignoring higher-order moments. For general non-Gaussian distributions, computing the Wasserstein distance typically requires costly linear programming-based algorithms whose complexity scales poorly with dimensionality, making them impractical in high dimensions Cuturi2013-bj. From this perspective, the Gaussian characterization is essential for efficient computation in high-dimensional spaces, and its computational and analytical benefits outweigh the limitation of not explicitly modeling higher-order moments. ##### Information Captured by First- and Second-order Statistics Furthermore, focusing on first- and second-order statistics lets us efficiently capture much of the information in the data distribution. In particular, it is known that the $L_{2}$-Wasserstein distance computed using only up to second-order statistics can provide a universal lower bound on the true $L_{2}$-Wasserstein distance between arbitrary distributions Cuesta-Albertos1996-bd. In this sense, introducing a Gaussian characterization is not merely a heuristic simplification; rather, it recovers information that is theoretically guaranteed in terms of the Wasserstein distance up to second-order statistics. Moreover, such characterizations are widely used in practice. For example, the Fréchet Inception Distance (FID), a standard metric for evaluating generative models, also relies on the $L_{2}$-Wasserstein distance between Gaussian approximations characterized by first- and second-order statistics, and has been successfully used across a broad range of distributions Heusel2017-ic. Taken together, these theoretical guarantees and empirical precedents support the use of second-order statistics-based Wasserstein characterization as a principled and practically effective approach. ## Appendix B Proof that SOCM Satisfies Desirable Properties In this section, we prove that SOCM, as defined in Eq. ([4](#S4.E4)), satisfies the desirable properties (a)–(e) introduced in § [4.2.3](#S4.SS2.SSS3). Specifically, we show that $$ $\mathrm{SOCM}(d_{\mu},d_{\Sigma})=(1-d_{\mu})d_{\Sigma}$ (17) $$ satisfies all properties (a)–(e): - (a) $\displaystyle d_{\mu}=0\land d_{\Sigma}=1\Leftrightarrow\mathrm{SOCM}=1$ - (b) $\displaystyle d_{\mu}=1\lor d_{\Sigma}=0\Leftrightarrow\mathrm{SOCM}=0$ - (c) $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}\leq 0$ - (d) $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\Sigma}}\geq 0$ - (e) $\displaystyle\frac{\partial^{2}\,\mathrm{SOCM}}{\partial d_{\mu}\,\partial d_{\Sigma}}\leq 0$ ### B.1 Proof of Property (a) We prove property (a): $d_{\mu}=0\land d_{\Sigma}=1\Leftrightarrow\mathrm{SOCM}=1$. We first show $(d_{\mu}=0\land d_{\Sigma}=1\Rightarrow\mathrm{SOCM}=1)$. When $d_{\mu}=0$ and $d_{\Sigma}=1$, $$ $\displaystyle\mathrm{SOCM}(0,1)$ $\displaystyle=(1-0)\cdot 1$ (18) $\displaystyle=1.$ (19) $$ Therefore, $d_{\mu}=0\land d_{\Sigma}=1\Rightarrow\mathrm{SOCM}=1$ holds. Next, we show $(\mathrm{SOCM}=1\Rightarrow d_{\mu}=0\land d_{\Sigma}=1)$. Assume $\mathrm{SOCM}(d_{\mu},d_{\Sigma})=1$, i.e., $$ $(1-d_{\mu})d_{\Sigma}=1.$ (20) $$ Since $d_{\mu},d_{\Sigma}\in[0,1]$, we have $(1-d_{\mu})\in[0,1]$ and $d_{\Sigma}\in[0,1]$. Therefore, the above equation holds only when $1-d_{\mu}=1$ and $d_{\Sigma}=1$, which implies $d_{\mu}=0$ and $d_{\Sigma}=1$. Therefore, $\mathrm{SOCM}=1\Rightarrow d_{\mu}=0\land d_{\Sigma}=1$ holds. ### B.2 Proof of Property (b) We prove property (b): $d_{\mu}=1\lor d_{\Sigma}=0\Leftrightarrow\mathrm{SOCM}=0$. We first show $(d_{\mu}=1\lor d_{\Sigma}=0\Rightarrow\mathrm{SOCM}=0)$. When $d_{\mu}=1$, $$ $\displaystyle\mathrm{SOCM}(1,d_{\Sigma})$ $\displaystyle=(1-1)\cdot d_{\Sigma}$ (21) $\displaystyle=0.