Johannes Lederer

Mike Laszkiewicz, Johannes Lederer, Asja Fischer

Learning the tail behavior of a distribution is a notoriously difficult problem. By definition, the number of samples from the tail is small, and deep generative models, such as normalizing flows, tend to concentrate on learning the body of the distribution. In this paper, we focus on improving the ability of normalizing flows to correctly capture the tail behavior and, thus, form more accurate models. We prove that the marginal tailedness of an autoregressive flow can be controlled via the tailedness of the marginals of its base distribution. This theoretical insight leads us to a novel type of flows based on flexible base distributions and data-driven linear layers. An empirical analysis shows that the proposed method improves on the accuracy -- especially on the tails of the distribution -- and is able to generate heavy-tailed data. We demonstrate its application on a weather and climate example, in which capturing the tail behavior is essential.

Johannes Lederer

Neural networks are becoming increasingly popular in applications, but our mathematical understanding of their potential and limitations is still limited. In this paper, we further this understanding by developing statistical guarantees for sparse deep learning. In contrast to previous work, we consider different types of sparsity, such as few active connections, few active nodes, and other norm-based types of sparsity. Moreover, our theories cover important aspects that previous theories have neglected, such as multiple outputs, regularization, and l2-loss. The guarantees have a mild dependence on network widths and depths, which means that they support the application of sparse but wide and deep networks from a statistical perspective. Some of the concepts and tools that we use in our derivations are uncommon in deep learning and, hence, might be of additional interest.

Mike Laszkiewicz, Denis Lukovnikov, Johannes Lederer et al.

In this work, we propose a set-membership inference attack for generative models using deep image watermarking techniques. In particular, we demonstrate how conditional sampling from a generative model can reveal the watermark that was injected into parts of the training data. Our empirical results demonstrate that the proposed watermarking technique is a principled approach for detecting the non-consensual use of image data in training generative models.

Ayla Jungbluth, Johannes Lederer

Many methods for time-series forecasting are known in classical statistics, such as autoregression, moving averages, and exponential smoothing. The DeepAR framework is a novel, recent approach for time-series forecasting based on deep learning. DeepAR has shown very promising results already. However, time series often have change points, which can degrade the DeepAR's prediction performance substantially. This paper extends the DeepAR framework by detecting and including those change points. We show that our method performs as well as standard DeepAR when there are no change points and considerably better when there are change points. More generally, we show that the batch size provides an effective and surprisingly simple way to deal with change points in DeepAR, Transformers, and other modern forecasting models.

Benedikt Lütke Schwienhorst, Nadja Klein, Johannes Lederer

Score matching is an alternative to maximum likelihood estimation when the normalizing constant is unknown or too costly to evaluate. However, vanilla score matching has shown to be inefficient relative to maximum likelihood estimation for multimodal distributions with well-separated modes, which are commonly encountered in practical applications. We compare a novel diffusion-based denoising score matching estimator (DDSME) to the vanilla score matching estimator (SME) in this scenario. In particular, we prove statistical guarantees for both estimators, showing that the error bound for the vanilla SME worsens when the separation between the modes increases, which can be avoided in case of the DDSME with suitable hyperparameter tuning. This provides a novel theoretical explanation for the superior behavior of diffusion-based score matching over the vanilla version.

Somnath Chakraborty, Johannes Lederer, Rainer von Sachs

Motivated by a variety of applications, high-dimensional time series have become an active topic of research. In particular, several methods and finite-sample theories for individual stable autoregressive processes with known lag have become available very recently. We, instead, consider multiple stable autoregressive processes that share an unknown lag. We use information across the different processes to simultaneously select the lag and estimate the parameters. We prove that the estimated process is stable, and we establish rates for the forecasting error that can outmatch the known rate in our setting. Our insights on the lag selection and the stability are also of interest for the case of individual autoregressive processes.

