Ievgen Redko

Brandon Amos, Samuel Cohen, Giulia Luise et al.

We study the use of amortized optimization to predict optimal transport (OT) maps from the input measures, which we call Meta OT. This helps repeatedly solve similar OT problems between different measures by leveraging the knowledge and information present from past problems to rapidly predict and solve new problems. Otherwise, standard methods ignore the knowledge of the past solutions and suboptimally re-solve each problem from scratch. We instantiate Meta OT models in discrete and continuous settings between grayscale images, spherical data, classification labels, and color palettes and use them to improve the computational time of standard OT solvers. Our source code is available at http://github.com/facebookresearch/meta-ot

Quang Huy Tran, Hicham Janati, Nicolas Courty et al. · harvard, mit

Optimal transport (OT) compares probability distributions by computing a meaningful alignment between their samples. CO-optimal transport (COOT) takes this comparison further by inferring an alignment between features as well. While this approach leads to better alignments and generalizes both OT and Gromov-Wasserstein distances, we provide a theoretical result showing that it is sensitive to outliers that are omnipresent in real-world data. This prompts us to propose unbalanced COOT for which we provably show its robustness to noise in the compared datasets. To the best of our knowledge, this is the first such result for OT methods in incomparable spaces. With this result in hand, we provide empirical evidence of this robustness for the challenging tasks of heterogeneous domain adaptation with and without varying proportions of classes and simultaneous alignment of samples and features across single-cell measurements.

Quentin Bouniot, Ievgen Redko, Anton Mallasto et al.

In the last decade, we have witnessed the introduction of several novel deep neural network (DNN) architectures exhibiting ever-increasing performance across diverse tasks. Explaining the upward trend of their performance, however, remains difficult as different DNN architectures of comparable depth and width -- common factors associated with their expressive power -- may exhibit a drastically different performance even when trained on the same dataset. In this paper, we introduce the concept of the non-linearity signature of DNN, the first theoretically sound solution for approximately measuring the non-linearity of deep neural networks. Built upon a score derived from closed-form optimal transport mappings, this signature provides a better understanding of the inner workings of a wide range of DNN architectures and learning paradigms, with a particular emphasis on the computer vision task. We provide extensive experimental results that highlight the practical usefulness of the proposed non-linearity signature and its potential for long-reaching implications. The code for our work is available at https://github.com/qbouniot/AffScoreDeep

Ambroise Odonnat, Vasilii Feofanov, Ievgen Redko

Self-training is a well-known approach for semi-supervised learning. It consists of iteratively assigning pseudo-labels to unlabeled data for which the model is confident and treating them as labeled examples. For neural networks, softmax prediction probabilities are often used as a confidence measure, although they are known to be overconfident, even for wrong predictions. This phenomenon is particularly intensified in the presence of sample selection bias, i.e., when data labeling is subject to some constraint. To address this issue, we propose a novel confidence measure, called $\mathcal{T}$-similarity, built upon the prediction diversity of an ensemble of linear classifiers. We provide the theoretical analysis of our approach by studying stationary points and describing the relationship between the diversity of the individual members and their performance. We empirically demonstrate the benefit of our confidence measure for three different pseudo-labeling policies on classification datasets of various data modalities. The code is available at https://github.com/ambroiseodt/tsim.

Pinar Demetci, Quang Huy Tran, Ievgen Redko et al. · harvard, mit

Gromov-Wasserstein distance has found many applications in machine learning due to its ability to compare measures across metric spaces and its invariance to isometric transformations. However, in certain applications, this invariance property can be too flexible, thus undesirable. Moreover, the Gromov-Wasserstein distance solely considers pairwise sample similarities in input datasets, disregarding the raw feature representations. We propose a new optimal transport-based distance, called Augmented Gromov-Wasserstein, that allows for some control over the level of rigidity to transformations. It also incorporates feature alignments, enabling us to better leverage prior knowledge on the input data for improved performance. We present theoretical insights into the proposed metric. We then demonstrate its usefulness for single-cell multi-omic alignment tasks and a transfer learning scenario in machine learning.

Ambroise Odonnat, Laetitia Chapel, Romain Tavenard et al.

