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Transformer models have revolutionized machine learning, yet the underpinnings behind their success are only beginning to be understood. In this work, we analyze transformers through the geometry of attention maps, treating them as weighted graphs and focusing on Ricci curvature, a metric linked to spectral properties and system robustness. We prove that lower Ricci curvature, indicating lower system robustness, leads to faster convergence of gradient descent during training. We also show that a higher frequency of positive curvature values enhances robustness, revealing a trade-off between performance and robustness. Building on this, we propose a regularization method to adjust the curvature distribution and provide experimental results supporting our theoretical predictions while offering insights into ways to improve transformer training and robustness. The geometric perspective provided in our paper offers a versatile framework for both understanding and improving the behavior of transformers. 

Backdoor attacks pose a critical threat by embedding hidden triggers into inputs, causing models to misclassify them into target labels. While extensive research has focused on mitigating these attacks in object recognition models through weight fine-tuning, much less attention has been given to detecting backdoored samples directly. Given the vast datasets used in training, manual inspection for backdoor triggers is impractical, and even state-of-the-art defense mechanisms fail to fully neutralize their impact. To address this gap, we introduce a groundbreaking method to detect unseen backdoored images during both training and inference. Leveraging the transformative success of prompt tuning in Vision Language Models (VLMs), our approach trains learnable text prompts to differentiate clean images from those with hidden backdoor triggers. Experiments demonstrate the exceptional efficacy of this method, achieving an impressive average accuracy of 86% across two renowned datasets for detecting unseen backdoor triggers, establishing a new standard in backdoor defense. 

Abstract In this paper, we show how relaxation techniques can be used to establish the existence of an optimal contract in the presence of information asymmetry. The method we illustrate was initially motivated by the problem of designing optimal brokerage fees, but it does apply to other optimal contract problems in which (i) the agent controls linearly the drift of a diffusion process, (ii) the direct dependence of the principal’s and the agent’s objectives on the strategy of the agent is of a special form, and (iii) the space of admissible contracts is compact. This method is then applied to establish the existence of an optimal brokerage fee in a market model with a private trading signal observed by the broker’s client, but not by the broker. 

Representation learning in high-dimensional spaces faces significant robustness challenges with noisy inputs, particularly with heavy-tailed noise. Arguing that topological data analysis (TDA) offers a solution, we leverage TDA to enhance representation stability in neural networks. Our theoretical analysis establishes conditions under which incorporating topological summaries improves robustness to input noise, especially for heavy-tailed distributions. Extending these results to representation-balancing methods used in causal inference, we propose the *Topology-Aware Treatment Effect Estimation* (TATEE) framework, through which we demonstrate how topological awareness can lead to learning more robust representations. A key advantage of this approach is that it requires no ground-truth or validation data, making it suitable for observational settings common in causal inference. The method remains computationally efficient with overhead scaling linearly with data size while staying constant in input dimension. Through extensive experiments with -stable noise distributions, we validate our theoretical results, demonstrating that TATEE consistently outperforms existing methods across noise regimes. This work extends stability properties of topological summaries to representation learning via a tractable framework scalable for high-dimensional inputs, providing insights into how it can enhance robustness, with applications extending to domains facing challenges with noisy data, such as causal inference.