Title: Small Singular Values Matter: A Random Matrix Analysis of  Transformer Models
Venue: NeurIPS 2025
Accepted: yes
Keywords: Random matrix theory

RMT

Llama 3

Spectra

singular value decomposition
Abstract: This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum.
Using  Random Matrix Theory (RMT) as a zero information hypothesis, we associate agreement with RMT as evidence of randomness and deviations as evidence for learning. 
Surprisingly, we observe pronounced departures from RMT not only among the largest singular values - the usual outliers - but also among the smallest ones. 
A comparison of the associated singular vectors with the eigenvectors of the activation covariance matrices shows that there is considerable overlap wherever  RMT is violated. Thus, significant directions in the data are captured by small singular values and their vectors as well as by the large ones. We confirm this empirically: zeroing out the singular values that deviate from RMT raises language-model perplexity far more than removing values from the bulk, and after fine-tuning the smallest decile can be the third most influential part of the spectrum. To explain how vectors linked to small singular values can carry more information than those linked to larger values, we propose a linear random-matrix model.  Our findings highlight the overlooked importance of the low end of the spectrum and provide theoretical and practical guidance for SVD-based pruning and compression of large language models.

Small Singular Values Matter: A Random Matrix Analysis of Transformer Models

Abstract

This work analyzes singular-value spectra of weight matrices in pretrained transformer models to understand how information is stored at both ends of the spectrum. Using Random Matrix Theory (RMT) as a zero information hypothesis, we associate agreement with RMT as evidence of randomness and deviations as evidence for learning. Surprisingly, we observe pronounced departures from RMT not only among the largest singular values - the usual outliers - but also among the smallest ones. A comparison of the associated singular vectors with the eigenvectors of the activation covariance matrices shows that there is considerable overlap wherever RMT is violated. Thus, significant directions in the data are captured by small singular values and their vectors as well as by the large ones. We confirm this empirically: zeroing out the singular values that deviate from RMT raises language-model perplexity far more than removing values from the bulk, and after fine-tuning the smallest decile can be the third most influential part of the spectrum. To explain how vectors linked to small singular values can carry more information than those linked to larger values, we propose a linear random-matrix model. Our findings highlight the overlooked importance of the low end of the spectrum and provide theoretical and practical guidance for SVD-based pruning and compression of large language models.

1 Introduction

Large language models (LLMs) have become foundational in deep learning, revolutionizing natural language processing tasks such as translation, text classification, and question answering [27, 45, 48, 51]. Despite the well-documented success [29], a thorough theoretical understanding of their inner workings remains incomplete. Although researchers have investigated various facets of LLMs [37], fundamental questions persist about how these models encode information and the specific roles of their components.

One promising approach for gaining deeper insights is the application of random matrix theory (RMT), which has proved effective for identifying structural properties and information density in neural networks [31, 43, 44]. In particular, RMT analysis of the spectrum of weight matrices can help determine where information is located within models. When networks are randomly initialized, the weight distributions precisely matches RMT predictions. After training, deviations from these predictions reveal how model parameters have adapted.

Matthias Thamm

Building on these insights, we use RMT to pinpoint regions in LLMs where relevant features are encoded, by identifying deviations from the RMT-predicted spectrum. We study the singular value spectra of weight matrices from three pretrained models: Bert 1 [26], Pythia 2 [9], and Llama-8B 3 [14]. We identify the regions lying outside the theoretically predicted Marchenko-Pastur spectrum [30] as areas of feature learning by comparing the corresponding singular vectors with the covariance matrix of the layer activations, finding strong similarity. Interestingly, this phenomenon is not only present for the largest but also for the smallest singular values. This similarity stays consistent across different blocks of the transformer architectures and holds for all three models we examine. When removing groups of singular values (and associated vectors) from these models, performance degrades most significantly for the smallest and largest singular values that violate RMT properties.

Additionally, we contribute to the ongoing discussion about removing small singular values in LLMs. Previous studies suggest that small singular values can be relevant for generalization [23], while other work indicates potential benefits from removing them [41]. However, the most common perspective is that they are negligible, as their removal is the optimal low rank W ′ solution for weights W under the L2 norm | W ′ -W | 2 [15]. We reconcile these perspectives by showing in which matrix types small singular values are important, and that the potential damage that is done by removing the smallest singular values from a pretrained transformer can be recovered by a fine-tuning step. Our results are of crucial relevance to any researcher doing SVD-based pruning with LLMs. All code to generate the figures is open source and available under [2].

2 Related Work

RMT has been widely used as a calculational tool for performing statistical averages in the analysis of machine learning models. Early applications of RMT to neural networks, such as [35], analyzed the spectral properties of loss surfaces in deep learning, providing insights into learning dynamics. Building on this foundation, Baskerville et al. [6] derived universal aspects of outliers in loss surfaces. Beyond its role in statistical analysis, RMT has been proposed as a tool for analyzing trained network weight matrices [34]. In [31], RMT was applied to weight matrices by examining the learning dynamics of image recognition models through their spectra. Following up on this work, Martin et al. [32] suggested that large outliers in the singular value spectrum are indicative of well-trained matrices. Further studies [28, 44] reinforced RMT's utility in understanding how networks evolve during training. They demonstrated that deviations from RMT predictions indicate where feature learning occurs, as opposed to lazy learning [12], where weights remain close to their initial random state. These findings underscore RMT's potential for identifying regions of learned features without the need for training data.