Mahsa Taheri, Fang Xie, Johannes Lederer

Since statistical guarantees for neural networks are usually restricted to global optima of intricate objective functions, it is unclear whether these theories explain the performances of actual outputs of neural network pipelines. The goal of this paper is, therefore, to bring statistical theory closer to practice. We develop statistical guarantees for shallow linear neural networks that coincide up to logarithmic factors with the global optima but apply to stationary points and the points nearby. These results support the common notion that neural networks do not necessarily need to be optimized globally from a mathematical perspective. We then extend our statistical guarantees to shallow ReLU neural networks, assuming the first layer weight matrices are nearly identical for the stationary network and the target. More generally, despite being limited to shallow neural networks for now, our theories make an important step forward in describing the practical properties of neural networks in mathematical terms.

Simon Damm, Mike Laszkiewicz, Johannes Lederer et al.

Recent advances in multimodal foundation models have set new standards in few-shot anomaly detection. This paper explores whether high-quality visual features alone are sufficient to rival existing state-of-the-art vision-language models. We affirm this by adapting DINOv2 for one-shot and few-shot anomaly detection, with a focus on industrial applications. We show that this approach does not only rival existing techniques but can even outmatch them in many settings. Our proposed vision-only approach, AnomalyDINO, follows the well-established patch-level deep nearest neighbor paradigm, and enables both image-level anomaly prediction and pixel-level anomaly segmentation. The approach is methodologically simple and training-free and, thus, does not require any additional data for fine-tuning or meta-learning. The approach is methodologically simple and training-free and, thus, does not require any additional data for fine-tuning or meta-learning. Despite its simplicity, AnomalyDINO achieves state-of-the-art results in one- and few-shot anomaly detection (e.g., pushing the one-shot performance on MVTec-AD from an AUROC of 93.1% to 96.6%). The reduced overhead, coupled with its outstanding few-shot performance, makes AnomalyDINO a strong candidate for fast deployment, e.g., in industrial contexts.

Ali Mohaddes, Johannes Lederer

The notion of group invariance helps neural networks in recognizing patterns and features under geometric transformations. Group convolutional neural networks enhance traditional convolutional neural networks by incorporating group-based geometric structures into their design. This research studies affine invariance on continuous-domain convolutional neural networks. Despite other research considering isometric invariance or similarity invariance, we focus on the full structure of affine transforms generated by the group of all invertible $2 \times 2$ real matrices (generalized linear group $\mathrm{GL}_2(\mathbb{R})$). We introduce a new criterion to assess the invariance of two signals under affine transformations. The input image is embedded into the affine Lie group $G_2 = \mathbb{R}^2 \ltimes \mathrm{GL}_2(\mathbb{R})$ to facilitate group convolution operations that respect affine invariance. Then, we analyze the convolution of embedded signals over $G_2$. In sum, our research could eventually extend the scope of geometrical transformations that usual deep-learning pipelines can handle.

Gitte Kremling, Jeffrey Näf, Johannes Lederer

The prevalence of missing values in data science poses a substantial risk to any further analyses. Despite a wealth of research, principled nonparametric methods to deal with general non-monotone missingness are still scarce. Instead, ad-hoc imputation methods are often used, for which it remains unclear whether the correct distribution can be recovered. In this paper, we propose FLOWGEM, a principled iterative method for generating a complete dataset from a dataset with values Missing at Random (MAR). Motivated by convergence results of the ignoring maximum likelihood estimator, our approach minimizes the expected Kullback-Leibler (KL) divergence between the observed data distribution and the distribution of the generated sample over different missingness patterns. To minimize the KL divergence, we employ a discretized particle evolution of the corresponding Wasserstein Gradient Flow, where the velocity field is approximated using a local linear estimator of the density ratio. This construction yields a data generation scheme that iteratively transports an initial particle ensemble toward the target distribution. Simulation studies and real-data benchmarks demonstrate that FLOWGEM achieves state-of-the-art performance across a range of settings, including the challenging case of non-monotonic MAR mechanisms. Together, these results position FLOWGEM as a principled and practical alternative to existing imputation methods, and a decisive step towards closing the gap between theoretical rigor and empirical performance.

