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Vardan Papyan

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16 papers

2 author rows

NeurIPS Conference 2025 Conference Paper

Attention Sinks: A 'Catch, Tag, Release' Mechanism for Embeddings

Stephen Zhang
Mustafa Khan
Vardan Papyan

Large language models (LLMs) often concentrate their attention on a few specific tokens referred to as attention sinks. Common examples include the first token, a prompt-independent sink, and punctuation tokens, which are prompt-dependent. While the tokens causing the sinks often lack direct semantic meaning, the presence of the sinks is critical for model performance, particularly under model compression and KV-caching. Despite their ubiquity, the function, semantic role, and origin of attention sinks—especially those beyond the first token—remain poorly understood. In this work, we conduct a comprehensive investigation demonstrating that attention sinks: catch a sequence of tokens, tag them using a common direction in embedding space, and release them back into the residual stream, where tokens are later retrieved based on the tags they have acquired. Probing experiments reveal these tags carry semantically meaningful information, such as the truth of a statement. These findings extend to reasoning models, where the mechanism spans more heads and explains greater variance in embeddings, or recent models with query-key normalization, where sinks remain just as prevalent. To encourage future theoretical analysis, we introduce a minimal problem which can be solved through the 'catch, tag, release' mechanism, and where it emerges through training.

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ICLR Conference 2025 Conference Paper

Breach By A Thousand Leaks: Unsafe Information Leakage in 'Safe' AI Responses

David Glukhov
Ziwen Han
Ilia Shumailov
Vardan Papyan
Nicolas Papernot

Vulnerability of Frontier language models to misuse has prompted the development of safety measures like filters and alignment training seeking to ensure safety through robustness to adversarially crafted prompts. We assert that robustness is fundamentally insufficient for ensuring safety goals due to inferential threats from dual-intent queries, with current defenses and evaluations failing to account for these risks. To quantify these risks, we introduce a new safety evaluation framework based on $\textit{impermissible information leakage}$ of model outputs and demonstrate how our proposed question-decomposition attack can extract dangerous knowledge from a censored LLM more effectively than traditional jailbreaking. Underlying our proposed evaluation method is a novel information-theoretic threat model of $\textit{inferential adversaries}$, distinguished from $\textit{security adversaries}$, such as jailbreaks, in that success involves inferring impermissible knowledge from victim outputs as opposed to forcing explicitly impermissible victim outputs. Through our information-theoretic framework, we show that ensuring safety against inferential adversaries requires defenses which bound impermissible information leakage, and, such defenses inevitably incur safety-utility trade-offs.

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ICLR Conference 2025 Conference Paper

OATS: Outlier-Aware Pruning Through Sparse and Low Rank Decomposition

Stephen Zhang
Vardan Papyan

The recent paradigm shift to large-scale foundation models has brought about a new era for deep learning that, while has found great success in practice, has also been plagued by prohibitively expensive costs in terms of high memory consumption and compute. To mitigate these issues, there has been a concerted effort in post-hoc neural network pruning techniques that do not require costly retraining. Despite the considerable progress being made, existing methods often exhibit a steady drop in model performance as the compression increases. In this paper, we present a novel approach to compressing large transformers, coined OATS, that compresses the model weights by approximating each weight matrix as the sum of a sparse matrix and a low-rank matrix. Prior to the decomposition, the weights are first scaled by the second moment of their input embeddings, so as to ensure the preservation of outlier features recently observed in large transformer models. Without retraining, OATS achieves state-of-the-art performance when compressing large language models, such as Llama-3 and Phi-3, and vision transformers, such as Google's ViT and DINOv2, by up to $60\\%$, all while speeding up the model's inference on a CPU by up to $1.37\times$ compared to prior pruning methods.