Transformers present unique challenges in understanding information storage. Prior work [25, 38] has shown that different matrix types specialize in storing distinct types of knowledge, while Aken et al. [1] examined how semantic information is encoded in neuron activations. In [39], LLMs are compressed by a low-rank approximation of the weights that optimizes for a minimal change in the activations. [49] compress LLMs based on a similar metric but transform the data on which the activations are computed first, as they find that the rank of the singular values does not represent their relevance otherwise. To overcome such issues with SVD-based approximations, several works [11, 16, 47, 50] focused on quantizing weights instead of creating low-rank approximations.

The low-rank structure of features in neural networks has been explored in [20]. Yu and Wu [52] highlight that even though the feature matrices of transformers are often low rank, their weight matrices are not, revealing a complex relationship between representations and parameters. Positional encodings, crucial to transformer performance, have also been studied for their role in shaping the learned feature space [46].

3 Notation and Spectral Properties of LLMs

Large language models (LLMs) are typically composed of three main parts: an initial embedding layer, a repeated stack of transformer blocks, and a final output layer. Each transformer block

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B ( l ) , where l ∈ [1 , N ] denotes the block index, has the same internal structure and weight matrix shapes. In this work, we use a consistent naming convention for the weight matrices within each transformer block. The attention sub-layer uses the Query matrix W ( l ) Q , the Key matrix W ( l ) K , the Value matrix W ( l ) V , and the Attention-Output matrix W ( l ) O . These matrices form the multi-head attention mechanism, which can be written as

Following the attention sub-layer, the feedforward sub-layer typically employs an Up-Projection matrix W ( l ) U and a Down-Projection matrix W ( l ) D . Depending on the architecture, there may also be a Gate-Projection matrix W ( l ) G . In the case of a Gated Linear Unit (GLU), these matrices are related by

where ⊙ denotes the Hadamard product. If no GLU is used, W U and W D are combined with a nonlinearity to form a standard feedforward layer.

To analyze these weight matrices, we use a singular value decomposition (SVD) to factor each weight matrix into its singular values and singular vectors. For a matrix W ∈ R m × n , the SVD is

where U ∈ R m × m and V ∈ R n × n are orthogonal matrices containing the left and right singular vectors, respectively, and S ∈ R m × n is a diagonal matrix holding the non-negative singular values.

When q = n/m stays constant, m,n →∞ and the entries of W are i.i.d. with finite variance σ and zero mean, the distribution of its singular values follows the Marchenko-Pastur (MP) law [30]

where n ≥ m without loss of generality. In large, randomly initialized neural networks, the singular values of weight matrices typically approximate this distribution. After training, deviations from the MP law indicate learned structure, often manifested as outliers beyond the MP bounds [ ν -, ν + ] . Past work suggests that such outliers can signal effective feature learning [32, 44].

When plotting the theoretical Marchenko-Pastur curve of an empirical matrix, we estimate the standard deviation of the underlying matrix as described in [4]. This is used in Figure 1 where we illustrate the Marchenko-Pastur distribution together with the spectra of trained and random weight matrices. We see that for random weight matrices of BERT size, where the smaller dimension has N = 768 , the theoretical prediction agrees perfectly with the empirical spectrum as shown in

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panels (a) and (c). When training weight matrices, outliers emerge as shown in panels (b) and (d) for the Key and Up-Projection matrix of Pythia. An important observation is that in the case of square matrices, outliers can only emerge towards larger values. In the case of non-square matrices, the theoretical spectrum has a minimal value that is strictly larger than zero, allowing for outliers towards small singular values. Neglecting these values to construct a low rank matrix W ′ from W is often considered optimal as the norm | W ′ -W | 2 is altered the least. However, we will show in the following that these singular values contribute significantly more to the performance of the network than larger singular values that remain in the bulk.

4 Overlap of Features and Weights

The following chapter shows how the singular vectors that correspond to outliers of the spectrumsmall and large- can be related to important directions of the data matrix. This is done by computing the covariance matrix of the activations that enter each matrix in each block and comparing the eigenvectors of this covariance matrix to the singular vectors of the corresponding weight matrix.

Formally we define the activation covariance matrix C ( W ( ℓ ) ) of matrix W ( ℓ ) using the input activations h ( W ( ℓ ) ) to matrix W ( ℓ ) and the average input activation entering that matrix ¯ h ( W ( ℓ ) ) . Using the index i as a token index that runs over the dataset, we compute

where we dropped the explicit dependence on W ( ℓ ) for cleaner notation. The resulting matrix C is symmetric, and therefore has an orthonormal set of eigenvectors f i with eigenvalues λ i .

Since the transformation of matrix W is given as W h + b , where W has singular value decomposition W = U S V ⊤ , the activations h are mapped onto the space spanned by the right singular vectors V . To see whether a particular eigenvector f j of the activation covariance matrix aligns with one of the right singular vectors v k of W , we compute

This measure quantifies the extent to which a singular vector captures a specific direction of the data represented by the activation covariance matrix. In the following, we estimate C from the WikiText [33] dataset. The appendix shows additional results from the BookCorpus [53] datasets.