$ (22) $$ Also, when $d_{\Sigma}=0$, $$ $\displaystyle\mathrm{SOCM}(d_{\mu},0)$ $\displaystyle=(1-d_{\mu})\cdot 0$ (23) $\displaystyle=0.$ (24) $$ Therefore, $d_{\mu}=1\lor d_{\Sigma}=0\Rightarrow\mathrm{SOCM}=0$ holds. Next, we show $(\mathrm{SOCM}=0\Rightarrow d_{\mu}=1\lor d_{\Sigma}=0)$. Assume $\mathrm{SOCM}(d_{\mu},d_{\Sigma})=0$, i.e., $$ $(1-d_{\mu})d_{\Sigma}=0.$ (25) $$ This holds when $1-d_{\mu}=0$ or $d_{\Sigma}=0$, i.e., when $d_{\mu}=1$ or $d_{\Sigma}=0$. Therefore, $\mathrm{SOCM}=0\Rightarrow d_{\mu}=1\lor d_{\Sigma}=0$ holds. ### B.3 Proof of Property (c) We prove property (c): $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}\leq 0$. Taking the partial derivative of $\mathrm{SOCM}(d_{\mu},d_{\Sigma})=(1-d_{\mu})d_{\Sigma}$ with respect to $d_{\mu}$, $$ $\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}=-d_{\Sigma}.$ (26) $$ Since $d_{\Sigma}\in[0,1]$, we have $-d_{\Sigma}\leq 0$. Therefore, $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}\leq 0$ holds. ### B.4 Proof of Property (d) We prove property (d): $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\Sigma}}\geq 0$. Taking the partial derivative of $\mathrm{SOCM}(d_{\mu},d_{\Sigma})=(1-d_{\mu})d_{\Sigma}$ with respect to $d_{\Sigma}$, $$ $\frac{\partial\,\mathrm{SOCM}}{\partial d_{\Sigma}}=1-d_{\mu}.$ (27) $$ Since $d_{\mu}\in[0,1]$, we have $1-d_{\mu}\geq 0$. Therefore, $\displaystyle\frac{\partial\,\mathrm{SOCM}}{\partial d_{\Sigma}}\geq 0$ holds. ### B.5 Proof of Property (e) We prove property (e): $\displaystyle\frac{\partial^{2}\,\mathrm{SOCM}}{\partial d_{\mu}\,\partial d_{\Sigma}}\leq 0$. From the proof of property (c), $$ $\frac{\partial\,\mathrm{SOCM}}{\partial d_{\mu}}=-d_{\Sigma}.$ (28) $$ Taking the partial derivative with respect to $d_{\Sigma}$, $$ $\frac{\partial^{2}\,\mathrm{SOCM}}{\partial d_{\mu}\,\partial d_{\Sigma}}=-1\leq 0.$ (29) $$ Therefore, property (e) holds. ### B.6 Generalization More generally, for any $\alpha>0$ and $\beta>0$, the following form also satisfies all properties (a)–(e): $$ $\mathrm{SOCM}_{\alpha,\beta}(d_{\mu},d_{\Sigma})=(1-d_{\mu})^{\alpha}d_{\Sigma}^{\beta}.$ (30) $$ The proof follows analogously, replacing $1-d_{\mu}$ and $d_{\Sigma}$ with their respective powers. Our adopted form in Eq. ([4](#S4.E4)) corresponds to the simple case $\alpha=\beta=1$. ## Appendix C Proof of Normalization Property We prove that under the normalization defined in § [5](#S5), the relationship $\bm{\mu}(\bm{X}_{i}^{\text{norm}})=\bm{\mu}(\bm{X}_{i})/\|\bm{\mu}(\bm{X}_{i})\|_{2}$ holds. ##### Statement Given a token embedding list $\bm{X}_{i}=[\bm{x}_{i,1},\ldots,\bm{x}_{i,n_{i}}]\in\mathbb{R}^{d\times n_{i}}$ and its normalized version $$ $\bm{X}_{i}^{\text{norm}}=\left[\frac{\bm{x}_{i,1}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}},\ldots,\frac{\bm{x}_{i,n_{i}}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}\right]\in\mathbb{R}^{d\times n_{i}},$ (31) $$ we show that $\bm{\mu}(\bm{X}_{i}^{\text{norm}})=\bm{\mu}(\bm{X}_{i})/\|\bm{\mu}(\bm{X}_{i})\|_{2}$. ##### Proof By the definition of mean pooling, we have $$ $\displaystyle\bm{\mu}(\bm{X}_{i}^{\text{norm}})$ $\displaystyle=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\frac{\bm{x}_{i,j}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}.