arXiv subjects

Bernhard Schölkopf

Publications and source records attributed to Bernhard Schölkopf.

At least 451 records · Page 25Linked to original sources

arXiv2017

Distilling Information Reliability and Source Trustworthiness from Digital Traces

Online knowledge repositories typically rely on their users or dedicated editors to evaluate the reliability of their content. These evaluations can be viewed as noisy measurements of both information reliability and information source trustworthiness. Can we leverage these noisy evaluations, often biased, to distill a robust, unbiased and interpretable measure of both notions? In this paper, we argue that the temporal traces left by these noisy evaluations give cues on the reliability of the information and the trustworthiness of the sources. Then, we propose a temporal point process modeling framework that links these temporal traces to robust, unbiased and interpretable notions of information reliability and source trustworthiness. Furthermore, we develop an efficient convex optimization procedure to learn the parameters of the model from historical traces. Experiments on real-world data gathered from Wikipedia and Stack Overflow show that our modeling framework accurately predicts evaluation events, provides an interpretable measure of information reliability and source trustworthiness, and yields interesting insights about real-world events.

cs.SI↗

arXiv2017

Discriminative Transfer Learning for General Image Restoration

Recently, several discriminative learning approaches have been proposed for effective image restoration, achieving convincing trade-off between image quality and computational efficiency. However, these methods require separate training for each restoration task (e.g., denoising, deblurring, demosaicing) and problem condition (e.g., noise level of input images). This makes it time-consuming and difficult to encompass all tasks and conditions during training. In this paper, we propose a discriminative transfer learning method that incorporates formal proximal optimization and discriminative learning for general image restoration. The method requires a single-pass training and allows for reuse across various problems and conditions while achieving an efficiency comparable to previous discriminative approaches. Furthermore, after being trained, our model can be easily transferred to new likelihood terms to solve untrained tasks, or be combined with existing priors to further improve image restoration quality.

cs.CV↗

arXiv2016

Kernel-based Tests for Joint Independence

We investigate the problem of testing whether $d$ random variables, which may or may not be continuous, are jointly (or mutually) independent. Our method builds on ideas of the two variable Hilbert-Schmidt independence criterion (HSIC) but allows for an arbitrary number of variables. We embed the $d$-dimensional joint distribution and the product of the marginals into a reproducing kernel Hilbert space and define the $d$-variable Hilbert-Schmidt independence criterion (dHSIC) as the squared distance between the embeddings. In the population case, the value of dHSIC is zero if and only if the $d$ variables are jointly independent, as long as the kernel is characteristic. Based on an empirical estimate of dHSIC, we define three different non-parametric hypothesis tests: a permutation test, a bootstrap test and a test based on a Gamma approximation. We prove that the permutation test achieves the significance level and that the bootstrap test achieves pointwise asymptotic significance level as well as pointwise asymptotic consistency (i.e., it is able to detect any type of fixed dependence in the large sample limit). The Gamma approximation does not come with these guarantees; however, it is computationally very fast and for small $d$, it performs well in practice. Finally, we apply the test to a problem in causal discovery.

math.ST↗

arXiv2016

Consistent Kernel Mean Estimation for Functions of Random Variables

We provide a theoretical foundation for non-parametric estimation of functions of random variables using kernel mean embeddings. We show that for any continuous function $f$, consistent estimators of the mean embedding of a random variable $X$ lead to consistent estimators of the mean embedding of $f(X)$. For Matérn kernels and sufficiently smooth functions we also provide rates of convergence. Our results extend to functions of multiple random variables. If the variables are dependent, we require an estimator of the mean embedding of their joint distribution as a starting point; if they are independent, it is sufficient to have separate estimators of the mean embeddings of their marginal distributions. In either case, our results cover both mean embeddings based on i.i.d. samples as well as "reduced set" expansions in terms of dependent expansion points. The latter serves as a justification for using such expansions to limit memory resources when applying the approach as a basis for probabilistic programming.

stat.ML↗

arXiv2016

The population of long-period transiting exoplanets

The Kepler Mission has discovered thousands of exoplanets and revolutionized our understanding of their population. This large, homogeneous catalog of discoveries has enabled rigorous studies of the occurrence rate of exoplanets and planetary systems as a function of their physical properties. However, transit surveys like Kepler are most sensitive to planets with orbital periods much shorter than the orbital periods of Jupiter and Saturn, the most massive planets in our Solar System. To address this deficiency, we perform a fully automated search for long-period exoplanets with only one or two transits in the archival Kepler light curves. When applied to the $\sim 40,000$ brightest Sun-like target stars, this search produces 16 long-period exoplanet candidates. Of these candidates, 6 are novel discoveries and 5 are in systems with inner short-period transiting planets. Since our method involves no human intervention, we empirically characterize the detection efficiency of our search. Based on these results, we measure the average occurrence rate of exoplanets smaller than Jupiter with orbital periods in the range 2-25 years to be $2.0\pm0.7$ planets per Sun-like star.