Figure 2 shows the overlap computed based on Eq. (7) in comparison to the spectrum of our three transformers. As the displayed matrices of Bert, Pythia, and Llama are square, we find outliers from the Marchenko-Pastur distribution only on the right side of the spectrum (upper panels). The bottom panels show the overlap O k of each singular vector at the position where the corresponding singular value is in the spectrum. We observe that singular vectors corresponding to outliers of the spectrum have a strong correspondence to one of the eigenvectors of the activation covariance matrix. We can

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interpret these singular vectors as picking up a direction of the data with high variance and weighting it highly. For reference, the grey lines display the 3 σ interval when assuming singular vectors from a random matrix. We find several statistically highly significant points above these lines.

In Figure 3 we show rectangular matrices of all three transformers, where in this case, outliers of the spectrum can emerge towards large values and small values. Interestingly, we observe that singular vectors of both groups have strong correspondence to an eigenvector, indicating that the singular vectors corresponding to the smallest singular values encode properties of the activation covariance matrix and hence of the data! This is an exceptional result as small singular values contribute the least to the overall matrix with respect to the L2 norm, and they are the first ones to be discarded when pruning naively. The results are again statistically highly significant even with respect to a 3 σ interval.

The two plots illustrated the relation between small singular values, and the data for specific matrices. To further look into this phenomenon, we present all matrix types of block 10 for both Pythia and Llama in Figure 4. Other blocks are displayed in the appendix. We see that for Pythia (upper 2 panels) Query, Key, and Value matrix exhibit increased overlap only for their largest singular values. The Attention-Output matrix has no significant overlap with the activation covariance matrix, a phenomenon that holds for BERT, Pythia, and Llama, as elaborated in the appendix. The Up and Down-Projection matrices exhibit significantly increased overlaps for both their largest and smallest singular values. However, we also find singular values with small overlaps in this region. This is different for Llama's Up and Gated-Projection matrix, where the small and large singular values significantly overlap with the covariance matrix. Interestingly, the singular vectors of the Down-Projection matrix have no increased overlap with any of the eigenvectors from the activation covariance matrix. We suspect that the placement of this matrix at the end of the Gated Linear Unit-rather than after a separate nonlinearity-may render certain weights redundant, leading to no significant overlap. Surprisingly, we find that the Key and Value matrix do not exhibit large overlaps for their smallest singular values while being non-square matrices. We speculate that this arises from their indirect relation to the activations of the previous layer, which are processed in the Attention mechanism. The development of the overlap as training progresses can be found in the appendix.

5 Contribution of Singular Values to Model Performance

In the preceding section, we observed a strong overlap between the singular vectors of weight matrices - particularly in regions outside the Marchenko-Pastur (MP) boundary - and the eigenvectors of the activation covariance matrix. To further evaluate the importance of these singular values and their corresponding vectors, we conduct experiments in which we selectively remove certain groups of singular values. Specifically, removing a singular value ν r from a weight matrix W involves zeroing

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out ν r in the diagonal matrix S of its singular value decomposition,

We then reconstruct the weight matrix ˜ W using the original singular vectors. To compare the relative impact of different parts of the spectrum, we group the rank-ordered singular values of each matrix into ten equal-sized deciles, with the smallest 10% in the first decile and the largest 10% in the tenth. We apply this procedure by choosing one matrix type (e.g., Query) and zeroing out one of the ten deciles in all weight matrices of that type. We then measure the resulting increase in ∆ Perplexity = PPL( f ˜ W ) -PPL( f W ) , on the WikiText dataset. The results are shown in Figure 5, where removing the largest singular values consistently causes a large perplexity increase in all matrix types for both Pythia and Llama. This is expected as the largest singular values and corresponding vectors are the directions that describe the largest percentage of the variance of the next layers' activations.

When going through the deciles from large to small, we observe that the contribution of the deciles decreases monotonically for all square matrices. For the non-square matrices, the largest nine deciles also follow this trend. However, for non-square matrices, the smallest decile always contributes more than some of the larger deciles. This observation underscores that valuable information can be encoded at the lower end of the spectrum, potentially explaining why standard pruning strategies often struggle with transformers [3].

These findings are further backed up by Llama results on the GSM8K benchmark, which tests basic math problems. These results are depicted in Table 1, where decile one corresponds to the smallest and decile ten to the largest singular values. For the Down-Projection matrix, we find that the smallest singular values (decile 1) are the second most important decile, which is in excellent agreement with our previous results for this rectangular matrix (compare Figure 5). In contrast, considering the quadratic Attention-Output matrix, the smallest singular values are one of the least important deciles, in agreement with theoretical expectations and the previous observations. The smallest singular values of the non-quadratic Gate-Projection matrix are the fourth important decile, also in excellent agreement with the results of Figure 5. For the quadratic Query matrix, only the two largest deciles appear to be important. Further results for the HumanEval benchmark show a similar pattern and are displayed in the Appendix.