The smoothness of the transformer architecture has been extensively studied in the context of generalization, training stability, and adversarial robustness. However, its role in transfer learning remains poorly understood. In this paper, we analyze the ability of vision transformer components to adapt their outputs to changes in inputs, or, in other words, their plasticity. Defined as an average rate of change, it captures the sensitivity to input perturbation; in particular, a high plasticity implies low smoothness. We demonstrate through theoretical analysis and comprehensive experiments that this perspective provides principled guidance in choosing the components to prioritize during adaptation. A key takeaway for practitioners is that the high plasticity of the attention modules and feedforward layers consistently leads to better finetuning performance. Our findings depart from the prevailing assumption that smoothness is desirable, offering a novel perspective on the functional properties of transformers. The code is available at https://github.com/ambroiseodt/vit-plasticity.

Vasilii Feofanov, Romain Ilbert, Malik Tiomoko et al.

Foundation models, while highly effective, are often resource-intensive, requiring substantial inference time and memory. This paper addresses the challenge of making these models more accessible with limited computational resources by exploring dimensionality reduction techniques. Our goal is to enable users to run large pre-trained foundation models on standard GPUs without sacrificing performance. We investigate classical methods such as Principal Component Analysis alongside neural network-based adapters, aiming to reduce the dimensionality of multivariate time series data while preserving key features. Our experiments show up to a 10x speedup compared to the baseline model, without performance degradation, and enable up to 4.5x more datasets to fit on a single GPU, paving the way for more user-friendly and scalable foundation models.

Romain Ilbert, Ambroise Odonnat, Vasilii Feofanov et al.

Transformer-based architectures achieved breakthrough performance in natural language processing and computer vision, yet they remain inferior to simpler linear baselines in multivariate long-term forecasting. To better understand this phenomenon, we start by studying a toy linear forecasting problem for which we show that transformers are incapable of converging to their true solution despite their high expressive power. We further identify the attention of transformers as being responsible for this low generalization capacity. Building upon this insight, we propose a shallow lightweight transformer model that successfully escapes bad local minima when optimized with sharpness-aware optimization. We empirically demonstrate that this result extends to all commonly used real-world multivariate time series datasets. In particular, SAMformer surpasses current state-of-the-art methods and is on par with the biggest foundation model MOIRAI while having significantly fewer parameters. The code is available at https://github.com/romilbert/samformer.

Oussama Zekri, Abdelhakim Benechehab, Ievgen Redko

Large-language models are notoriously famous for their impressive performance across a wide range of tasks. One surprising example of such impressive performance is a recently identified capacity of LLMs to understand the governing principles of dynamical systems satisfying the Markovian property. In this paper, we seek to explore this direction further by studying the dynamics of stochastic gradient descent in convex and non-convex optimization. By leveraging the theoretical link between the SGD and Markov chains, we show a remarkable zero-shot performance of LLMs in predicting the local minima to which SGD converges for previously unseen starting points. On a more general level, we inquire about the possibility of using LLMs to perform zero-shot randomized trials for larger deep learning models used in practice.

Vasilii Feofanov, Songkang Wen, Marius Alonso et al.

In recent years, there has been increasing interest in developing foundation models for time series data that can generalize across diverse downstream tasks. While numerous forecasting-oriented foundation models have been introduced, there is a notable scarcity of models tailored for time series classification. To address this gap, we present Mantis, a new open-source foundation model for time series classification based on the Vision Transformer (ViT) architecture that has been pre-trained using a contrastive learning approach. Our experimental results show that Mantis outperforms existing foundation models both when the backbone is frozen and when fine-tuned, while achieving the lowest calibration error. In addition, we propose several adapters to handle the multivariate setting, reducing memory requirements and modeling channel interdependence.

Théo Gnassounou, Yessin Moakher, Shifeng Xie et al.

Foundation models for time series are emerging as powerful general-purpose backbones, yet their potential for domain-specific biomedical signals such as electroencephalography (EEG) remains rather unexplored. In this work, we investigate the applicability a recently proposed time series classification foundation model, to a different EEG tasks such as motor imagery classification and sleep stage prediction. We test two pretraining regimes: (a) pretraining on heterogeneous real-world time series from multiple domains, and (b) pretraining on purely synthetic data. We find that both variants yield strong performance, consistently outperforming EEGNet, a widely used convolutional baseline, and CBraMod, the most recent EEG-specific foundation model. These results suggest that generalist time series foundation models, even when pretrained on data of non-neural origin or on synthetic signals, can transfer effectively to EEG. Our findings highlight the promise of leveraging cross-domain pretrained models for brain signal analysis, suggesting that EEG may benefit from advances in the broader time series literature.