Gitte Kremling, Francesco Iafrate, Mahsa Taheri et al.

Score-based generative modeling, implemented through probability flow ODEs, has shown impressive results in numerous practical settings. However, most convergence guarantees rely on restrictive regularity assumptions on the target distribution -- such as strong log-concavity or bounded support. This work establishes non-asymptotic convergence bounds in the 2-Wasserstein distance for a general class of probability flow ODEs under considerably weaker assumptions: weak log-concavity and Lipschitz continuity of the score function. Our framework accommodates non-log-concave distributions, such as Gaussian mixtures, and explicitly accounts for initialization errors, score approximation errors, and effects of discretization via an exponential integrator scheme. Bridging a key theoretical challenge in diffusion-based generative modeling, our results extend convergence theory to more realistic data distributions and practical ODE solvers. We provide concrete guarantees for the efficiency and correctness of the sampling algorithm, complementing the empirical success of diffusion models with rigorous theory. Moreover, from a practical perspective, our explicit rates might be helpful in choosing hyperparameters, such as the step size in the discretization.

Mahsa Taheri, Johannes Lederer

Diffusion models are one of the key architectures of generative AI. Their main drawback, however, is the computational costs. This study indicates that the concept of sparsity, well known especially in statistics, can provide a pathway to more efficient diffusion pipelines. Our mathematical guarantees prove that sparsity can reduce the input dimension's influence on the computational complexity to that of a much smaller intrinsic dimension of the data. Our empirical findings confirm that inducing sparsity can indeed lead to better samples at a lower cost.

Mike Laszkiewicz, Imant Daunhawer, Julia E. Vogt et al.

Recent years have witnessed a rapid development of deep generative models for creating synthetic media, such as images and videos. While the practical applications of these models in everyday tasks are enticing, it is crucial to assess the inherent risks regarding their fairness. In this work, we introduce a comprehensive framework for benchmarking the performance and fairness of conditional generative models. We develop a set of metrics$\unicode{x2013}$inspired by their supervised fairness counterparts$\unicode{x2013}$to evaluate the models on their fairness and diversity. Focusing on the specific application of image upsampling, we create a benchmark covering a wide variety of modern upsampling methods. As part of the benchmark, we introduce UnfairFace, a subset of FairFace that replicates the racial distribution of common large-scale face datasets. Our empirical study highlights the importance of using an unbiased training set and reveals variations in how the algorithms respond to dataset imbalances. Alarmingly, we find that none of the considered methods produces statistically fair and diverse results. All experiments can be reproduced using our provided repository.

Mike Laszkiewicz, Jonas Ricker, Johannes Lederer et al.

Recent breakthroughs in generative modeling have sparked interest in practical single-model attribution. Such methods predict whether a sample was generated by a specific generator or not, for instance, to prove intellectual property theft. However, previous works are either limited to the closed-world setting or require undesirable changes to the generative model. We address these shortcomings by, first, viewing single-model attribution through the lens of anomaly detection. Arising from this change of perspective, we propose FLIPAD, a new approach for single-model attribution in the open-world setting based on final-layer inversion and anomaly detection. We show that the utilized final-layer inversion can be reduced to a convex lasso optimization problem, making our approach theoretically sound and computationally efficient. The theoretical findings are accompanied by an experimental study demonstrating the effectiveness of our approach and its flexibility to various domains.

Rebecca Marion, Johannes Lederer, Bernadette Govaerts et al.

Sparse linear prediction methods suffer from decreased prediction accuracy when the predictor variables have cluster structure (e.g. there are highly correlated groups of variables). To improve prediction accuracy, various methods have been proposed to identify variable clusters from the data and integrate cluster information into a sparse modeling process. But none of these methods achieve satisfactory performance for prediction, variable selection and variable clustering simultaneously. This paper presents Variable Cluster Principal Component Regression (VC-PCR), a prediction method that supervises variable selection and variable clustering in order to solve this problem. Experiments with real and simulated data demonstrate that, compared to competitor methods, VC-PCR achieves better prediction, variable selection and clustering performance when cluster structure is present.