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We provide further experiments for removing deciles from all weight matrix types for LLaMA 3.1 8B Chat on the RULER [22] benchmark for a context length of 8192. The tasks needle in a haystack (niah) were evaluated with several values to be extracted (niah multiV) and for several queries (niah multiQ). We also conducted experiments on value tracking (vt), common word extraction (cwe), and frequent words extraction (fwe). The results when removing singular values from all layers simultaneously are shown in table 2. We find that the smallest singular values are even the second most important decile in this task.

For Bert we additionally conduct fine-tuning experiments on BoolQ [13], rte [8, 19, 21] and SST2 [42]. We compare two scenarios: (i) removing a particular decile of singular values before fine-tuning and then fine-tuning the altered model, versus (ii) fine-tuning the model first and then removing the singular value decile. Now, we are removing deciles from all weight matrices simultaneously to compare to previous literature results and estimate the 3 σ interval from the standard deviation of six runs without pruning. Figure 6 shows the results when removing singular value deciles from Bert, where in case (i) -left pannel, prune first-, removing small singular values reduces the accuracy slightly but statistically significantly in the case of RTE and SST22. For BoolQ, the lost knowledge was either recovered or not necessary. By contrast, in case (ii) -right panel, fine-tune first-, removing the same small singular values degrades accuracy significantly in all cases, showcasing their surprising relevance. Large singular values (decile ten) are consistently important, as removing them drops performance to near random guessing (off-scale points in the right panel). This finding provides insights into a recent debate about the importance of small singular values in transformers: some argue that they are essential for good performance [23], while others report performance gains from their removal [41]. Our results clarify this conflict by showing that small singular values do matter,

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but mainly once the model has been fine-tuned. Hsu et al. [23], fine-tune before pruning, and observed that small singular values are critical, while Sharma et al. [41] found gains from pruning them, and evaluating the model without additional fine-tuning. We interpret this behavior as evidence that fine-tuning - and potentially alignment- relies on small singular values and their associated vectors. Since alignment may degrade performance on reasoning tasks [36], removing these directions can improve task metrics. However, doing so after alignment could unintentionally erase that alignment.

6 Minimal Random Matrix Theory Model

The aim of this section is to provide a minimal RMT model for the occurrence of small singular value outliers. For this, we consider a simple two-layer linear student model, trained on Gaussian inputs, and labels produced by a linear rank-1 teacher. The effects of non-trivial correlations in the noise depend on whether the noise has predominantly small or large covariance in the direction of the outlier.

A common model to describe weight matrices W consists of a low-rank matrix W 0 encoding the rule and a noise matrix X as W = W 0 + X . Under the common assumption [24, 43] of white noise, ⟨ X ⟩ = 0 , Cov( X ) = ✶ , singular values of W 0 can only be outliers in W if they lie above the Marchenko-Pastur bulk of X , i.e. above the BBP critical value [5, 7, 10]. In the following, we present a model with a non-trivial covariance of the noise, which can have a singular value of W below the MP bulk.

We start from a teacher-student setup with teacher function σ T ( ξ ) = u ⊤ λ ξ with normalized u ∈ R N × 1 and λ ∈ R . The student is a linear two-layer network σ S ( ξ ) = v ⊤ W ξ with normalized, fixed second-layer weight v ∈ R K × 1 and trained weight matrix W ∈ R K × N of aspect ratio q = K/N < 1 . Here, v may be interpreted as a learned, stabilized weight singular vector in a subsequent layer of a real network. We show in the following that this model explains the occurrence of small outliers below the MP bulk, and that it relates the amount of information contained in the outlier singular vector to the distance of the outlier to the bulk.

Weconsider a mean-squared-error loss function L ( ξ ) = ( u ⊤ λ ξ -v ⊤ W ξ ) 2 / 2 for inputs ξ ∈ N (0 , 1) , giving rise to a generalization error

Here we adopted the notation A 2 = A ⊤ A . We study the weight matrix ensemble

where βα →∞ corresponds to perfect learning, and the α term has the role of a prior for weight matrices at initialization or an L 2 regularization strength. In an annealed approximation, β corresponds to the number of examples shown to the network divided by the input dimension [17]. Here, the mean is W 0 = αβλ 1+ αβ vu ⊤ , and the correlation matrix is given by

Weight matrices of this matrix ensemble have the form W = W 0 + X , with a random matrix X of zero mean and row covariance matrix Σ (while columns are uncorrelated). For 0 < (1 + αβ ) -1 < 1 , the noise matrix X has an outlier singular value below its Marchenko-Pastur bulk. This stabilizes a small singular value outlier of W if λ < √ α (1 -q ) at [18]

For the case of perfect learning, i.e. β → ∞ , the outlier singular value approaches λ √ 1 -αq α -λ 2 , only differing from λ of W 0 by the level repulsion with the bulk.

In Fig. 7, we depict ⟨ ν min ⟩ (left panel) and the singular vector overlaps (right panel) ⟨| ˜ u · u |⟩ (blue) and ⟨| ˜ v · v |⟩ (red) of the corresponding singular vectors ˜ u , ˜ v of W = ˜ V ˜ S ˜ U ⊤ as a function of 1 / (1 + αβ ) . We find that ideal learning, 1 / (1 + αβ ) → 0 , corresponds to an outlier maximally separated from the MP bulk with best overlap of the corresponding singular vectors of W with those of W 0 . For poorer learning, the singular value ν min moves towards the bulk and the vector overlap ⟨| ˜ u · u |⟩ approaches the expected overlap between random vectors (blue, dashed line), when outlier and MP bulk fuse at β bulk = α 2 ( 1 -√ α 2 -4 √ qαλ 2 ) / 2 λ 2 .

