Conformal prediction is a theoretically grounded framework for constructing predictive intervals. We study conformal prediction with missing values in the covariates -- a setting that brings new challenges to uncertainty quantification. We first show that the marginal coverage guarantee of conformal prediction holds on imputed data for any missingness distribution and almost all imputation functions. However, we emphasize that the average coverage varies depending on the pattern of missing values: conformal methods tend to construct prediction intervals that under-cover the response conditionally to some missing patterns. This motivates our novel generalized conformalized quantile regression framework, missing data augmentation, which yields prediction intervals that are valid conditionally to the patterns of missing values, despite their exponential number. We then show that a universally consistent quantile regression algorithm trained on the imputed data is Bayes optimal for the pinball risk, thus achieving valid coverage conditionally to any given data point. Moreover, we examine the case of a linear model, which demonstrates the importance of our proposal in overcoming the heteroskedasticity induced by missing values. Using synthetic and data from critical care, we corroborate our theory and report improved performance of our methods.
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One of the most challenging fields where Artificial Intelligence (AI) can be applied is lung cancer research, specifically non-small cell lung cancer (NSCLC). In particular, overall survival (OS), the time between diagnosis and death, is a vital indicator of patient status, enabling tailored treatment and improved OS rates. In this analysis, there are two challenges to take into account. First, few studies effectively exploit the information available from each patient, leveraging both uncensored (i.e., dead) and censored (i.e., survivors) patients, considering also the events' time. Second, the handling of incomplete data is a common issue in the medical field. This problem is typically tackled through the use of imputation methods. Our objective is to present an AI model able to overcome these limits, effectively learning from both censored and uncensored patients and their available features, for the prediction of OS for NSCLC patients. We present a novel approach to survival analysis with missing values in the context of NSCLC, which exploits the strengths of the transformer architecture to account only for available features without requiring any imputation strategy. By making use of ad-hoc losses for OS, it is able to account for both censored and uncensored patients, as well as changes in risks over time. We compared our method with state-of-the-art models for survival analysis coupled with different imputation strategies. We evaluated the results obtained over a period of 6 years using different time granularities obtaining a Ct-index, a time-dependent variant of the C-index, of 71.97, 77.58 and 80.72 for time units of 1 month, 1 year and 2 years, respectively, outperforming all state-of-the-art methods regardless of the imputation method used.
The dictionary learning problem can be viewed as a data-driven process to learn a suitable transformation so that data is sparsely represented directly from example data. In this paper, we examine the problem of learning a dictionary that is invariant under a pre-specified group of transformations. Natural settings include Cryo-EM, multi-object tracking, synchronization, pose estimation, etc. We specifically study this problem under the lens of mathematical representation theory. Leveraging the power of non-abelian Fourier analysis for functions over compact groups, we prescribe an algorithmic recipe for learning dictionaries that obey such invariances. We relate the dictionary learning problem in the physical domain, which is naturally modelled as being infinite dimensional, with the associated computational problem, which is necessarily finite dimensional. We establish that the dictionary learning problem can be effectively understood as an optimization instance over certain matrix orbitopes having a particular block-diagonal structure governed by the irreducible representations of the group of symmetries. This perspective enables us to introduce a band-limiting procedure which obtains dimensionality reduction in applications. We provide guarantees for our computational ansatz to provide a desirable dictionary learning outcome. We apply our paradigm to investigate the dictionary learning problem for the groups SO(2) and SO(3). While the SO(2)-orbitope admits an exact spectrahedral description, substantially less is understood about the SO(3)-orbitope. We describe a tractable spectrahedral outer approximation of the SO(3)-orbitope, and contribute an alternating minimization paradigm to perform optimization in this setting. We provide numerical experiments to highlight the efficacy of our approach in learning SO(3)-invariant dictionaries, both on synthetic and on real world data.
We present a nonparametric model-agnostic framework for building prediction intervals of insurance claims, with finite sample statistical guarantees, extending the technique of split conformal prediction to the domain of two-stage frequency-severity modeling. The effectiveness of the framework is showcased with simulated and real datasets. When the underlying severity model is a random forest, we extend the two-stage split conformal prediction procedure, showing how the out-of-bag mechanism can be leveraged to eliminate the need for a calibration set and to enable the production of prediction intervals with adaptive width.
The aim of this paper is to develop estimation and inference methods for the drift parameters of multivariate L\'evy-driven continuous-time autoregressive processes of order $p\in\mathbb{N}$. Starting from a continuous-time observation of the process, we develop consistent and asymptotically normal maximum likelihood estimators. We then relax the unrealistic assumption of continuous-time observation by considering natural discretizations based on a combination of Riemann-sum, finite difference, and thresholding approximations. The resulting estimators are also proven to be consistent and asymptotically normal under a general set of conditions, allowing for both finite and infinite jump activity in the driving L\'evy process. When discretizing the estimators, allowing for irregularly spaced observations is of great practical importance. In this respect, CAR($p$) models are not just relevant for "true" continuous-time processes: a CAR($p$) specification provides a natural continuous-time interpolation for modeling irregularly spaced data - even if the observed process is inherently discrete. As a practically relevant application, we consider the setting where the multivariate observation is known to possess a graphical structure. We refer to such a process as GrCAR and discuss the corresponding drift estimators and their properties. The finite sample behavior of all theoretical asymptotic results is empirically assessed by extensive simulation experiments.