Abdelhakim Benechehab, Youssef Attia El Hili, Ambroise Odonnat et al.

The emerging zero-shot capabilities of Large Language Models (LLMs) have led to their applications in areas extending well beyond natural language processing tasks. In reinforcement learning, while LLMs have been extensively used in text-based environments, their integration with continuous state spaces remains understudied. In this paper, we investigate how pre-trained LLMs can be leveraged to predict in context the dynamics of continuous Markov decision processes. We identify handling multivariate data and incorporating the control signal as key challenges that limit the potential of LLMs' deployment in this setup and propose Disentangled In-Context Learning (DICL) to address them. We present proof-of-concept applications in two reinforcement learning settings: model-based policy evaluation and data-augmented off-policy reinforcement learning, supported by theoretical analysis of the proposed methods. Our experiments further demonstrate that our approach produces well-calibrated uncertainty estimates. We release the code at https://github.com/abenechehab/dicl.

Renchunzi Xie, Ambroise Odonnat, Vasilii Feofanov et al.

Estimating the test performance of a model, possibly under distribution shift, without having access to the ground-truth labels is a challenging, yet very important problem for the safe deployment of machine learning algorithms in the wild. Existing works mostly rely on information from either the outputs or the extracted features of neural networks to estimate a score that correlates with the ground-truth test accuracy. In this paper, we investigate -- both empirically and theoretically -- how the information provided by the gradients can be predictive of the ground-truth test accuracy even under distribution shifts. More specifically, we use the norm of classification-layer gradients, backpropagated from the cross-entropy loss after only one gradient step over test data. Our intuition is that these gradients should be of higher magnitude when the model generalizes poorly. We provide the theoretical insights behind our approach and the key ingredients that ensure its empirical success. Extensive experiments conducted with various architectures on diverse distribution shifts demonstrate that our method significantly outperforms current state-of-the-art approaches. The code is available at https://github.com/Renchunzi-Xie/GdScore

Simon Roschmann, Quentin Bouniot, Vasilii Feofanov et al.

Time series classification is a fundamental task in healthcare and industry, yet the development of time series foundation models (TSFMs) remains limited by the scarcity of publicly available time series datasets. In this work, we propose Time Vision Transformer (TiViT), a framework that converts time series into images to leverage the representational power of frozen Vision Transformers (ViTs) pretrained on large-scale image datasets. First, we theoretically motivate our approach by analyzing the 2D patching of ViTs for time series, showing that it can increase the number of label-relevant tokens and reduce the sample complexity. Second, we empirically demonstrate that TiViT achieves state-of-the-art performance on standard time series classification benchmarks by utilizing the hidden representations of large OpenCLIP models. We explore the structure of TiViT representations and find that intermediate layers with high intrinsic dimension are the most effective for time series classification. Finally, we assess the alignment between TiViT and TSFM representation spaces and identify a strong complementarity, with further performance gains achieved by combining their features. Our findings reveal a new direction for reusing vision representations in a non-visual domain. Code is available at https://github.com/ExplainableML/TiViT.

Oliver Struckmeier, Ievgen Redko, Anton Mallasto et al.

Optimal transport (OT) is a powerful geometric tool used to compare and align probability measures following the least effort principle. Despite its widespread use in machine learning (ML), OT problem still bears its computational burden, while at the same time suffering from the curse of dimensionality for measures supported on general high-dimensional spaces. In this paper, we propose to tackle these challenges using representation learning. In particular, we seek to learn an embedding space such that the samples of the two input measures become alignable in it with a simple affine mapping that can be calculated efficiently in closed-form. We then show that such approach leads to results that are comparable to solving the original OT problem when applied to the transfer learning task on which many OT baselines where previously evaluated in both homogeneous and heterogeneous DA settings. The code for our contribution is available at \url{https://github.com/Oleffa/LaOT}.

Shifeng Xie, Vasilii Feofanov, Marius Alonso et al.