$ (32) $$ Since $\|\bm{\mu}(\bm{X}_{i})\|_{2}$ is a scalar independent of the summation index $j$, we can factor it out: $$ $\displaystyle\bm{\mu}(\bm{X}_{i}^{\text{norm}})$ $\displaystyle=\frac{1}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}\cdot\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\bm{x}_{i,j}.$ (33) $$ The remaining summation is precisely the definition of $\bm{\mu}(\bm{X}_{i})$: $$ $\displaystyle\bm{\mu}(\bm{X}_{i}^{\text{norm}})$ $\displaystyle=\frac{1}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}\cdot\bm{\mu}(\bm{X}_{i})$ (34) $\displaystyle=\frac{\bm{\mu}(\bm{X}_{i})}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}.$ (35) $$ Furthermore, this implies that $\|\bm{\mu}(\bm{X}_{i}^{\text{norm}})\|_{2}=1$, which is the desired property for computing SOCM as defined in § [4.2.1](#S4.SS2.SSS1). ## Appendix D Implementation Details This section provides detailed information about the implementation of our experiments described in § [5](#S5), § [6](#S6), and § [7](#S7). ### D.1 Dataset Details ##### Preprocessing For comparison across different models and datasets, we did not use any task-specific prefixes (e.g., query:, passage:) when encoding texts. Note that some text embedders are designed to utilize such prefixes to distinguish between different text types Wang2022-gw; Li2023-oy. ##### Dataset URLs Table [3](#A4.T3) shows the URLs for the datasets used in our experiments. **Table 3: URLs for datasets used in our experiments.** | Dataset | URL | | --- | --- | | Wikipedia | [https://huggingface.co/datasets/princeton-nlp/datasets-for-simcse/resolve/main/wiki1m_for_simcse.txt](https://huggingface.co/datasets/princeton-nlp/datasets-for-simcse/resolve/main/wiki1m_for_simcse.txt) | | MS MARCO | [https://huggingface.co/datasets/mteb/msmarco](https://huggingface.co/datasets/mteb/msmarco) | | MS MARCO Hard Negatives | [https://huggingface.co/datasets/sentence-transformers/msmarco-co-condenser-margin-mse-sym-mnrl-mean-v1](https://huggingface.co/datasets/sentence-transformers/msmarco-co-condenser-margin-mse-sym-mnrl-mean-v1) | | MTEB (eng, v2) | [https://github.com/embeddings-benchmark/mteb](https://github.com/embeddings-benchmark/mteb) | ##### Language In our experiments, we used English-language datasets. ### D.2 Model Details **Table 4: Details of the Hugging Face models used in our experiments, with parameter counts (in millions) taken from the corresponding backbone model sizes.** | Model | model_name | Params (M) | | --- | --- | --- | | BERT | bert-base-uncased | 110 | | Unsup-SimCSE-mean | h-tomo/unsup-simcse-bert-base-uncased-mean | 110 | | $\text{E5}_{\text{base}}$ | intfloat/e5-base-v2 | 110 | | $\text{GTE}_{\text{base}}$ | thenlper/gte-base | 110 | | MiniLM | microsoft/MiniLM-L12-H384-uncased | 30 | | all-MiniLM-L12-v2 | sentence-transformers/all-MiniLM-L12-v2 | 30 | | $\text{E5}_{\text{small}}$ | intfloat/e5-small-v2 | 30 | | $\text{GTE}_{\text{small}}$ | thenlper/gte-small | 30 | | MPNet | microsoft/mpnet-base | 109 | | all-mpnet-base-v2 | sentence-transformers/all-mpnet-base-v2 | 109 | | nomic-bert-2048 | nomic-ai/nomic-bert-2048 | 137 | | nomic-embed-text-v1.5 | nomic-ai/nomic-embed-text-v1.5 | 137 | Table [4](#A4.T4) lists the specific model identifiers from Hugging Face for each model used in our experiments. For most models, we used publicly available pre-trained checkpoints. For Unsupervised SimCSE Gao2021-ds, we trained the model ourselves using the original codebase, as the unsupervised checkpoint was not publicly available at the time of our experiments. We followed the default training configuration provided in the official implementation, using bert-base-uncased as the backbone