Mike Laszkiewicz, Johannes Lederer, Asja Fischer

Normalizing flows, which learn a distribution by transforming the data to samples from a Gaussian base distribution, have proven powerful density approximations. But their expressive power is limited by this choice of the base distribution. We, therefore, propose to generalize the base distribution to a more elaborate copula distribution to capture the properties of the target distribution more accurately. In a first empirical analysis, we demonstrate that this replacement can dramatically improve the vanilla normalizing flows in terms of flexibility, stability, and effectivity for heavy-tailed data. Our results suggest that the improvements are related to an increased local Lipschitz-stability of the learned flow.

Leni Ven, Johannes Lederer

Deep learning requires several design choices, such as the nodes' activation functions and the widths, types, and arrangements of the layers. One consideration when making these choices is the vanishing-gradient problem, which is the phenomenon of algorithms getting stuck at suboptimal points due to small gradients. In this paper, we revisit the vanishing-gradient problem in the context of sigmoid-type activation. We use mathematical arguments to highlight two different sources of the phenomenon, namely large individual parameters and effects across layers, and to illustrate two simple remedies, namely regularization and rescaling. We then demonstrate the effectiveness of the two remedies in practice. In view of the vanishing-gradient problem being a main reason why tanh and other sigmoid-type activation has become much less popular than relu-type activation, our results bring sigmoid-type activation back to the table.

Shih-Ting Huang, Johannes Lederer

Deep learning systems are typically designed to perform for a wide range of test inputs. For example, deep learning systems in autonomous cars are supposed to deal with traffic situations for which they were not specifically trained. In general, the ability to cope with a broad spectrum of unseen test inputs is called generalization. Generalization is definitely important in applications where the possible test inputs are known but plentiful or simply unknown, but there are also cases where the possible inputs are few and unlabeled but known beforehand. For example, medicine is currently interested in targeting treatments to individual patients; the number of patients at any given time is usually small (typically one), their diagnoses/responses/... are still unknown, but their general characteristics (such as genome information, protein levels in the blood, and so forth) are known before the treatment. We propose to call deep learning in such applications targeted deep learning. In this paper, we introduce a framework for targeted deep learning, and we devise and test an approach for adapting standard pipelines to the requirements of targeted deep learning. The approach is very general yet easy to use: it can be implemented as a simple data-preprocessing step. We demonstrate on a variety of real-world data that our approach can indeed render standard deep learning faster and more accurate when the test inputs are known beforehand.

Shih-Ting Huang, Johannes Lederer

Data used in deep learning is notoriously problematic. For example, data are usually combined from diverse sources, rarely cleaned and vetted thoroughly, and sometimes corrupted on purpose. Intentional corruption that targets the weak spots of algorithms has been studied extensively under the label of "adversarial attacks." In contrast, the arguably much more common case of corruption that reflects the limited quality of data has been studied much less. Such "random" corruptions are due to measurement errors, unreliable sources, convenience sampling, and so forth. These kinds of corruption are common in deep learning, because data are rarely collected according to strict protocols -- in strong contrast to the formalized data collection in some parts of classical statistics. This paper concerns such corruption. We introduce an approach motivated by very recent insights into median-of-means and Le Cam's principle, we show that the approach can be readily implemented, and we demonstrate that it performs very well in practice. In conclusion, we believe that our approach is a very promising alternative to standard parameter training based on least-squares and cross-entropy loss.