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ICML Conference 2025 Conference Paper

On the Importance of Gaussianizing Representations

Daniel Eftekhari
Vardan Papyan

The normal distribution plays a central role in information theory – it is at the same time the best-case signal and worst-case noise distribution, has the greatest representational capacity of any distribution, and offers an equivalence between uncorrelatedness and independence for joint distributions. Accounting for the mean and variance of activations throughout the layers of deep neural networks has had a significant effect on facilitating their effective training, but seldom has a prescription for precisely what distribution these activations should take, and how this might be achieved, been offered. Motivated by the information-theoretic properties of the normal distribution, we address this question and concurrently present normality normalization: a novel normalization layer which encourages normality in the feature representations of neural networks using the power transform and employs additive Gaussian noise during training. Our experiments comprehensively demonstrate the effectiveness of normality normalization, in regards to its generalization performance on an array of widely used model and dataset combinations, its strong performance across various common factors of variation such as model width, depth, and training minibatch size, its suitability for usage wherever existing normalization layers are conventionally used, and as a means to improving model robustness to random perturbations.

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ICLR Conference 2025 Conference Paper

Transformer Block Coupling and its Correlation with Generalization in LLMs

Murdock Aubry
Haoming Meng
Anton Sugolov
Vardan Papyan

Large Language Models (LLMs) have made significant strides in natural language processing, and a precise understanding of the internal mechanisms driving their success is essential. In this work, we analyze the trajectories of token embeddings as they pass through transformer blocks, linearizing the system along these trajectories through their Jacobian matrices. By examining the relationships between these block Jacobians, we uncover the phenomenon of **transformer block coupling** in a multitude of LLMs, characterized by the coupling of their top singular vectors across tokens and depth. Our findings reveal that coupling *positively correlates* with model performance, and that this relationship is stronger than with other hyperparameters such as parameter count, model depth, and embedding dimension. We further investigate how these properties emerge during training, observing a progressive development of coupling, increased linearity, and layer-wise exponential growth in token trajectories. Additionally, experiments with Vision Transformers (ViTs) corroborate the emergence of coupling and its relationship with generalization, reinforcing our findings in LLMs. Collectively, these insights offer a novel perspective on token interactions in transformers, opening new directions for studying their mechanisms as well as improving training and generalization.

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NeurIPS Conference 2024 Conference Paper

Linguistic Collapse: Neural Collapse in (Large) Language Models

Robert Wu
Vardan Papyan

Neural collapse ($\mathcal{NC}$) is a phenomenon observed in classification tasks where top-layer representations collapse into their class means, which become equinorm, equiangular and aligned with the classifiers. These behaviors -- associated with generalization and robustness -- would manifest under specific conditions: models are trained towards zero loss, with noise-free labels belonging to balanced classes, which do not outnumber the model's hidden dimension. Recent studies have explored $\mathcal{NC}$ in the absence of one or more of these conditions to extend and capitalize on the associated benefits of ideal geometries. Language modeling presents a curious frontier, as \textit{training by token prediction} constitutes a classification task where none of the conditions exist: the vocabulary is imbalanced and exceeds the embedding dimension; different tokens might correspond to similar contextual embeddings; and large language models (LLMs) in particular are typically only trained for a few epochs. This paper empirically investigates the impact of scaling the architectures and training of causal language models (CLMs) on their progression towards $\mathcal{NC}$. We find that $\mathcal{NC}$ properties that develop with scale (and regularization) are linked to generalization. Moreover, there is evidence of some relationship between $\mathcal{NC}$ and generalization independent of scale. Our work thereby underscores the generality of $\mathcal{NC}$ as it extends to the novel and more challenging setting of language modeling. Downstream, we seek to inspire further research on the phenomenon to deepen our understanding of LLMs -- and neural networks at large -- and improve existing architectures based on $\mathcal{NC}$-related properties. Our code is hosted on GitHub: (https: //github. com/rhubarbwu/linguistic-collapse).