model and training on English Wikipedia with the default hyperparameters. ## Appendix E Additional Results for § 5 To validate the generalizability of our findings, we conducted additional experiments on MS MARCO. ##### Experimental Setting We followed the same experimental procedure as described in § [5](#S5). Specifically, we randomly sampled 1,000 texts from MS MARCO passages and generated 499,500 text pairs by comparing these texts pairwise. We computed SOCM values for the same set of models used in § [5](#S5). In addition, we computed SOCM for 50,000 query-negative passage pairs from MS MARCO hard negatives. ##### Results **Table 5: Average SOCM values for each model on text pairs from MS MARCO passages. For text encoders derived from backbone models, values in parentheses show the change in SOCM. Bold values indicate a reduction.** | Model | Avg. SOCM $\downarrow$ | | --- | --- | | BERT | $0.491$ | | $\rightarrow\text{Unsup-SimCSE-mean}$ | $\bm{0.269}$ $(-0.222)$ | | $\rightarrow\text{E5}_{\text{base}}$ | $\bm{0.043}$ $(-0.448)$ | | $\rightarrow\text{GTE}_{\text{base}}$ | $\bm{0.025}$ $(-0.466)$ | | MiniLM | $0.289$ | | $\rightarrow\text{all-MiniLM-L12-v2}$ | $0.348$ $\hskip 3.41432pt(+0.059)$ | | $\rightarrow\text{E5}_{\text{small}}$ | $\bm{0.085}$ $(-0.204)$ | | $\rightarrow\text{GTE}_{\text{small}}$ | $\bm{0.055}$ $(-0.234)$ | | MPNet | $0.106$ | | $\rightarrow\text{all-mpnet-base-v2}$ | $\bm{0.094}$ $(-0.012)$ | | nomic-bert-2048 | $0.110$ | | $\rightarrow\text{nomic-embed-text-v1.5}$ | $0.133$ $\hskip 3.41432pt(+0.023)$ | **Table 6: Average SOCM values for each model on query–negative passage pairs from MS MARCO hard negatives. For text encoders derived from backbone models, values in parentheses show the change in SOCM. Bold values indicate a reduction.** | Model | Avg. SOCM $\downarrow$ | | --- | --- | | BERT | $0.480$ | | $\rightarrow\text{Unsup-SimCSE-mean}$ | $\bm{0.257}$ $(-0.224)$ | | $\rightarrow\text{E5}_{\text{base}}$ | $\bm{0.036}$ $(-0.445)$ | | $\rightarrow\text{GTE}_{\text{base}}$ | $\bm{0.017}$ $(-0.464)$ | | MiniLM | $0.340$ | | $\rightarrow\text{all-MiniLM-L12-v2}$ | $\bm{0.330}$ $(-0.010)$ | | $\rightarrow\text{E5}_{\text{small}}$ | $\bm{0.087}$ $(-0.253)$ | | $\rightarrow\text{GTE}_{\text{small}}$ | $\bm{0.048}$ $(-0.292)$ | | MPNet | $0.129$ | | $\rightarrow\text{all-mpnet-base-v2}$ | $\bm{0.093}$ $(-0.036)$ | | nomic-bert-2048 | $0.131$ | | $\rightarrow\text{nomic-embed-text-v1.5}$ | $\bm{0.104}$ $(-0.027)$ | Table [5](#A5.T5) shows the average SOCM values for each model on MS MARCO, and Table [6](#A5.T6) shows those on MS MARCO hard negatives. In both cases, we observed similar trends to the Wikipedia results (§ [5](#S5)). ## Appendix F Proof of Theorem 1 Recall that for any matrix $\bm{M}\in\mathbb{R}^{d\times n}$, $$ $S(\bm{M})=\frac{1}{n}\sum_{j=1}^{n}\|\bm{m}_{j}-\bm{\mu}(\bm{M})\|_{2}^{2}=\frac{1}{n}\|\bm{M}\bm{P}\|_{F}^{2},$ (36) $$ where $\bm{P}=\bm{I}_{n}-\frac{1}{n}\bm{1}\bm{1}^{\top}$ is the centering matrix. We first control the spread of $\bm{Z}=\bm{W}^{o}\bm{W}^{v}\bm{H}\bm{A}^{\top}$. Since $\bm{P}$ is symmetric, $$ $\bm{Z}\bm{P}=\bm{W}^{o}\bm{W}^{v}\bm{H}\bm{A}^{\top}\bm{P}=\bm{W}^{o}\bm{W}^{v}\bm{H}(\bm{P}\bm{A})^{\top}.$ (37) $$ Hence, by the operator norm inequality $\|\bm{AB}\|_{F}\leq\|\bm{A}\|_{\mathrm{op}}\|\bm{B}\|_{F}$, $$ $\displaystyle\mathbb{E}_{\bm{H}}[S(\bm{Z})]$ $\displaystyle=\frac{1}{n}\mathbb{E}_{\bm{H}}\bigl[\|\bm{Z}\bm{P}\|_{F}^{2}\bigr]$ $\displaystyle=\frac{1}{n}\mathbb{E}_{\bm{H}}\bigl[\|\bm{W}^{o}\bm{W}^{v}\bm{H}(\bm{P}\bm{A})^{\top}\|_{F}^{2}\bigr]$ $\displaystyle\leq\frac{\|\bm{W}^{o}\bm{W}^{v}\|_{\mathrm{op}}^{2}}{n}\mathbb{E}_{\bm{H}}\bigl[\|\bm{H}(\bm{P}\bm{A})^{\top}\|_{F}^{2}\bigr].