astro-ph.EP↗

arXiv2016

Recovery of non-linear cause-effect relationships from linearly mixed neuroimaging data

Causal inference concerns the identification of cause-effect relationships between variables. However, often only linear combinations of variables constitute meaningful causal variables. For example, recovering the signal of a cortical source from electroencephalography requires a well-tuned combination of signals recorded at multiple electrodes. We recently introduced the MERLiN (Mixture Effect Recovery in Linear Networks) algorithm that is able to recover, from an observed linear mixture, a causal variable that is a linear effect of another given variable. Here we relax the assumption of this cause-effect relationship being linear and present an extended algorithm that can pick up non-linear cause-effect relationships. Thus, the main contribution is an algorithm (and ready to use code) that has broader applicability and allows for a richer model class. Furthermore, a comparative analysis indicates that the assumption of linear cause-effect relationships is not restrictive in analysing electroencephalographic data.

stat.ME↗

arXiv2016

Screening Rules for Convex Problems

We propose a new framework for deriving screening rules for convex optimization problems. Our approach covers a large class of constrained and penalized optimization formulations, and works in two steps. First, given any approximate point, the structure of the objective function and the duality gap is used to gather information on the optimal solution. In the second step, this information is used to produce screening rules, i.e. safely identifying unimportant weight variables of the optimal solution. Our general framework leads to a large variety of useful existing as well as new screening rules for many applications. For example, we provide new screening rules for general simplex and $L_1$-constrained problems, Elastic Net, squared-loss Support Vector Machines, minimum enclosing ball, as well as structured norm regularized problems, such as group lasso.

math.OC↗

arXiv2016

End-to-End Learning for Image Burst Deblurring

We present a neural network model approach for multi-frame blind deconvolution. The discriminative approach adopts and combines two recent techniques for image deblurring into a single neural network architecture. Our proposed hybrid-architecture combines the explicit prediction of a deconvolution filter and non-trivial averaging of Fourier coefficients in the frequency domain. In order to make full use of the information contained in all images in one burst, the proposed network embeds smaller networks, which explicitly allow the model to transfer information between images in early layers. Our system is trained end-to-end using standard backpropagation on a set of artificially generated training examples, enabling competitive performance in multi-frame blind deconvolution, both with respect to quality and runtime.

cs.CV↗

arXiv2016

Depth Estimation Through a Generative Model of Light Field Synthesis

Light field photography captures rich structural information that may facilitate a number of traditional image processing and computer vision tasks. A crucial ingredient in such endeavors is accurate depth recovery. We present a novel framework that allows the recovery of a high quality continuous depth map from light field data. To this end we propose a generative model of a light field that is fully parametrized by its corresponding depth map. The model allows for the integration of powerful regularization techniques such as a non-local means prior, facilitating accurate depth map estimation.

cs.CV↗

arXiv2016

Discovery and Visualization of Nonstationary Causal Models

It is commonplace to encounter nonstationary data, of which the underlying generating process may change over time or across domains. The nonstationarity presents both challenges and opportunities for causal discovery. In this paper we propose a principled framework to handle nonstationarity, and develop some methods to address three important questions. First, we propose an enhanced constraint-based method to detect variables whose local mechanisms are nonstationary and recover the skeleton of the causal structure over observed variables. Second, we present a way to determine some causal directions by taking advantage of information carried by changing distributions. Third, we develop a method for visualizing the nonstationarity of causal modules. Experimental results on various synthetic and real-world data sets are presented to demonstrate the efficacy of our methods.