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ICML Conference 2024 Conference Paper

Out of the Ordinary: Spectrally Adapting Regression for Covariate Shift

Benjamin Eyre
Elliot Creager
David Madras
Vardan Papyan
Richard S. Zemel

Designing deep neural network classifiers that perform robustly on distributions differing from the available training data is an active area of machine learning research. However, out-of-distribution generalization for regression—the analogous problem for modeling continuous targets—remains relatively unexplored. To tackle this problem, we return to first principles and analyze how the closed-form solution for Ordinary Least Squares (OLS) regression is sensitive to covariate shift. We characterize the out-of-distribution risk of the OLS model in terms of the eigenspectrum decomposition of the source and target data. We then use this insight to propose a method called Spectral Adapted Regressor (SpAR) for adapting the weights of the last layer of a pre-trained neural regression model to perform better on input data originating from a different distribution. We demonstrate how this lightweight spectral adaptation procedure can improve out-of-distribution performance for synthetic and real-world datasets.

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ICML Conference 2024 Conference Paper

Position: Fundamental Limitations of LLM Censorship Necessitate New Approaches

David Glukhov
Ilia Shumailov
Yarin Gal
Nicolas Papernot
Vardan Papyan

Large language models (LLMs) have exhibited impressive capabilities in comprehending complex instructions. However, their blind adherence to provided instructions has led to concerns regarding risks of malicious use. Existing defence mechanisms, such as model fine-tuning or output censorship methods have proven to be fallible at ensuring that LLMs do not return semantically impermissible responses. We present fundamental limitations of verifying the semantic properties of LLM outputs and identifying compositional threats, illustrating inherent challenges of current approaches to censoring LLM outputs. Specifically, we demonstrate that semantic censorship can be perceived as an undecidable problem, and semantic properties of LLM outputs can become impossible to verify when the LLM is capable of providing "encrypted" outputs. We further show challenges of censorship can extend beyond just semantic censorship, as attackers can reconstruct impermissible outputs from a collection of permissible ones. Consequently, we call for a re-evaluation of the problem of censorship and its goals, stressing the need for new definitions and approaches to censorship. In addition, we provide an initial attempt toward achieving this goal through syntactic censorship, drawing from a security perspective to design censorship methods that can provide guarantees.

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ICLR Conference 2024 Conference Paper

Pushing Boundaries: Mixup's Influence on Neural Collapse

Quinn LeBlanc Fisher
Haoming Meng
Vardan Papyan

Mixup is a data augmentation strategy that employs convex combinations of training instances and their respective labels to improve the robustness and calibration of deep neural networks. Despite its widespread adoption, the nuanced mechanisms that underpin its success are not entirely understood. The observed phenomenon of Neural Collapse, where the last-layer activations and classifier of deep networks converge to a simplex equiangular tight frame (ETF), provides a compelling motivation to explore whether mixup induces alternative geometric configurations and whether those could explain its success. In this study, we delve into the last-layer activations of training data for deep networks subjected to mixup, aiming to uncover insights into its operational efficacy. Our investigation, spanning various architectures and dataset pairs, reveals that mixup's last-layer activations predominantly converge to a distinctive configuration different than one might expect. In this configuration, activations from mixed-up examples of identical classes align with the classifier, while those from different classes delineate channels along the decision boundary. These findings are unexpected, as mixed-up features are not simple convex combinations of feature class means (as one might get, for example, by training mixup with the mean squared error loss). By analyzing this distinctive geometric configuration, we elucidate the mechanisms by which mixup enhances model calibration. To further validate our empirical observations, we conduct a theoretical analysis under the assumption of an unconstrained features model, utilizing the mixup loss. Through this, we characterize and derive the optimal last-layer features under the assumption that the classifier forms a simplex ETF.

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ICML Conference 2024 Conference Paper

Sparsest Models Elude Pruning: An Exposé of Pruning's Current Capabilities

Stephen Zhang
Vardan Papyan

Pruning has emerged as a promising approach for compressing large-scale models, yet its effectiveness in recovering the sparsest of models has not yet been explored. We conducted an extensive series of 485, 838 experiments, applying a range of state-of-the-art pruning algorithms to a synthetic dataset we created, named the Cubist Spiral. Our findings reveal a significant gap in performance compared to ideal sparse networks, which we identified through a novel combinatorial search algorithm. We attribute this performance gap to current pruning algorithms’ poor behaviour under overparameterization, their tendency to induce disconnected paths throughout the network, and their propensity to get stuck at suboptimal solutions, even when given the optimal width and initialization. This gap is concerning, given the simplicity of the network architectures and datasets used in our study. We hope that our research encourages further investigation into new pruning techniques that strive for true network sparsity.