$ (38) $$ Write $\bm{H}=\bm{\eta}\bm{1}^{\top}+\bm{E}$, where the columns of $\bm{E}$ are i.i.d. $\mathcal{N}(\bm{0},c\bm{I}_{d})$. Since each row of $\bm{A}$ sums to one (i.e., $\bm{A}\bm{1}=\bm{1}$), we have $(\bm{P}\bm{A})\bm{1}=\bm{P}(\bm{A}\bm{1})=\bm{P}\bm{1}=\bm{0}$, and thus $\bm{1}^{\top}(\bm{P}\bm{A})^{\top}=\bm{0}^{\top}$. Therefore, the mean part vanishes: $$ $\bm{H}(\bm{P}\bm{A})^{\top}=\bm{E}(\bm{P}\bm{A})^{\top}.$ (39) $$ By the isotropy of the Gaussian noise, $\mathbb{E}_{\bm{H}}\bigl[\|\bm{E}\bm{B}\|_{F}^{2}\bigr]=cd\,\|\bm{B}\|_{F}^{2}$ for any $\bm{B}\in\mathbb{R}^{n\times n}$. Taking $\bm{B}=(\bm{P}\bm{A})^{\top}$, $$ $\mathbb{E}_{\bm{H}}[S(\bm{Z})]\leq\frac{\|\bm{W}^{o}\bm{W}^{v}\|_{\mathrm{op}}^{2}}{n}\cdot cd\,\|\bm{P}\bm{A}\|_{F}^{2}.$ (40) $$ On the other hand, since $\|\bm{P}\|_{F}^{2}=\mathrm{tr}(\bm{P})=n-1$, $$ $\displaystyle\mathbb{E}_{\bm{H}}[S(\bm{H})]$ $\displaystyle=\frac{1}{n}\mathbb{E}_{\bm{H}}[\|\bm{E}\bm{P}\|_{F}^{2}]$ $\displaystyle=\frac{cd}{n}\|\bm{P}\|_{F}^{2}$ $\displaystyle=\frac{cd(n-1)}{n}.$ (41) $$ Combining the above two inequalities, $$ $\displaystyle\mathbb{E}_{\bm{H}}[S(\bm{Z})]$ $\displaystyle\leq\|\bm{W}^{o}\bm{W}^{v}\|_{\mathrm{op}}^{2}\frac{\|\bm{P}\bm{A}\|_{F}^{2}}{n-1}\,\mathbb{E}_{\bm{H}}[S(\bm{H})]$ $\displaystyle=\lambda\,\mathbb{E}_{\bm{H}}[S(\bm{H})].$ (42) $$ By Definition [1](#Thmdefinition1) and the hypothesis $\lambda<1$, Next, we control the spread of $\bm{Y}=\bm{Z}+\bm{H}$. Using $\bm{Y}\bm{P}=\bm{Z}\bm{P}+\bm{H}\bm{P}$ and the Minkowski inequality, $$ $\displaystyle\sqrt{\mathbb{E}_{\bm{H}}[S(\bm{Y})]}$ $\displaystyle=\frac{1}{\sqrt{n}}\Bigl(\mathbb{E}_{\bm{H}}\bigl[\|\bm{Y}\bm{P}\|_{F}^{2}\bigr]\Bigr)^{1/2}$ $\displaystyle\leq\frac{1}{\sqrt{n}}\Bigl(\mathbb{E}_{\bm{H}}\bigl[\|\bm{Z}\bm{P}\|_{F}^{2}\bigr]\Bigr)^{1/2}$ $\displaystyle\quad+\frac{1}{\sqrt{n}}\Bigl(\mathbb{E}_{\bm{H}}\bigl[\|\bm{H}\bm{P}\|_{F}^{2}\bigr]\Bigr)^{1/2}$ $\displaystyle=\sqrt{\mathbb{E}_{\bm{H}}[S(\bm{Z})]}+\sqrt{\mathbb{E}_{\bm{H}}[S(\bm{H})]}$ $\displaystyle\leq(1+\sqrt{\lambda})\sqrt{\mathbb{E}_{\bm{H}}[S(\bm{H})]}.$ (43) $$ Squaring both sides yields $$ $\mathbb{E}_{\bm{H}}[S(\bm{Y})]\leq(1+\sqrt{\lambda})^{2}\,\mathbb{E}_{\bm{H}}[S(\bm{H})].$ (44) $$ Therefore, $$ $\frac{\mathbb{E}_{\bm{H}}[S(\bm{Y})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{Y})\|_{2}^{2}]}\leq(1+\sqrt{\lambda})^{2}\frac{\mathbb{E}_{\bm{H}}[S(\bm{H})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{Z}+\bm{H})\|_{2}^{2}]}.$ (45) $$ By Definition [2](#Thmdefinition2), it holds that $r=\frac{\mathbb{E}_{\bm{H}}[S(\bm{H})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{Z}+\bm{H})\|_{2}^{2}]}$, hence $$ $\frac{\mathbb{E}_{\bm{H}}[S(\bm{Y})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{Y})\|_{2}^{2}]}\leq(1+\sqrt{\lambda})^{2}r=O(r),$ (46) $$ where the hidden constant depends only on $\lambda$. Finally, Definition [3](#Thmdefinition3) gives $$ $\displaystyle\frac{\mathbb{E}_{\bm{H}}[S(\bm{X})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{X})\|_{2}^{2}]}$ $\displaystyle\leq C\,\frac{\mathbb{E}_{\bm{H}}[S(\bm{Y})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{Y})\|_{2}^{2}]}$ $\displaystyle\leq C(1+\sqrt{\lambda})^{2}r.$ (47) $$ We therefore conclude that $$ $\frac{\mathbb{E}_{\bm{H}}[S(\bm{X})]}{\mathbb{E}_{\bm{H}}[\|\bm{\mu}(\bm{X})\|_{2}^{2}]}=O(rC)\quad(r,C\to 0).