Johannes Lederer

Activation functions shape the outputs of artificial neurons and, therefore, are integral parts of neural networks in general and deep learning in particular. Some activation functions, such as logistic and relu, have been used for many decades. But with deep learning becoming a mainstream research topic, new activation functions have mushroomed, leading to confusion in both theory and practice. This paper provides an analytic yet up-to-date overview of popular activation functions and their properties, which makes it a timely resource for anyone who studies or applies neural networks.

Johannes Lederer

We analyze the optimization landscapes of deep learning with wide networks. We highlight the importance of constraints for such networks and show that constraint -- as well as unconstraint -- empirical-risk minimization over such networks has no confined points, that is, suboptimal parameters that are difficult to escape from. Hence, our theories substantiate the common belief that wide neural networks are not only highly expressive but also comparably easy to optimize.

Johannes Lederer

It has been observed that certain loss functions can render deep-learning pipelines robust against flaws in the data. In this paper, we support these empirical findings with statistical theory. We especially show that empirical-risk minimization with unbounded, Lipschitz-continuous loss functions, such as the least-absolute deviation loss, Huber loss, Cauchy loss, and Tukey's biweight loss, can provide efficient prediction under minimal assumptions on the data. More generally speaking, our paper provides theoretical evidence for the benefits of robust loss functions in deep learning.

Sarah Friedrich, Gerd Antes, Sigrid Behr et al.

The research on and application of artificial intelligence (AI) has triggered a comprehensive scientific, economic, social and political discussion. Here we argue that statistics, as an interdisciplinary scientific field, plays a substantial role both for the theoretical and practical understanding of AI and for its future development. Statistics might even be considered a core element of AI. With its specialist knowledge of data evaluation, starting with the precise formulation of the research question and passing through a study design stage on to analysis and interpretation of the results, statistics is a natural partner for other disciplines in teaching, research and practice. This paper aims at contributing to the current discussion by highlighting the relevance of statistical methodology in the context of AI development. In particular, we discuss contributions of statistics to the field of artificial intelligence concerning methodological development, planning and design of studies, assessment of data quality and data collection, differentiation of causality and associations and assessment of uncertainty in results. Moreover, the paper also deals with the equally necessary and meaningful extension of curricula in schools and universities.

Mohamed Hebiri, Johannes Lederer

Sparsity has become popular in machine learning, because it can save computational resources, facilitate interpretations, and prevent overfitting. In this paper, we discuss sparsity in the framework of neural networks. In particular, we formulate a new notion of sparsity that concerns the networks' layers and, therefore, aligns particularly well with the current trend toward deep networks. We call this notion layer sparsity. We then introduce corresponding regularization and refitting schemes that can complement standard deep-learning pipelines to generate more compact and accurate networks.

Mahsa Taheri, Fang Xie, Johannes Lederer

Neural networks have become standard tools in the analysis of data, but they lack comprehensive mathematical theories. For example, there are very few statistical guarantees for learning neural networks from data, especially for classes of estimators that are used in practice or at least similar to such. In this paper, we develop a general statistical guarantee for estimators that consist of a least-squares term and a regularizer. We then exemplify this guarantee with $\ell_1$-regularization, showing that the corresponding prediction error increases at most sub-linearly in the number of layers and at most logarithmically in the total number of parameters. Our results establish a mathematical basis for regularized estimation of neural networks, and they deepen our mathematical understanding of neural networks and deep learning more generally.

Mike Laszkiewicz, Asja Fischer, Johannes Lederer

Many Machine Learning algorithms are formulated as regularized optimization problems, but their performance hinges on a regularization parameter that needs to be calibrated to each application at hand. In this paper, we propose a general calibration scheme for regularized optimization problems and apply it to the graphical lasso, which is a method for Gaussian graphical modeling. The scheme is equipped with theoretical guarantees and motivates a thresholding pipeline that can improve graph recovery. Moreover, requiring at most one line search over the regularization path, the calibration scheme is computationally more efficient than competing schemes that are based on resampling. Finally, we show in simulations that our approach can improve on the graph recovery of other approaches considerably.