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NeurIPS Conference 2023 Conference Paper

Residual Alignment: Uncovering the Mechanisms of Residual Networks

Jianing Li
Vardan Papyan

The ResNet architecture has been widely adopted in deep learning due to its significant boost to performance through the use of simple skip connections, yet the underlying mechanisms leading to its success remain largely unknown. In this paper, we conduct a thorough empirical study of the ResNet architecture in classification tasks by linearizing its constituent residual blocks using Residual Jacobians and measuring their singular value decompositions. Our measurements ([code](https: //colab. research. google. com/drive/1yKjEg2yF616tnZFAfuN0aQ-E9v3JmyjN? usp=sharing)) reveal a process called Residual Alignment (RA) characterized by four properties: - **(RA1): ** intermediate representations of a given input are *equispaced* on a *line*, embedded in high dimensional space, as observed by Gai and Zhang [2021]; - **(RA2): ** top left and right singular vectors of Residual Jacobians align with each other and across different depths; - **(RA3): ** Residual Jacobians are at most rank $C$ for fully-connected ResNets, where $C$ is the number of classes; and- **(RA4): ** top singular values of Residual Jacobians scale inversely with depth. RA consistently occurs in models that generalize well, in both fully-connected and convolutional architectures, across various depths and widths, for varying numbers of classes, on all tested benchmark datasets, but ceases to occur once the skip connections are removed. It also provably occurs in a novel mathematical model we propose. This phenomenon reveals a strong alignment between residual branches of a ResNet (RA2+4), imparting a highly rigid geometric structure to the intermediate representations as they progress *linearly* through the network (RA1) up to the final layer, where they undergo Neural Collapse.

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ICLR Conference 2022 Conference Paper

Neural Collapse Under MSE Loss: Proximity to and Dynamics on the Central Path

X. Y. Han
Vardan Papyan
David L. Donoho

The recently discovered Neural Collapse (NC) phenomenon occurs pervasively in today's deep net training paradigm of driving cross-entropy (CE) loss towards zero. During NC, last-layer features collapse to their class-means, both classifiers and class-means collapse to the same Simplex Equiangular Tight Frame, and classifier behavior collapses to the nearest-class-mean decision rule. Recent works demonstrated that deep nets trained with mean squared error (MSE) loss perform comparably to those trained with CE. As a preliminary, we empirically establish that NC emerges in such MSE-trained deep nets as well through experiments on three canonical networks and five benchmark datasets. We provide, in a Google Colab notebook, PyTorch code for reproducing MSE-NC and CE-NC: https://colab.research.google.com/github/neuralcollapse/neuralcollapse/blob/main/neuralcollapse.ipynb. The analytically-tractable MSE loss offers more mathematical opportunities than the hard-to-analyze CE loss, inspiring us to leverage MSE loss towards the theoretical investigation of NC. We develop three main contributions: (I) We show a new decomposition of the MSE loss into (A) terms directly interpretable through the lens of NC and which assume the last-layer classifier is exactly the least-squares classifier; and (B) a term capturing the deviation from this least-squares classifier. (II) We exhibit experiments on canonical datasets and networks demonstrating that term-(B) is negligible during training. This motivates us to introduce a new theoretical construct: the central path, where the linear classifier stays MSE-optimal for feature activations throughout the dynamics. (III) By studying renormalized gradient flow along the central path, we derive exact dynamics that predict NC.