Time series foundation models (TSFMs) have recently gained significant attention due to their strong zero-shot capabilities and widespread real-world applications. Such models typically require a computationally costly pretraining on large-scale, carefully curated collections of real-world sequences. To allow for a sample-efficient pretraining of TSFMs, we propose CauKer, a novel algorithm designed to generate diverse, causally coherent synthetic time series with realistic trends, seasonality, and nonlinear interactions. CauKer combines Gaussian Process (GP) kernel composition with Structural Causal Models (SCM) to produce data for sample-efficient pretraining of state-of-the-art classification TSFMs having different architectures and following different pretraining approaches. Additionally, our experiments reveal that CauKer-generated datasets exhibit clear scaling laws for both dataset size (10K to 10M samples) and model capacity (1M to 783M parameters), unlike real-world datasets, which display irregular scaling behavior.

Ambroise Odonnat, Vasilii Feofanov, Laetitia Chapel et al.

Recent studies have observed that intermediate layers of foundation models often yield more discriminative representations than the final layer. While initially attributed to autoregressive pretraining, this phenomenon has also been identified in models trained via supervised and discriminative self-supervised objectives. In this paper, we conduct a comprehensive study to analyze the behavior of intermediate layers in pretrained vision transformers. Through extensive linear probing experiments across a diverse set of image classification benchmarks, we find that distribution shift between pretraining and downstream data is the primary cause of performance degradation in deeper layers. Furthermore, we perform a fine-grained analysis at the module level. Our findings reveal that standard probing of transformer block outputs is suboptimal; instead, probing the activation within the feedforward network yields the best performance under significant distribution shift, whereas the normalized output of the multi-head self-attention module is optimal when the shift is weak.

Romain Ilbert, Malik Tiomoko, Cosme Louart et al.

In this paper, we introduce a novel theoretical framework for multi-task regression, applying random matrix theory to provide precise performance estimations, under high-dimensional, non-Gaussian data distributions. We formulate a multi-task optimization problem as a regularization technique to enable single-task models to leverage multi-task learning information. We derive a closed-form solution for multi-task optimization in the context of linear models. Our analysis provides valuable insights by linking the multi-task learning performance to various model statistics such as raw data covariances, signal-generating hyperplanes, noise levels, as well as the size and number of datasets. We finally propose a consistent estimation of training and testing errors, thereby offering a robust foundation for hyperparameter optimization in multi-task regression scenarios. Experimental validations on both synthetic and real-world datasets in regression and multivariate time series forecasting demonstrate improvements on univariate models, incorporating our method into the training loss and thus leveraging multivariate information.

Quang Huy Tran, Hicham Janati, Ievgen Redko et al.

Optimal transport (OT) theory underlies many emerging machine learning (ML) methods nowadays solving a wide range of tasks such as generative modeling, transfer learning and information retrieval. These latter works, however, usually build upon a traditional OT setup with two distributions, while leaving a more general multi-marginal OT formulation somewhat unexplored. In this paper, we study the multi-marginal OT (MMOT) problem and unify several popular OT methods under its umbrella by promoting structural information on the coupling. We show that incorporating such structural information into MMOT results in an instance of a different of convex (DC) programming problem allowing us to solve it numerically. Despite high computational cost of the latter procedure, the solutions provided by DC optimization are usually as qualitative as those obtained using currently employed optimization schemes.

Charlotte Laclau, Ievgen Redko, Manvi Choudhary et al.

Machine learning and data mining algorithms have been increasingly used recently to support decision-making systems in many areas of high societal importance such as healthcare, education, or security. While being very efficient in their predictive abilities, the deployed algorithms sometimes tend to learn an inductive model with a discriminative bias due to the presence of this latter in the learning sample. This problem gave rise to a new field of algorithmic fairness where the goal is to correct the discriminative bias introduced by a certain attribute in order to decorrelate it from the model's output. In this paper, we study the problem of fairness for the task of edge prediction in graphs, a largely underinvestigated scenario compared to a more popular setting of fair classification. To this end, we formulate the problem of fair edge prediction, analyze it theoretically, and propose an embedding-agnostic repairing procedure for the adjacency matrix of an arbitrary graph with a trade-off between the group and individual fairness. We experimentally show the versatility of our approach and its capacity to provide explicit control over different notions of fairness and prediction accuracy.