$ (48) $$ This completes the proof. ## Appendix G Proof of Theorem 2 Let $t_{1},t_{2}$ be arbitrary texts, and let $\bm{X}_{i}=\bm{f}(t_{i})$ for $i=1,2$. By definition of $\bm{\mu}(\cdot)$, $$ $\bm{\mu}(\bm{X}_{i}^{\text{norm}})=\frac{\bm{\mu}(\bm{X}_{i})}{\|\bm{\mu}(\bm{X}_{i})\|_{2}},$ (49) $$ hence $\|\bm{\mu}(\bm{X}_{i}^{\text{norm}})\|_{2}=1$. Since normalization scales each token embedding by the scalar $1/\|\bm{\mu}(\bm{X}_{i})\|_{2}$, the covariance matrix scales quadratically: $$ $\bm{\Sigma}(\bm{X}_{i}^{\text{norm}})=\frac{1}{\|\bm{\mu}(\bm{X}_{i})\|_{2}^{2}}\bm{\Sigma}(\bm{X}_{i}).$ (50) $$ Using $S(\bm{X})=\mathrm{tr}(\bm{\Sigma}(\bm{X}))$, the assumption gives $$ $\mathrm{tr}\!\left(\bm{\Sigma}(\bm{X}_{i}^{\text{norm}})\right)=\frac{S(\bm{X}_{i})}{\|\bm{\mu}(\bm{X}_{i})\|_{2}^{2}}<\varepsilon\qquad(i=1,2).$ (51) $$ Since $(1-d_{\mu})\leq 1$, we have $$ $\mathrm{SOCM}\leq d_{\Sigma}.$ (52) $$ By definition of $d_{\Sigma}$, $$ $\displaystyle d_{\Sigma}$ $\displaystyle=\frac{1}{4}\mathrm{tr}\Bigl(\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})+\bm{\Sigma}(\bm{X}_{2}^{\text{norm}})$ $\displaystyle\quad-2\Bigl(\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})^{1/2}\bm{\Sigma}(\bm{X}_{2}^{\text{norm}})$ $\displaystyle\qquad\qquad\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})^{1/2}\Bigr)^{1/2}\Bigr).$ (53) $$ Since $\bigl(\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})^{1/2}\bm{\Sigma}(\bm{X}_{2}^{\text{norm}})\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})^{1/2}\bigr)^{1/2}$ is positive semidefinite, its trace is nonnegative, and therefore $$ $d_{\Sigma}\leq\frac{1}{4}\mathrm{tr}\!\left(\bm{\Sigma}(\bm{X}_{1}^{\text{norm}})\right)+\frac{1}{4}\mathrm{tr}\!\left(\bm{\Sigma}(\bm{X}_{2}^{\text{norm}})\right)<\frac{\varepsilon}{2}.$ (54) $$ Hence, $$ $\mathrm{SOCM}\leq d_{\Sigma}<\frac{\varepsilon}{2},$ (55) $$ which gives $\mathrm{SOCM}=O(\varepsilon)$ as $\varepsilon\to 0$. This completes the proof. ## Appendix H Additional Results for § 6 ### H.1 Results for Additional Models We provide additional results for the layer-wise analysis of token embedding concentration described in § [6](#S6). In addition to the results for BERT and GTE${}_{\text{base}}$ reported in § [6](#S6), Figures [6](#A8.F6)–[10](#A8.F10) show the layer-wise averages of $\lambda$, $r$, $C$, and $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ for the remaining model pairs: E5${}_{\text{base}}$ and BERT (Figure [6](#A8.F6)), Unsupervised SimCSE and BERT (Figure [7](#A8.F7)), GTE${}_{\text{small}}$ and MiniLM (Figure [8](#A8.F8)), E5${}_{\text{small}}$ and MiniLM (Figure [9](#A8.F9)), and all-MiniLM-L12-v2 and MiniLM (Figure [10](#A8.F10)). These results tend to show similar trends to those observed for BERT and GTE${}_{\text{base}}$ in § [6](#S6). Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x9.png Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x13.png Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x17.png Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x21.png Figure: (a) Avg. $\lambda$. Refer to caption: 2604.27398v1/x25.png ### H.2 Within-Text Token Embedding Concentration In § [6](#S6), we connected our findings to the anisotropy of token embeddings within each text reported by Xiao2023-vu. As anisotropy is commonly measured using cosine similarity (Xiao2023-vu), we quantify within-text token embedding concentration via the layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$, where $\bm{X}=[\bm{x}_{1},\ldots,\bm{x}_{n}]$ denotes the token embeddings of a text, and each average is taken over all texts in the dataset.