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JMLR Journal 2020 Journal Article

Traces of Class/Cross-Class Structure Pervade Deep Learning Spectra

Vardan Papyan

Numerous researchers recently applied empirical spectral analysis to the study of modern deep learning classifiers. We identify and discuss an important formal class/cross-class structure and show how it lies at the origin of the many visually striking features observed in deep neural network spectra, some of which were reported in recent articles, others are unveiled here for the first time. These include spectral outliers, “spikes”, and small but distinct continuous distributions, “bumps”, often seen beyond the edge of a “main bulk”. [abs] [ pdf ][ bib ] &copy JMLR 2020. ( edit, beta )

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ICML Conference 2019 Conference Paper

Measurements of Three-Level Hierarchical Structure in the Outliers in the Spectrum of Deepnet Hessians

Vardan Papyan

We expose a structure in deep classifying neural networks in the derivative of the logits with respect to the parameters of the model, which is used to explain the existence of outliers in the spectrum of the Hessian. Previous works decomposed the Hessian into two components, attributing the outliers to one of them, the so-called Covariance of gradients. We show this term is not a Covariance but a second moment matrix, i. e. , it is influenced by means of gradients. These means possess an additive two-way structure that is the source of the outliers in the spectrum. This structure can be used to approximate the principal subspace of the Hessian using certain "averaging" operations, avoiding the need for high-dimensional eigenanalysis. We corroborate this claim across different datasets, architectures and sample sizes.

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NeurIPS Conference 2018 Conference Paper

Neural Proximal Gradient Descent for Compressive Imaging

Morteza Mardani
Qingyun Sun
David Donoho
Vardan Papyan
Hatef Monajemi
Shreyas Vasanawala
John Pauly

Recovering high-resolution images from limited sensory data typically leads to a serious ill-posed inverse problem, demanding inversion algorithms that effectively capture the prior information. Learning a good inverse mapping from training data faces severe challenges, including: (i) scarcity of training data; (ii) need for plausible reconstructions that are physically feasible; (iii) need for fast reconstruction, especially in real-time applications. We develop a successful system solving all these challenges, using as basic architecture the repetitive application of alternating proximal and data fidelity constraints. We learn a proximal map that works well with real images based on residual networks with recurrent blocks. Extensive experiments are carried out under different settings: (a) reconstructing abdominal MRI of pediatric patients from highly undersampled k-space data and (b) super-resolving natural face images. Our key findings include: 1. a recurrent ResNet with a single residual block (10-fold repetition) yields an effective proximal which accurately reveals MR image details. 2. Our architecture significantly outperforms conventional non-recurrent deep ResNets by 2dB SNR; it is also trained much more rapidly. 3. It outperforms state-of-the-art compressed-sensing Wavelet-based methods by 4dB SNR, with 100x speedups in reconstruction time.

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JMLR Journal 2017 Journal Article

Convolutional Neural Networks Analyzed via Convolutional Sparse Coding

Vardan Papyan
Yaniv Romano
Michael Elad

Convolutional neural networks (CNN) have led to many state-of- the-art results spanning through various fields. However, a clear and profound theoretical understanding of the forward pass, the core algorithm of CNN, is still lacking. In parallel, within the wide field of sparse approximation, Convolutional Sparse Coding (CSC) has gained increasing attention in recent years. A theoretical study of this model was recently conducted, establishing it as a reliable and stable alternative to the commonly practiced patch-based processing. Herein, we propose a novel multi-layer model, ML-CSC, in which signals are assumed to emerge from a cascade of CSC layers. This is shown to be tightly connected to CNN, so much so that the forward pass of the CNN is in fact the thresholding pursuit serving the ML-CSC model. This connection brings a fresh view to CNN, as we are able to attribute to this architecture theoretical claims such as uniqueness of the representations throughout the network, and their stable estimation, all guaranteed under simple local sparsity conditions. Lastly, identifying the weaknesses in the above pursuit scheme, we propose an alternative to the forward pass, which is connected to deconvolutional and recurrent networks, and also has better theoretical guarantees. [abs] [ pdf ][ bib ] &copy JMLR 2017. ( edit, beta )

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