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Given a sequence of observable variables $\{(x_1, y_1), \ldots, (x_n, y_n)\}$, the conformal prediction method estimates a confidence set for $y_{n+1}$ given $x_{n+1}$ that is valid for any finite sample size by merely assuming that the joint distribution of the data is permutation invariant. Although attractive, computing such a set is computationally infeasible in most regression problems. Indeed, in these cases, the unknown variable $y_{n+1}$ can take an infinite number of possible candidate values, and generating conformal sets requires retraining a predictive model for each candidate. In this paper, we focus on a sparse linear model with only a subset of variables for prediction and use numerical continuation techniques to approximate the solution path efficiently. The critical property we exploit is that the set of selected variables is invariant under a small perturbation of the input data. Therefore, it is sufficient to enumerate and refit the model only at the change points of the set of active features and smoothly interpolate the rest of the solution via a Predictor-Corrector mechanism. We show how our path-following algorithm accurately approximates conformal prediction sets and illustrate its performance using synthetic and real data examples.

The two-trials rule for drug approval requires "at least two adequate and well-controlled studies, each convincing on its own, to establish effectiveness". This is usually employed by requiring two significant pivotal trials and is the standard regulatory requirement to provide evidence for a new drug's efficacy. However, there is need to develop suitable alternatives to this rule for a number of reasons, including the possible availability of data from more than two trials. I consider the case of up to 3 studies and stress the importance to control the partial Type-I error rate, where only some studies have a true null effect, while maintaining the overall Type-I error rate of the two-trials rule, where all studies have a null effect. Some less-known $p$-value combination methods are useful to achieve this: Pearson's method, Edgington's method and the recently proposed harmonic mean $\chi^2$-test. I study their properties and discuss how they can be extended to a sequential assessment of success while still ensuring overall Type-I error control. I compare the different methods in terms of partial Type-I error rate, project power and the expected number of studies required. Edgington's method is eventually recommended as it is easy to implement and communicate, has only moderate partial Type-I error rate inflation but substantially increased project power.

This article investigates uncertainty quantification of the generalized linear lasso~(GLL), a popular variable selection method in high-dimensional regression settings. In many fields of study, researchers use data-driven methods to select a subset of variables that are most likely to be associated with a response variable. However, such variable selection methods can introduce bias and increase the likelihood of false positives, leading to incorrect conclusions. In this paper, we propose a post-selection inference framework that addresses these issues and allows for valid statistical inference after variable selection using GLL. We show that our method provides accurate $p$-values and confidence intervals, while maintaining high statistical power. In a second stage, we focus on the sparse logistic regression, a popular classifier in high-dimensional statistics. We show with extensive numerical simulations that SIGLE is more powerful than state-of-the-art PSI methods. SIGLE relies on a new method to sample states from the distribution of observations conditional on the selection event. This method is based on a simulated annealing strategy whose energy is given by the first order conditions of the logistic lasso.

The triple-differences (TD) design is a popular identification strategy for causal effects in settings where researchers do not believe the parallel trends assumption of conventional difference-in-differences (DiD) is satisfied. TD designs augment the conventional 2x2 DiD with a "placebo" stratum -- observations that are nested in the same units and time periods but are known to be entirely unaffected by the treatment. However, many TD applications go beyond this simple 2x2x2 and use observations on many units in many "placebo" strata across multiple time periods. A popular estimator for this setting is the triple-differences regression (TDR) fixed-effects estimator -- an extension of the common "two-way fixed effects" estimator for DiD. This paper decomposes the TDR estimator into its component two-group/two-period/two-strata triple-differences and illustrates how interpreting this parameter causally in settings with arbitrary staggered adoption requires strong effect homogeneity assumptions as many placebo DiDs incorporate observations under treatment. The decomposition clarifies the implied identifying variation behind the triple-differences regression estimator and suggests researchers should be cautious when implementing these estimators in settings more complex than the 2x2x2 case. Alternative approaches that only incorporate "clean placebos" such as direct imputation of the counterfactual may be more appropriate. The paper concludes by demonstrating the utility of this imputation estimator in an application of the "gravity model" to the estimation of the effect of the WTO/GATT on international trade.

Electroencephalogram (EEG) signals reflect brain activity across different brain states, characterized by distinct frequency distributions. Through multifractal analysis tools, we investigate the scaling behaviour of different classes of EEG signals and artifacts. We show that brain states associated to sleep and general anaesthesia are not in general characterized by scale invariance. The lack of scale invariance motivates the development of artifact removal algorithms capable of operating independently at each scale. We examine here the properties of the wavelet quantile normalization algorithm, a recently introduced adaptive method for real-time correction of transient artifacts in EEG signals. We establish general results regarding the regularization properties of the WQN algorithm, showing how it can eliminate singularities introduced by artefacts, and we compare it to traditional thresholding algorithms. Furthermore, we show that the algorithm performance is independent of the wavelet basis. We finally examine its continuity and boundedness properties and illustrate its distinctive non-local action on the wavelet coefficients through pathological examples.