Nina Vesseron, Ievgen Redko, Charlotte Laclau

The theoretical analysis of deep neural networks (DNN) is arguably among the most challenging research directions in machine learning (ML) right now, as it requires from scientists to lay novel statistical learning foundations to explain their behaviour in practice. While some success has been achieved recently in this endeavour, the question on whether DNNs can be analyzed using the tools from other scientific fields outside the ML community has not received the attention it may well have deserved. In this paper, we explore the interplay between DNNs and game theory (GT), and show how one can benefit from the classic readily available results from the latter when analyzing the former. In particular, we consider the widely studied class of congestion games, and illustrate their intrinsic relatedness to both linear and non-linear DNNs and to the properties of their loss surface. Beyond retrieving the state-of-the-art results from the literature, we argue that our work provides a very promising novel tool for analyzing the DNNs and support this claim by proposing concrete open problems that can advance significantly our understanding of DNNs when solved.

Quentin Bouniot, Ievgen Redko, Romaric Audigier et al.

In this paper, we consider the framework of multi-task representation (MTR) learning where the goal is to use source tasks to learn a representation that reduces the sample complexity of solving a target task. We start by reviewing recent advances in MTR theory and show that they can provide novel insights for popular meta-learning algorithms when analyzed within this framework. In particular, we highlight a fundamental difference between gradient-based and metric-based algorithms in practice and put forward a theoretical analysis to explain it. Finally, we use the derived insights to improve the performance of meta-learning methods via a new spectral-based regularization term and confirm its efficiency through experimental studies on few-shot classification benchmarks. To the best of our knowledge, this is the first contribution that puts the most recent learning bounds of MTR theory into practice for the task of few-shot classification.

Charlotte Laclau, Franck Iutzeler, Ievgen Redko

In this paper, we introduce and formalize a rank-one partitioning learning paradigm that unifies partitioning methods that proceed by summarizing a data set using a single vector that is further used to derive the final clustering partition. Using this unification as a starting point, we propose a novel algorithmic solution for the partitioning problem based on rank-one matrix factorization and denoising of piecewise constant signals. Finally, we propose an empirical demonstration of our findings and demonstrate the robustness of the proposed denoising step. We believe that our work provides a new point of view for several unsupervised learning techniques that helps to gain a deeper understanding about the general mechanisms of data partitioning.

Ievgen Redko, Emilie Morvant, Amaury Habrard et al.

All famous machine learning algorithms that comprise both supervised and semi-supervised learning work well only under a common assumption: the training and test data follow the same distribution. When the distribution changes, most statistical models must be reconstructed from newly collected data, which for some applications can be costly or impossible to obtain. Therefore, it has become necessary to develop approaches that reduce the need and the effort to obtain new labeled samples by exploiting data that are available in related areas, and using these further across similar fields. This has given rise to a new machine learning framework known as transfer learning: a learning setting inspired by the capability of a human being to extrapolate knowledge across tasks to learn more efficiently. Despite a large amount of different transfer learning scenarios, the main objective of this survey is to provide an overview of the state-of-the-art theoretical results in a specific, and arguably the most popular, sub-field of transfer learning, called domain adaptation. In this sub-field, the data distribution is assumed to change across the training and the test data, while the learning task remains the same. We provide a first up-to-date description of existing results related to domain adaptation problem that cover learning bounds based on different statistical learning frameworks.

Ievgen Redko, Titouan Vayer, Rémi Flamary et al.

Optimal transport (OT) is a powerful geometric and probabilistic tool for finding correspondences and measuring similarity between two distributions. Yet, its original formulation relies on the existence of a cost function between the samples of the two distributions, which makes it impractical when they are supported on different spaces. To circumvent this limitation, we propose a novel OT problem, named COOT for CO-Optimal Transport, that simultaneously optimizes two transport maps between both samples and features, contrary to other approaches that either discard the individual features by focusing on pairwise distances between samples or need to model explicitly the relations between them. We provide a thorough theoretical analysis of our problem, establish its rich connections with other OT-based distances and demonstrate its versatility with two machine learning applications in heterogeneous domain adaptation and co-clustering/data summarization, where COOT leads to performance improvements over the state-of-the-art methods.