(^5^55Note that this quantity is not identical to the anisotropy measure used by Xiao2023-vu.) A higher value indicates greater concentration, consistent with higher within-text anisotropy as described by Xiao2023-vu. Figures [11](#A8.F11)–[16](#A8.F16) show these results for the following model pairs: GTE${}_{\text{base}}$ and BERT (Figure [11](#A8.F11)), E5${}_{\text{base}}$ and BERT (Figure [12](#A8.F12)), Unsupervised SimCSE and BERT (Figure [13](#A8.F13)), GTE${}_{\text{small}}$ and MiniLM (Figure [14](#A8.F14)), E5${}_{\text{small}}$ and MiniLM (Figure [15](#A8.F15)), and all-MiniLM-L12-v2 and MiniLM (Figure [16](#A8.F16)). Fine-tuned text encoders tend to show higher within-text average cosine similarity than their corresponding backbone models, particularly in the later layers. This trend is consistent with the lower SOCM and smaller $S(\bm{X})/\|\bm{\mu}(\bm{X})\|_{2}^{2}$ observed in fine-tuned text encoders (§ [6](#S6)), supporting the connection between within-text token embedding concentration and robustness against collapse by mean pooling. Figure: Figure 11: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for BERT and GTE${}_{\text{base}}$ on the Wikipedia dataset. Refer to caption: 2604.27398v1/x29.png Figure: Figure 12: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for BERT and E5${}_{\text{base}}$ on the Wikipedia dataset. Refer to caption: 2604.27398v1/x30.png Figure: Figure 13: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for BERT and Unsupervised SimCSE on the Wikipedia dataset. Refer to caption: 2604.27398v1/x31.png Figure: Figure 14: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for MiniLM and GTE${}_{\text{small}}$ on the Wikipedia dataset. Refer to caption: 2604.27398v1/x32.png Figure: Figure 15: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for MiniLM and E5${}_{\text{small}}$ on the Wikipedia dataset. Refer to caption: 2604.27398v1/x33.png Figure: Figure 16: Layer-wise average of $\frac{1}{n^{2}}\sum_{j,k}\cos(\bm{x}_{j},\bm{x}_{k})$ for MiniLM and all-MiniLM-L12-v2 on the Wikipedia dataset. Refer to caption: 2604.27398v1/x34.png ## Appendix I Analysis of d μ d_{\mu} and d Σ d_{\Sigma} We examine the components $d_{\mu}$ and $d_{\Sigma}$ that constitute SOCM to understand the variation in SOCM across model pairs observed in § [5](#S5). ##### Observations Figure: Figure 17: Scatter plots of $d_{\mu}$ and $d_{\Sigma}$ for the models examined in § [5](#S5) on Wikipedia. Each point represents a text pair, colored by SOCM. The top and right marginal histograms show the distributions of $d_{\mu}$ and $d_{\Sigma}$. Refer to caption: 2604.27398v1/x35.png Figure [17](#A9.F17) shows scatter plots of $d_{\mu}$ and $d_{\Sigma}$ for all examined models on the Wikipedia dataset. Across all model pairs, fine-tuned text encoders tend to exhibit small $d_{\Sigma}$ values. ##### Connection to Token Embedding Concentration This tendency is consistent with the results of § [6](#S6), which showed that fine-tuned text encoders tend to concentrate token embeddings around their within-text mean. When token embeddings concentrate, their within-text covariance $\bm{\Sigma}(\bm{X})$ becomes small. This directly suppresses $d_{\Sigma}$: since $d_{\Sigma}$ is the scaled Bures-Wasserstein distance between $\bm{\Sigma}(\bm{X}_{1})$ and $\bm{\Sigma}(\bm{X}_{2})$, it satisfies $d_{\Sigma}\leq\bigl(\mathrm{tr}(\bm{\Sigma}(\bm{X}_{1}))+\mathrm{tr}(\bm{\Sigma}(\bm{X}_{2}))\bigr)/4$. Therefore, when both $\bm{\Sigma}(\bm{X}_{1})$ and $\bm{\Sigma}(\bm{X}_{2})$ are small, $d_{\Sigma}$ is bounded to be small as well, regardless of how different the two covariance matrices are from each other. ## Appendix J Trace Bound under Normalization In this section, we prove that the assumption $\mathrm{tr}(\bm{\Sigma}(\bm{X}_{i}))\leq 2$ in § [4.2.1](#S4.SS2.SSS1) holds under the normalization procedure described in § [5](#S5) and certain assumptions about the model architecture. ### J.1 Setup and Notation We consider token embeddings output from a LayerNorm layer Devlin2019-mb. Let $\bm{y}_{i,j}\in\mathbb{R}^{d}$ denote the hidden state before LayerNorm for the $j$-th token in text $i$. The $k$-th dimension of the token embedding after LayerNorm is given by: $$ $x_{i,j,k}=\gamma_{k}\cdot\frac{y_{i,j,k}-m_{i,j}}{s_{i,j}}+\beta_{k},$ (56) $$ where $\gamma_{k},\beta_{k}\in\mathbb{R}$ are learnable parameters for dimension $k$, and $m_{i,j}$ and $s_{i,j}$ are the mean and standard deviation computed across dimensions: $$ $\displaystyle m_{i,j}$ $\displaystyle=\frac{1}{d}\sum_{k=1}^{d}y_{i,j,k},$ (57) $\displaystyle s_{i,j}$ $\displaystyle=\sqrt{\frac{1}{d}\sum_{k=1}^{d}(y_{i,j,k}-m_{i,j})^{2}}.$ (58) $$ Let $\bm{x}_{i,j}=[x_{i,j,1},\ldots,x_{i,j,d}]^{\top}\in\mathbb{R}^{d}$ denote the token embedding after LayerNorm. The token embedding list before normalization is $\bm{X}_{i}=[\bm{x}_{i,1},\ldots,\bm{x}_{i,n_{i}}]$ with $\bm{\mu}(\bm{X}_{i})=\frac{1}{n_{i}}\sum_{j=1}^{n_{i}}\bm{x}_{i,j}$. After normalization by the mean norm (as described in § [5](#S5)), we obtain the normalized token embedding list $\bm{X}_{i}^{\text{norm}}=[\bm{x}_{i,1}^{\text{norm}},\ldots,\bm{x}_{i,n_{i}}^{\text{norm}}]$, where: $$ $\bm{x}_{i,j}^{\text{norm}}=\frac{\bm{x}_{i,j}}{\|\bm{\mu}(\bm{X}_{i})\|_{2}}.$ (59) $$ ### J.2 Assumptions We make the following assumptions: ##### Assumption 1 LayerNorm parameters are shared across all dimensions, i.e., $\gamma_{1}=\cdots=\gamma_{d}=\gamma$ and $\beta_{1}=\cdots=\beta_{d}=\beta$. This assumption reflects the initialization of LayerNorm parameters in typical Transformer models Devlin2019-mb, where $\gamma_{1}=\cdots=\gamma_{d}=1$ and $\beta_{1}=\cdots=\beta_{d}=0$. ##### Assumption 2 Token embeddings within the same text exhibit sufficient similarity. Specifically, the expected cosine similarity between token embeddings within the same text satisfies $\mathbb{E}_{j0.$ (74) $$ Furthermore, this function converges to 2 as $n_{i}\to\infty$: $$ $\lim_{n_{i}\to\infty}\frac{2(n_{i}-1)}{n_{i}+2}=2.$ (75) $$ Therefore, $\mathrm{tr}(\bm{\Sigma}(\bm{X}_{i}^{\text{norm}}))\leq 2$ holds under Assumptions 1 and 2. ## Appendix K Computational Resources All experiments in this paper were conducted using a single NVIDIA RTX 6000 Ada graphics card. In § [5](#S5), computing SOCM values for all text pairs required approximately 2 hours per model. The analyses with MTEB (eng, v2) in § [7](#S7) required approximately 6 hours per model. ## Appendix L Use of AI Assistants In preparing this paper, we utilized AI assistants (Claude, ChatGPT) to support various aspects of the writing and implementation process. These tools were employed for tasks such as code debugging, language polishing, formatting assistance, and generating visualizations. However, all research ideas, methodological designs, experimental analyses, and scientific interpretations presented in this work are entirely our own. The AI assistants served solely as technical aids and did not contribute to the conceptual or intellectual content of this research. \CJK@envEnd