Renal cell carcinoma represents a significant global health challenge with a low survival rate. This research aimed to devise a comprehensive deep-learning model capable of predicting survival probabilities in patients with renal cell carcinoma by integrating CT imaging and clinical data and addressing the limitations observed in prior studies. The aim is to facilitate the identification of patients requiring urgent treatment. The proposed framework comprises three modules: a 3D image feature extractor, clinical variable selection, and survival prediction. The feature extractor module, based on the 3D CNN architecture, predicts the ISUP grade of renal cell carcinoma tumors linked to mortality rates from CT images. A selection of clinical variables is systematically chosen using the Spearman score and random forest importance score as criteria. A deep learning-based network, trained with discrete LogisticHazard-based loss, performs the survival prediction. Nine distinct experiments are performed, with varying numbers of clinical variables determined by different thresholds of the Spearman and importance scores. Our findings demonstrate that the proposed strategy surpasses the current literature on renal cancer prognosis based on CT scans and clinical factors. The best-performing experiment yielded a concordance index of 0.84 and an area under the curve value of 0.8 on the test cohort, which suggests strong predictive power. The multimodal deep-learning approach developed in this study shows promising results in estimating survival probabilities for renal cell carcinoma patients using CT imaging and clinical data. This may have potential implications in identifying patients who require urgent treatment, potentially improving patient outcomes. The code created for this project is available for the public on: \href{//github.com/Balasingham-AI-Group/Survival_CTplusClinical}{GitHub}

In this work, we investigate the interval generalized Sylvester matrix equation ${\bf{A}}X{\bf{B}}+{\bf{C}}X{\bf{D}}={\bf{F}}$ and develop some techniques for obtaining outer estimations for the so-called united solution set of this interval system. First, we propose a modified variant of the Krawczyk operator which causes reducing computational complexity to cubic, compared to Kronecker product form. We then propose an iterative technique for enclosing the solution set. These approaches are based on spectral decompositions of the midpoints of ${\bf{A}}$, ${\bf{B}}$, ${\bf{C}}$ and ${\bf{D}}$ and in both of them we suppose that the midpoints of ${\bf{A}}$ and ${\bf{C}}$ are simultaneously diagonalizable as well as for the midpoints of the matrices ${\bf{B}}$ and ${\bf{D}}$. Some numerical experiments are given to illustrate the performance of the proposed methods.

The growing energy and performance costs of deep learning have driven the community to reduce the size of neural networks by selectively pruning components. Similarly to their biological counterparts, sparse networks generalize just as well, if not better than, the original dense networks. Sparsity can reduce the memory footprint of regular networks to fit mobile devices, as well as shorten training time for ever growing networks. In this paper, we survey prior work on sparsity in deep learning and provide an extensive tutorial of sparsification for both inference and training. We describe approaches to remove and add elements of neural networks, different training strategies to achieve model sparsity, and mechanisms to exploit sparsity in practice. Our work distills ideas from more than 300 research papers and provides guidance to practitioners who wish to utilize sparsity today, as well as to researchers whose goal is to push the frontier forward. We include the necessary background on mathematical methods in sparsification, describe phenomena such as early structure adaptation, the intricate relations between sparsity and the training process, and show techniques for achieving acceleration on real hardware. We also define a metric of pruned parameter efficiency that could serve as a baseline for comparison of different sparse networks. We close by speculating on how sparsity can improve future workloads and outline major open problems in the field.

Most deep learning-based models for speech enhancement have mainly focused on estimating the magnitude of spectrogram while reusing the phase from noisy speech for reconstruction. This is due to the difficulty of estimating the phase of clean speech. To improve speech enhancement performance, we tackle the phase estimation problem in three ways. First, we propose Deep Complex U-Net, an advanced U-Net structured model incorporating well-defined complex-valued building blocks to deal with complex-valued spectrograms. Second, we propose a polar coordinate-wise complex-valued masking method to reflect the distribution of complex ideal ratio masks. Third, we define a novel loss function, weighted source-to-distortion ratio (wSDR) loss, which is designed to directly correlate with a quantitative evaluation measure. Our model was evaluated on a mixture of the Voice Bank corpus and DEMAND database, which has been widely used by many deep learning models for speech enhancement. Ablation experiments were conducted on the mixed dataset showing that all three proposed approaches are empirically valid. Experimental results show that the proposed method achieves state-of-the-art performance in all metrics, outperforming previous approaches by a large margin.