Limitations

We show empirical evidence that the singular vectors corresponding to the smallest singular values encode important directions in the data. In case of the Down-Projection matrix of Llama and the Attention-Output matrices we do not observe an enhanced overlap with the activation covariance matrix, while the smallest singular values of the Down-Projection matrix are still important for model performance, as the perplexity increases upon their removal. We argue that this is due to the lack of a non-linearity in these layers, however further research is needed to clarify this. Similarly, we argue that the laziness of the Attention-Output matrix is responsible for the lack of overlap with the activations, which needs to be verified in more detail. To further strengthen the validity of our theoretical model, the covariance of the noise present in multiple LLM weight matrices should be computed, which is only possible when doing multiple training runs.

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7 Conclusion

In this paper, we analyzed the singular value spectra and activation covariance matrices of Bert, Pythia, and Llama-8B models. We demonstrated that the spectra deviate from the Marchenko-Pastur distribution, not only for large singular values, but also for small singular values in the case of nonsquare matrices. The singular vectors corresponding to the singular values outside of the theoretically predicted spectrum were shown to be related to specific directions in the activations entering that matrix. When removing these singular values in deciles for Llama and Pythia, we found that while the largest singular values are the most important decile in all cases, the smallest singular values can even be the second most important decile. This importance is established for non-square matrices, where the smallest singular values fall outside of the Marchenko-Pastur region.

Furthermore, our fine-tuning results shed light on the debate over the relevance of small singular values in LLMs. While removing small singular values after fine-tuning drastically lowers model performance, removing them beforehand sometimes has no statistical effect on the final task, explaining some discrepancies in the literature. Using a random-matrix model we demonstrate that for singular values well below the MP-bulk, the associated singular vectors are more similar to the teacher vector than for singular values close to or inside the bulk. Thus, small singular values and associated vectors carry more information than larger ones.

In summary, this work demonstrates the importance of small singular values, showcasing when and where they are important, while additionally providing a theoretical basis for how this phenomenon can occur. We expect these insights to be crucial for future SVD-based pruning algorithms.

Acknowledgments and Disclosure of Funding

Computations for this work were done using resources of the Leipzig University Computing Center.

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Att.-Out

Up-Proj.

Down-Proj.

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2.5

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Gate-Proj.

Llama Block 5

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2.5

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Llama Block 15

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A Activation Covariance Overlap

We show additional results that demonstrate the generality of the observed overlaps between the right singular vectors of the weight matrix and the eigenvectors of the corresponding activation covariance matrix. The results for block 5 of Llama and Pythia are provided in Fig. 8, while the results for block 15 are shown in Fig. 9. When singular values deviate from the Marchenko-Pastur distribution, we again find a significant overlap of the corresponding singular vectors and the eigenvectors of the activation covariance matrix, particularly in the case of small singular values.

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T=143

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singular vector index singular vector index

singular vector index singular vector index

singular vector index singular vector index

To further study the outliers, in particular the ones corresponding to smaller singular values, we additionally analyze the temporal evolution. Fig. 10 shows the overlap of the activation covariance matrix and the weight matrix in the training of Pythia for block 10, computed on the WikiText dataset. As expected, there is no overlap at initialization as both the weight and the activation covariance matrix are random. Interestingly, already at T = 1 (corresponding to 1000 gradient updates), the model forms a significant overlap with the activation covariance matrix for all matrix types. In case of the Attention-Output matrix, this overlap is later reduced, while the overlap of Query, Key, and Value increases significantly, which we interpret as repositioning of the information in the weight matrices. The overlap of the singular vectors corresponding to small singular values emerges in the later stages of training for the Down-Projection matrix and increases drastically for the Up-Projection matrix. Considering that fine-tuning can significantly impact the smallest singular values, we speculate that the smallest singular values can be related to potentially more sophisticated concepts, which may be learned in the later stages of pre-training.

B Additional Results on Book Corpus

To show that the results presented in the manuscript are of a general form and do not depend on the WikiText dataset, we conduct additional experiments on the Book Corpus dataset. Figure 11 confirms that for all non-square matrices, the smallest singular values are more important than some of the larger deciles. In case of the Llama Down-Projection matrix, the smallest decile is even the second most important group. These results can be attributed to the same phenomenon as shown in Figure 12. The overlap between singular vectors and eigenvectors of the activation covariance is very similar to the one observed in the main manuscript, showcasing large overlaps for some of the smaller singular values.

C Special Feature of the Activation Covariance Basis

When computing the overlaps between a particular singular vector and the eigenvectors of the activation covariance basis, we defined the quantity

as the maximum coefficient when developing the singular vector v in the basis of the eigenvectors f j . If one of the 768 -4096 coefficients is large, we argue that these two vectors have a strong correspondence. A natural questions is the behavior of the quantity in a different basis. In principal there exist a trivial basis for which the overlap is exactly one, i.e. the singular vectors v j themselves.