In this paper, we present a model describing the collective motion of birds. We explore the dynamic relationship between followers and leaders, wherein a select few agents, known as leaders, can initiate spontaneous changes in direction without being influenced by external factors like predators. Starting at the microscopic level, we develop a kinetic model that characterizes the behaviour of large crowds with transient leadership. One significant challenge lies in managing topological interactions, as identifying nearest neighbors in extensive systems can be computationally expensive. To address this, we propose a novel stochastic particle method to simulate the mesoscopic dynamics and reduce the computational cost of identifying closer agents from quadratic to logarithmic complexity using a $k$-nearest neighbours search algorithm with a binary tree. Lastly, we conduct various numerical experiments for different scenarios to validate the algorithm's effectiveness and investigate collective dynamics in both two and three dimensions.
We study the problem of collaboratively learning least squares estimates for $m$ agents. Each agent observes a different subset of the features$\unicode{x2013}$e.g., containing data collected from sensors of varying resolution. Our goal is to determine how to coordinate the agents in order to produce the best estimator for each agent. We propose a distributed, semi-supervised algorithm Collab, consisting of three steps: local training, aggregation, and distribution. Our procedure does not require communicating the labeled data, making it communication efficient and useful in settings where the labeled data is inaccessible. Despite this handicap, our procedure is nearly asymptotically local minimax optimal$\unicode{x2013}$even among estimators allowed to communicate the labeled data such as imputation methods. We test our method on real and synthetic data.
In many modern statistical problems, the limited available data must be used both to develop the hypotheses to test, and to test these hypotheses-that is, both for exploratory and confirmatory data analysis. Reusing the same dataset for both exploration and testing can lead to massive selection bias, leading to many false discoveries. Selective inference is a framework that allows for performing valid inference even when the same data is reused for exploration and testing. In this work, we are interested in the problem of selective inference for data clustering, where a clustering procedure is used to hypothesize a separation of the data points into a collection of subgroups, and we then wish to test whether these data-dependent clusters in fact represent meaningful differences within the data. Recent work by Gao et al. [2022] provides a framework for doing selective inference for this setting, where a hierarchical clustering algorithm is used for producing the cluster assignments, which was then extended to k-means clustering by Chen and Witten [2022]. Both these works rely on assuming a known covariance structure for the data, but in practice, the noise level needs to be estimated-and this is particularly challenging when the true cluster structure is unknown. In our work, we extend this work to the setting of noise with unknown variance, and provide a selective inference method for this more general setting. Empirical results show that our new method is better able to maintain high power while controlling Type I error when the true noise level is unknown.
Hyperspectral Image (HSI)s cover hundreds or thousands of narrow spectral bands, conveying a wealth of spatial and spectral information. However, due to the instrumental errors and the atmospheric changes, the HSI obtained in practice are often contaminated by noise and dead pixels(lines), resulting in missing information that may severely compromise the subsequent applications. We introduce here a novel HSI missing pixel prediction algorithm, called Low Rank and Sparsity Constraint Plug-and-Play (LRS-PnP). It is shown that LRS-PnP is able to predict missing pixels and bands even when all spectral bands of the image are missing. The proposed LRS-PnP algorithm is further extended to a self-supervised model by combining the LRS-PnP with the Deep Image Prior (DIP), called LRS-PnP-DIP. In a series of experiments with real data, It is shown that the LRS-PnP-DIP either achieves state-of-the-art inpainting performance compared to other learning-based methods, or outperforms them.
One of the most challenging fields where Artificial Intelligence (AI) can be applied is lung cancer research, specifically non-small cell lung cancer (NSCLC). In particular, overall survival (OS) is a vital indicator of patient status, helping to identify subgroups with diverse survival probabilities, enabling tailored treatment and improved OS rates. In this analysis, there are two challenges to take into account. First, few studies effectively exploit the information available from each patient, leveraging both uncensored (i.e., dead) and censored (i.e., survivors) patients, considering also the death times. Second, the handling of incomplete data is a common issue in the medical field. This problem is typically tackled through the use of imputation methods. Our objective is to present an AI model able to overcome these limits, effectively learning from both censored and uncensored patients and their available features, for the prediction of OS for NSCLC patients. We present a novel approach to survival analysis in the context of NSCLC, which exploits the strengths of the transformer architecture accounting for only available features without requiring any imputation strategy. By making use of ad-hoc losses for OS, it accounts for both censored and uncensored patients, considering risks over time. We evaluated the results over a period of 6 years using different time granularities obtaining a Ct-index, a time-dependent variant of the C-index, of 71.97, 77.58 and 80.72 for time units of 1 month, 1 year and 2 years, respectively, outperforming all state-of-the-art methods regardless of the imputation method used.
Clustering is one of the most fundamental and wide-spread techniques in exploratory data analysis. Yet, the basic approach to clustering has not really changed: a practitioner hand-picks a task-specific clustering loss to optimize and fit the given data to reveal the underlying cluster structure. Some types of losses---such as k-means, or its non-linear version: kernelized k-means (centroid based), and DBSCAN (density based)---are popular choices due to their good empirical performance on a range of applications. Although every so often the clustering output using these standard losses fails to reveal the underlying structure, and the practitioner has to custom-design their own variation. In this work we take an intrinsically different approach to clustering: rather than fitting a dataset to a specific clustering loss, we train a recurrent model that learns how to cluster. The model uses as training pairs examples of datasets (as input) and its corresponding cluster identities (as output). By providing multiple types of training datasets as inputs, our model has the ability to generalize well on unseen datasets (new clustering tasks). Our experiments reveal that by training on simple synthetically generated datasets or on existing real datasets, we can achieve better clustering performance on unseen real-world datasets when compared with standard benchmark clustering techniques. Our meta clustering model works well even for small datasets where the usual deep learning models tend to perform worse.
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