Léo Gautheron, Ievgen Redko, Carole Lartizien

In this paper, we propose a new feature selection method for unsupervised domain adaptation based on the emerging optimal transportation theory. We build upon a recent theoretical analysis of optimal transport in domain adaptation and show that it can directly suggest a feature selection procedure leveraging the shift between the domains. Based on this, we propose a novel algorithm that aims to sort features by their similarity across the source and target domains, where the order is obtained by analyzing the coupling matrix representing the solution of the proposed optimal transportation problem. We evaluate our method on a well-known benchmark data set and illustrate its capability of selecting correlated features leading to better classification performances. Furthermore, we show that the proposed algorithm can be used as a pre-processing step for existing domain adaptation techniques ensuring an important speed-up in terms of the computational time while maintaining comparable results. Finally, we validate our algorithm on clinical imaging databases for computer-aided diagnosis task with promising results.

Georgios Balikas, Charlotte Laclau, Ievgen Redko et al.

Many information retrieval algorithms rely on the notion of a good distance that allows to efficiently compare objects of different nature. Recently, a new promising metric called Word Mover's Distance was proposed to measure the divergence between text passages. In this paper, we demonstrate that this metric can be extended to incorporate term-weighting schemes and provide more accurate and computationally efficient matching between documents using entropic regularization. We evaluate the benefits of both extensions in the task of cross-lingual document retrieval (CLDR). Our experimental results on eight CLDR problems suggest that the proposed methods achieve remarkable improvements in terms of Mean Reciprocal Rank compared to several baselines.

Ievgen Redko, Nicolas Courty, Rémi Flamary et al.

In this paper, we propose to tackle the problem of reducing discrepancies between multiple domains referred to as multi-source domain adaptation and consider it under the target shift assumption: in all domains we aim to solve a classification problem with the same output classes, but with labels' proportions differing across them. This problem, generally ignored in the vast majority papers on domain adaptation papers, is nevertheless critical in real-world applications, and we theoretically show its impact on the adaptation success. To address this issue, we design a method based on optimal transport, a theory that has been successfully used to tackle adaptation problems in machine learning. Our method performs multi-source adaptation and target shift correction simultaneously by learning the class probabilities of the unlabeled target sample and the coupling allowing to align two (or more) probability distributions. Experiments on both synthetic and real-world data related to satellite image segmentation task show the superiority of the proposed method over the state-of-the-art.

Charlotte Laclau, Ievgen Redko, Basarab Matei et al.

In this paper, we present a novel method for co-clustering, an unsupervised learning approach that aims at discovering homogeneous groups of data instances and features by grouping them simultaneously. The proposed method uses the entropy regularized optimal transport between empirical measures defined on data instances and features in order to obtain an estimated joint probability density function represented by the optimal coupling matrix. This matrix is further factorized to obtain the induced row and columns partitions using multiscale representations approach. To justify our method theoretically, we show how the solution of the regularized optimal transport can be seen from the variational inference perspective thus motivating its use for co-clustering. The algorithm derived for the proposed method and its kernelized version based on the notion of Gromov-Wasserstein distance are fast, accurate and can determine automatically the number of both row and column clusters. These features are vividly demonstrated through extensive experimental evaluations.

Ievgen Redko, Younès Bennani

The ability of a human being to extrapolate previously gained knowledge to other domains inspired a new family of methods in machine learning called transfer learning. Transfer learning is often based on the assumption that objects in both target and source domains share some common feature and/or data space. In this paper, we propose a simple and intuitive approach that minimizes iteratively the distance between source and target task distributions by optimizing the kernel target alignment (KTA). We show that this procedure is suitable for transfer learning by relating it to Hilbert-Schmidt Independence Criterion (HSIC) and Quadratic Mutual Information (QMI) maximization. We run our method on benchmark computer vision data sets and show that it can outperform some state-of-art methods.

Ievgen Redko, Amaury Habrard, Marc Sebban

Domain adaptation (DA) is an important and emerging field of machine learning that tackles the problem occurring when the distributions of training (source domain) and test (target domain) data are similar but different. Current theoretical results show that the efficiency of DA algorithms depends on their capacity of minimizing the divergence between source and target probability distributions. In this paper, we provide a theoretical study on the advantages that concepts borrowed from optimal transportation theory can bring to DA. In particular, we show that the Wasserstein metric can be used as a divergence measure between distributions to obtain generalization guarantees for three different learning settings: (i) classic DA with unsupervised target data (ii) DA combining source and target labeled data, (iii) multiple source DA. Based on the obtained results, we provide some insights showing when this analysis can be tighter than other existing frameworks.