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In Figure 13 we display the quantity

which is just the maximal component of each singular vector. For Pythia, M k is small for the singular vectors corresponding to largest singular values in all cases but the Down-Projection matrix, showcasing that the activation covariance basis is a special basis in this case. For Llama we observe a different picture where the new basis has almost a complementary behavior to the activation covariance eigenbasis. While the Attention-Output and Down-Projection matrix of Llama do not show increased overlaps O k , they appear to be strongly localized, indicated by large M k . This is an interesting additional observation and may explain the large perplexity increases when removing the smallest singular values of the Down-Projection matrix.

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D Lazy Learning Regime

It is possible to train neural networks in the lazy regime where the final weights of the trained model are very close to the initial ones [12]. By rescaling the input of the softmax function in the final layer by a constant α > 1

we achieve that very small changes in the output logits prior to the softmax function have a large effect on the output after the softmax function. To allow for learning with a usual learning rate, the loss is changed to

to incorporate the large differences in the output activations a L induced by small weight changes. When training such networks, previous studies [28, 44] demonstrated that RMT properties stay intact in the case of lazy learning, where weights remain close to their initial random state. However, large outliers (i.e. singular values outside of the MP-boundary [ ν -, ν + ] ) often reflect critical learned features leading to the results that models trained in the lazy regime generally perform worse than models trained in the rich or feature learning regime. In Figure 14 we compare the average singular value spectrum of all blocks to its initial state for both the Query and Attention-Output matrix. We find that for all three models, the Attention-Output matrix has significantly fewer outliers. Considering that there is no overlap with the activation covariance matrix either, we speculate that the Attention-Output matrix may be trained in the lazy regime.

E Computational Resources

The computational resources needed to compute the shown results are mostly defined from the perplexity computations. Computing spectra and the activation covariance matrices for all models can be estimated as less than 100 GPU hours on a V100. To compute the perplexity on subparts of the WikiText and BookCorpus dataset several times with different parts of the model removed, we estimate less than 1000 GPU hours.

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F Details on Random Matrix Model

As described in the main text, we consider the matrix ensemble

Using the expression for the generalization error Eq. (10), we first complete the square

with

Computing the inverse of Σ -1 yields Eq. (12).

We compute the partition function, i.e. the normalization, by performing the Gaussian integral

We can then compute the generalization error

The expectation value of the generalization error is minimal for largest separation ( 1 / (1 + αβ ) → 0 ) of the observed singular value ν min from the MP bulk of the spectrum of W , suggesting that moving singular values of relevant directions out of the bulk can occur naturally when minimizing the loss during training.

In addition, the expectation value of the small outlier singular value ⟨ ν min ⟩ , Eq. (13), can be analytically obtained in the limit N → ∞ , q = K/N = const . For this, the outlier eigenvalue η min ≡ α -α 2 β/ (1 + αβ ) + ( λαβ/ (1 + αβ )) 2 of Σ+ W ⊤ 0 W 0 is related to ⟨ ν min ⟩ 2 via the blue function [18],

where g Σ ( x ) = δ ( x -α ) is the density of Σ eigenvalues in the K →∞ limit, for which the single outlier contribution is negligible. Thus B ( y ) = 1 /y + qα/ (1 -αy ) and ⟨ ν min ⟩ follows from Eq. (25) as given in Eq. (13).

G Layer Specific Statistics

As a first step towards providing more comprehensive statistics for each layer, we present outlier counts for several layers of Llama3-8B in the format [number of left outliers | number of right outliers] in Table 3. We observe that for some matrices related to the attention mechanism (Query, Key, Value, Att.-Output), the number of outliers to the right of the spectral bulk is significantly reduced in layers closer to the output. However, the number of outliers to the left of the spectrum in rectangular matrices is slightly increased in later layers, with a peak in layer 19 in our example.

To also provide a metric on the layer-specific importance of the smallest singular values, we remove deciles from all matrices in that specific layer and measure the perplexity on the BookCorpus dataset to see whether a trend emerges. The base perplexity without removal is 6.0045, and the results are presented in Table 4. We see that removing singular values from a single layer has only a small effect on perplexity in general, a notable exception being the removal of the largest singular values from Layer 0. The second most important decile of Layer 0 is that of the smallest singular values. For the other layers studied, removal of the largest singular values has the largest effect on perplexity, in good agreement with the number of outliers in the first layer. Furthermore, layers 19 and 24 appear to be the least important ones with respect to perplexity, which is also in good agreement with the number of large outliers.

H Scaling Relations

To analyze the scale on which our results for small singular values hold, we compare the scaling of the number of outliers for various model sizes. In general, we know that larger weight matrices display less finite-size effects when comparing them to random matrix theory results derived for infinitely large matrices. To analyze whether there might be a finite size scaling, we compare our three models: Bert with 110M parameters (embedding dimension d = 768 ), Pythia with 506M parameters and d = 1024 , and Llama with 8.03B parameters and d = 4096 .

To quantify the number of singular value outliers, we considered the percentage of singular values smaller than the lower Marchenko-Pastur bound (averaged over all layers) in the matrix types where we showed that small singular values are important: For the Up-Projection matrices we find for Bert,

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Pythia, and Llama [3 . 8% , 5 . 9% , 3 . 2%] , and for the Down-Projection matrix [2 . 1% , 6 . 5% , 4 . 7%] . We do not observe a clear trend here, which might indicate that the number of relevant small singular values scales linearly with model size, i.e., is a constant in terms of percentages. However, it is challenging to derive scaling relations from only three models, especially since they are far from identical, and other details of training may be relevant too.