Shih-Ting Huang, Fang Xie, Johannes Lederer

Ridge estimators regularize the squared Euclidean lengths of parameters. Such estimators are mathematically and computationally attractive but involve tuning parameters that can be difficult to calibrate. In this paper, we show that ridge estimators can be modified such that tuning parameters can be avoided altogether. We also show that these modified versions can improve on the empirical prediction accuracies of standard ridge estimators combined with cross-validation, and we provide first theoretical guarantees.

Shih-Ting Huang, Yannick Düren, Kristoffer H. Hellton et al.

Personalized medicine has become an important part of medicine, for instance predicting individual drug responses based on genomic information. However, many current statistical methods are not tailored to this task, because they overlook the individual heterogeneity of patients. In this paper, we look at personalized medicine from a linear regression standpoint. We introduce an alternative version of the ridge estimator and target individuals by establishing a tuning parameter calibration scheme that minimizes prediction errors of individual patients. In stark contrast, classical schemes such as cross-validation minimize prediction errors only on average. We show that our pipeline is optimal in terms of oracle inequalities, fast, and highly effective both in simulations and on real data.

Rui Zhuang, Johannes Lederer

Maximum regularized likelihood estimators (MRLEs) are arguably the most established class of estimators in high-dimensional statistics. In this paper, we derive guarantees for MRLEs in Kullback-Leibler divergence, a general measure of prediction accuracy. We assume only that the densities have a convex parametrization and that the regularization is definite and positive homogenous. The results thus apply to a very large variety of models and estimators, such as tensor regression and graphical models with convex and non-convex regularized methods. A main conclusion is that MRLEs are broadly consistent in prediction - regardless of whether restricted eigenvalues or similar conditions hold.

Yunqi Bu, Johannes Lederer

In applications of graphical models, we typically have more information than just the samples themselves. A prime example is the estimation of brain connectivity networks based on fMRI data, where in addition to the samples themselves, the spatial positions of the measurements are readily available. With particular regard for this application, we are thus interested in ways to incorporate additional knowledge most effectively into graph estimation. Our approach to this is to make neighborhood selection receptive to additional knowledge by strengthening the role of the tuning parameters. We demonstrate that this concept (i) can improve reproducibility, (ii) is computationally convenient and efficient, and (iii) carries a lucid Bayesian interpretation. We specifically show that the approach provides effective estimations of brain connectivity graphs from fMRI data. However, providing a general scheme for the inclusion of additional knowledge, our concept is expected to have applications in a wide range of domains.

Wei Li, Johannes Lederer

Feature selection is a standard approach to understanding and modeling high-dimensional classification data, but the corresponding statistical methods hinge on tuning parameters that are difficult to calibrate. In particular, existing calibration schemes in the logistic regression framework lack any finite sample guarantees. In this paper, we introduce a novel calibration scheme for $\ell_1$-penalized logistic regression. It is based on simple tests along the tuning parameter path and is equipped with optimal guarantees for feature selection. It is also amenable to easy and efficient implementations, and it rivals or outmatches existing methods in simulations and real data applications.

Mahsa Taheri, Néhémy Lim, Johannes Lederer

Modern technologies are generating ever-increasing amounts of data. Making use of these data requires methods that are both statistically sound and computationally efficient. Typically, the statistical and computational aspects are treated separately. In this paper, we propose an approach to entangle these two aspects in the context of regularized estimation. Applying our approach to sparse and group-sparse regression, we show that it can improve on standard pipelines both statistically and computationally.

Johannes Lederer, Lu Yu, Irina Gaynanova

The abundance of high-dimensional data in the modern sciences has generated tremendous interest in penalized estimators such as the lasso, scaled lasso, square-root lasso, elastic net, and many others. In this paper, we establish a general oracle inequality for prediction in high-dimensional linear regression with such methods. Since the proof relies only on convexity and continuity arguments, the result holds irrespective of the design matrix and applies to a wide range of penalized estimators. Overall, the bound demonstrates that generic estimators can provide consistent prediction with any design matrix. From a practical point of view, the bound can help to identify the potential of specific estimators, and they can help to get a sense of the prediction accuracy in a given application.