I Reduction in the Space of Activations

In the main text, we address the relevance of small singular values and demonstrate a correspondence between their corresponding singular vectors and the eigenvectors of the activation covariance matrix. As modern reduction algorithms often aim to reduce the number of network parameters by reducing the dimension of the activation space [3, 40], it is also crucial to identify exactly to which eigenvectors these singular vectors correspond. This would provide additional information about the relevance of specific directions and, therefore, may enable the construction of more precise reduction algorithms.

While a detailed analysis would exceed the scope of this paper, Figure 15 provides an idea of what insights may be gained from such an analysis. Here, small indices (bottom left corner) correspond to eigenvectors ( y axis) and singular vectors ( x axis) for the largest eigenvalues and singular values, respectively. We observe that in the case of the Query matrix of Llama (left panel), the largest overlaps can be found between the eigenvectors corresponding to the smallest eigenvalues and singular vectors corresponding to the largest singular values, which is surprising as it indicates the relevance of directions in activation space which are small in terms of the variance that they carry. A similar observation can be found for the Up-Projection matrix of Llama (right panel), where both the eigenvectors for smallest and largest eigenvalues have an increased overlap with the singular vectors for the largest singular values. Surprisingly, there is a correspondence between vectors for the smallest singular values and smallest eigenvalues, which may indicate that they are very far from random noise. However, further investigations with a wider scope are necessary to provide a clear picture of this phenomenon.

J Additional Results on HumanEval

When removing deciles of singular values (starting with the smallest to the largest ones) from the Down-Projection matrix (Down-Proj.), the Attention-Output matrix (Att.-Out.), the Gate-Projection matrix (Gate-Proj.), or the Query matrix, we observe the performances displayed in Table 5 on the HumanEval benchmark. In excellent agreement with previous results, we find that the smallest singular values of the Down-Projection and Gate-Projection matrices are important, which is not the case for the smallest singular values of the quadratic matrices.

Peer reviews:
{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper uses the MP law to estimate what a typical distribution of singular values should look like. For trained matrices, outliers arise from the typical distribution, on both left and right side (small and large values) for non-square matrices. This reveals that there exist some small singular values that are of relevance to the training of the matrix - thus truncating singular values by magnitude (as is usually done for frobenius norm minimization, per Ben-Eckart) may not be a good idea.\n\nUsing activation covariance eigenvectors, correlations are measured with weight singular vectors. It turns out that indeed, singular vectors associated with those singular values falling outside the MP law do indeed excite activation eigenvectors.\n\nSTRENGTHS AND WEAKNESSES:\nNice paper giving a fresh perspective into singular values decomposition of weights. I like the departure from the usual frobenius norm minimization and the fact that the study compares trained matrices to random ones and essentially asks what different because such differences are likely what makes the model strong. Good introduction of interesting concepts.\n\nIn terms of weaknesses, there are two main issues:\n1) the paper does not showcase empirically how strong the proposed method derived from the study can be at compressing models. It would be good to see comparisons of the method to SOTA techniques in tensor decomposition such as ESPACE, BLAST, etc..\n2) Some minor issues that need answering. Please refer to the \"Questions\" part of the review.\n\nQUESTIONS:\n- It is interesting to study RMT applied to weight matrices. But what about activations, there are much more redundancies in activations which can be better exploited to compress a model using matrix factorization, as was shown in [1]. Can we perform a similar study for activations?\n[1]: \nSakr, Charbel, and Brucek Khailany. \"ESPACE: Dimensionality Reduction of Activations for Model Compression.\" Neurips 2024\n\n- For the spectral overlaps in Fig 2 & 3, it seems rather qualitative, and more like the authors are trying to to convince us that something is there by telling us it's there (namely by switching between blue and red colors). However, it does seem like some vectors in the blue region do exhibit strong overlap with the activation eigenvectors. I wonder if there is a more proper way to visualize these (maybe log plots). And please comment on why some blue singular vectors have such strong overlap.\n\n- In the experimental setup of section 5, why are deciles used? Why can't we just do the truncation as per the description of the earlier section, i.e., take out the singular values that supposedly arise from the randomness, and leave the outliers that supposedly came about during training? Such a 1-to-1 experiment correlating with the earlier theory would be interesting.\n\n\n- Not a question but the last sentence of the abstract is repeated twice.\n\nLIMITATIONS:\nYes, it's there on pages 8-9 just before the conclusion.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 5, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-29"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThis paper revisits transformer models via random matrix theory. Empirically, they found out that while small singular values typically represents \"outliers\", removing those information significantly impacted model performance, even more so than removing bulk values.\nAfter fine-tuning, the bottom decile can become the third–most influential spectral region. \nTo explain this, the paper proposes a linear teacher–student RMT model in which noise covariance induces small‐value outliers whose vectors carry more task‐relevant information the further they lie from the bulk.\n\nSTRENGTHS AND WEAKNESSES:\n## Strength\n1. I think the insights from this paper is novel. This paper highlights an overlooked spectral regime (small singular values) as carriers of learned features, overturning the conventional emphasis on large‐value outliers alone.\n2. I think the paper carries out several interesting potential directions in transformer theories, like a new perspective of RMT, analyzing small singular values etc.\n3. While the paper mainly present empirical findings, I do think it provides practical guidance, theoretical implications to the ML community.\n4. The paper is very well written, easy to follow for readers.\n\n## Weakness\n1. To my knowledge, the RMT model used in the paper assumes the inputs to follow Gaussian distribution, and with linear attention. The latter assumption does not seem to be general to all transformer variants.\n2. It would be nice to see some further explanation on the connection to \"lazy training\". So far, it is not clear to me why \"lazy training\" would lead to their empirical results.\n\nQUESTIONS:\n1. Can the authors provide further explanation on the connection to lazy training? (Appendix D) As to my knowledge, lazy training only seems to happen when one has infinitely wide networks.\n2. How does the phenomenon scale with model size? Are small-value outliers equally important in different sizes of models?\n\nLIMITATIONS:\nYes, it is right before the conclusion.", "TITLE": "Official Review", "RECOMMENDATION": 5, "REVIEWER_CONFIDENCE": 3, "CLARITY": 4, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-06-30"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nBy analyzing weight matrices through Random Matrix Theory, the study finds that both the largest and smallest singular values deviate from randomness, indicating they encode meaningful information. Removing non-random singular values greatly impacts model performance, highlighting the overlooked importance of small singular values and offering practical insights for compressing large language models.\n\nSTRENGTHS AND WEAKNESSES:\n**Strengths**\n\n1. Theoretical analysis is solid. The paper explain the phenomenon that small singular values that occur in practice also have a significant impact.\n\n2.  It has certain practical significance and provides insight for designing a more suitable pruning or LLM optimization scheme in the future.\n\n3. The article is well presented and easy to read.\n\n\n\n\n**Weakness**\n1.  Evaluation is limited. We know that PPL is often inaccurate, especially in long context tasks [1][2], so I strongly recommend the authors to evaluate tasks like needle in a haystack. \n\n[1] Can perplexity reflect large language model’s ability in long text understanding?  ICLR 2024.\n\n[2] Base of RoPE Bounds Context Length. NeurlPS 2024.\n\nQUESTIONS:\n1) I noticed that since small singular values have a significant impact, I speculate that amplifying them will be beneficial for the model's learning, and this seems to be related to the muon optimizer [1]. What is your opinion on this?\n\n2) About evaluation, multiple choice and PPLs are often misleading, and some studies have revealed that many components in the network contribute very little to multiple choice questions[2].  The impact on generating questions is quite different [3].  Therefore, in addition to needle in a haystack, I also suggest that the author measure more benchmarks for generative tasks.\n\n\n\n\n[1] https://kellerjordan.github.io/posts/muon/\n\n[2] Understanding the skill gap in recurrent language models: The role of the gather-and-aggregate mechanism.\n\n[3] Shortgpt: Layers in large language models are more redundant than you expect.\n\nLIMITATIONS:\nSee weakness.", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 2, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-01"}

{"IS_META_REVIEW": false, "comments": "SUMMARY:\nThe paper applies Random Matrix Theory to pretrained Transformer weights, showing that both large and small singular-value outliers—identified against the Marchenko–Pastur baseline—align with salient activation directions and materially affect performance. Experiments on BERT, Pythia and Llama plus a teacher-student RMT model support these claims and highlight the oft-overlooked importance of small singular values for pruning and fine-tuning.\n\nSTRENGTHS AND WEAKNESSES:\n### Strengths\n\n1. The paper proposes a novel and interpretable use of the Marchenko–Pastur law as a random baseline to dissect learned signal (outliers) from noise (bulk) in the weight spectra of LLMs, and supports it with overlap ( $O_k$ ) measures and pruning experiments.\n2. It rigorously distinguishes between square and rectangular matrices—showing how only the latter can exhibit small singular value outliers—and explains this asymmetry with a rank-1 RMT teacher–student model in Section 6.\n\n### Weaknesses\n\n1. The core experiments are conducted on older models (e.g., BERT, Pythia), which limits the relevance of the findings. Replacing these with more recent state-of-the-art models such as Qwen 2.5/3 would make the results significantly more meaningful.\n2. The impact of pruning order and fine-tuning is not yet validated on modern models, and results on real-world tasks such as GSM-8K or MT-Bench are missing.\n3. The analysis of outlier counts, overlap scores, and pruning sensitivity across layers is limited to selected blocks; more comprehensive layer-wise statistics are needed.\n4. The RMT model in Section 6 makes strong simplifying assumptions (e.g., linear two-layer, Gaussian inputs, rank-1 teacher), which may limit generalizability. Its assumptions and scope should be better discussed and tested numerically.\n\nQUESTIONS:\nNo major concerns at this stage.\n\nLIMITATIONS:\nYes", "TITLE": "Official Review", "RECOMMENDATION": 4, "REVIEWER_CONFIDENCE": 3, "CLARITY": 3, "IMPACT": 3, "ORIGINALITY": 3, "DATE": "2025-07-02"}