Jacob Bien, Irina Gaynanova, Johannes Lederer et al.

The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a non-convex optimization problem. This paper shows a remarkable result: despite the non-convexity of the TREX problem, there exists a polynomial-time algorithm that is guaranteed to find the global minimum. This result adds the TREX to a very short list of non-convex optimization problems that can be globally optimized (principal components analysis being a famous example). After deriving and developing this new approach, we demonstrate that (i) the ability of the preexisting TREX heuristic to reach the global minimum is strongly dependent on the difficulty of the underlying statistical problem, (ii) the new polynomial-time algorithm for TREX permits a novel variable ranking and selection scheme, (iii) this scheme can be incorporated into a rule that controls the false discovery rate (FDR) of included features in the model. To achieve this last aim, we provide an extension of the results of Barber & Candes (2015) to establish that the knockoff filter framework can be applied to the TREX. This investigation thus provides both a rare case study of a heuristic for non-convex optimization and a novel way of exploiting non-convexity for statistical inference.

Johannes Lederer, Christian Müller

We introduce Graphical TREX (GTREX), a novel method for graph estimation in high-dimensional Gaussian graphical models. By conducting neighborhood selection with TREX, GTREX avoids tuning parameters and is adaptive to the graph topology. We compare GTREX with standard methods on a new simulation set-up that is designed to assess accurately the strengths and shortcomings of different methods. These simulations show that a neighborhood selection scheme based on Lasso and an optimal (in practice unknown) tuning parameter outperforms other standard methods over a large spectrum of scenarios. Moreover, we show that GTREX can rival this scheme and, therefore, can provide competitive graph estimation without the need for tuning parameter calibration.

Johannes Lederer, Sergio Guadarrama

Sparse Filtering is a popular feature learning algorithm for image classification pipelines. In this paper, we connect the performance of Sparse Filtering with spectral properties of the corresponding feature matrices. This connection provides new insights into Sparse Filtering; in particular, it suggests early stopping of Sparse Filtering. We therefore introduce the Optimal Roundness Criterion (ORC), a novel stopping criterion for Sparse Filtering. We show that this stopping criterion is related with pre-processing procedures such as Statistical Whitening and demonstrate that it can make image classification with Sparse Filtering considerably faster and more accurate.

Johannes Lederer, Christian Müller

Lasso is a seminal contribution to high-dimensional statistics, but it hinges on a tuning parameter that is difficult to calibrate in practice. A partial remedy for this problem is Square-Root Lasso, because it inherently calibrates to the noise variance. However, Square-Root Lasso still requires the calibration of a tuning parameter to all other aspects of the model. In this study, we introduce TREX, an alternative to Lasso with an inherent calibration to all aspects of the model. This adaptation to the entire model renders TREX an estimator that does not require any calibration of tuning parameters. We show that TREX can outperform cross-validated Lasso in terms of variable selection and computational efficiency. We also introduce a bootstrapped version of TREX that can further improve variable selection. We illustrate the promising performance of TREX both on synthetic data and on a recent high-dimensional biological data set that considers riboflavin production in B. subtilis.

Arnak S. Dalalyan, Mohamed Hebiri, Johannes Lederer

Although the Lasso has been extensively studied, the relationship between its prediction performance and the correlations of the covariates is not fully understood. In this paper, we give new insights into this relationship in the context of multiple linear regression. We show, in particular, that the incorporation of a simple correlation measure into the tuning parameter can lead to a nearly optimal prediction performance of the Lasso even for highly correlated covariates. However, we also reveal that for moderately correlated covariates, the prediction performance of the Lasso can be mediocre irrespective of the choice of the tuning parameter. We finally show that our results also lead to near-optimal rates for the least-squares estimator with total variation penalty.