cs.AI↗

arXiv2016

A Causal, Data-Driven Approach to Modeling the Kepler Data

Astronomical observations are affected by several kinds of noise, each with its own causal source; there is photon noise, stochastic source variability, and residuals coming from imperfect calibration of the detector or telescope. The precision of NASA Kepler photometry for exoplanet science---the most precise photometric measurements of stars ever made---appears to be limited by unknown or untracked variations in spacecraft pointing and temperature, and unmodeled stellar variability. Here we present the Causal Pixel Model (CPM) for Kepler data, a data-driven model intended to capture variability but preserve transit signals. The CPM works at the pixel level so that it can capture very fine-grained information about the variation of the spacecraft. The CPM predicts each target pixel value from a large number of pixels of other stars sharing the instrument variabilities while not containing any information on possible transits in the target star. In addition, we use the target star's future and past (auto-regression). By appropriately separating, for each data point, the data into training and test sets, we ensure that information about any transit will be perfectly isolated from the model. The method has four hyper-parameters (the number of predictor stars, the auto-regressive window size, and two L2-regularization amplitudes for model components), which we set by cross-validation. We determine a generic set of hyper-parameters that works well for most of the stars and apply the method to a corresponding set of target stars. We find that we can consistently outperform (for the purposes of exoplanet detection) the Kepler Pre-search Data Conditioning (PDC) method for exoplanet discovery.

astro-ph.EP↗

arXiv2016

The Arrow of Time in Multivariate Time Series

We prove that a time series satisfying a (linear) multivariate autoregressive moving average (VARMA) model satisfies the same model assumption in the reversed time direction, too, if all innovations are normally distributed. This reversibility breaks down if the innovations are non-Gaussian. This means that under the assumption of a VARMA process with non-Gaussian noise, the arrow of time becomes detectable. Our work thereby provides a theoretic justification of an algorithm that has been used for inferring the direction of video snippets. We present a slightly modified practical algorithm that estimates the time direction for a given sample and prove its consistency. We further investigate how the performance of the algorithm depends on sample size, number of dimensions of the time series and the order of the process. An application to real world data from economics shows that considering multivariate processes instead of univariate processes can be beneficial for estimating the time direction. Our result extends earlier work on univariate time series. It relates to the concept of causal inference, where recent methods exploit non-Gaussianity of the error terms for causal structure learning.

math.ST↗

arXiv2016

Unifying distillation and privileged information

Distillation (Hinton et al., 2015) and privileged information (Vapnik & Izmailov, 2015) are two techniques that enable machines to learn from other machines. This paper unifies these two techniques into generalized distillation, a framework to learn from multiple machines and data representations. We provide theoretical and causal insight about the inner workings of generalized distillation, extend it to unsupervised, semisupervised and multitask learning scenarios, and illustrate its efficacy on a variety of numerical simulations on both synthetic and real-world data.

stat.ML↗

arXiv2016

Kernel Mean Shrinkage Estimators

A mean function in a reproducing kernel Hilbert space (RKHS), or a kernel mean, is central to kernel methods in that it is used by many classical algorithms such as kernel principal component analysis, and it also forms the core inference step of modern kernel methods that rely on embedding probability distributions in RKHSs. Given a finite sample, an empirical average has been used commonly as a standard estimator of the true kernel mean. Despite a widespread use of this estimator, we show that it can be improved thanks to the well-known Stein phenomenon. We propose a new family of estimators called kernel mean shrinkage estimators (KMSEs), which benefit from both theoretical justifications and good empirical performance. The results demonstrate that the proposed estimators outperform the standard one, especially in a "large d, small n" paradigm.

stat.ML↗

arXiv2015

Distinguishing cause from effect using observational data: methods and benchmarks

The discovery of causal relationships from purely observational data is a fundamental problem in science. The most elementary form of such a causal discovery problem is to decide whether X causes Y or, alternatively, Y causes X, given joint observations of two variables X, Y. An example is to decide whether altitude causes temperature, or vice versa, given only joint measurements of both variables. Even under the simplifying assumptions of no confounding, no feedback loops, and no selection bias, such bivariate causal discovery problems are challenging. Nevertheless, several approaches for addressing those problems have been proposed in recent years. We review two families of such methods: Additive Noise Methods (ANM) and Information Geometric Causal Inference (IGCI). We present the benchmark CauseEffectPairs that consists of data for 100 different cause-effect pairs selected from 37 datasets from various domains (e.g., meteorology, biology, medicine, engineering, economy, etc.) and motivate our decisions regarding the "ground truth" causal directions of all pairs. We evaluate the performance of several bivariate causal discovery methods on these real-world benchmark data and in addition on artificially simulated data. Our empirical results on real-world data indicate that certain methods are indeed able to distinguish cause from effect using only purely observational data, although more benchmark data would be needed to obtain statistically significant conclusions. One of the best performing methods overall is the additive-noise method originally proposed by Hoyer et al. (2009), which obtains an accuracy of 63+-10 % and an AUC of 0.74+-0.05 on the real-world benchmark. As the main theoretical contribution of this work we prove the consistency of that method.

cs.LG↗

arXiv2015

Causal Inference by Identification of Vector Autoregressive Processes with Hidden Components

A widely applied approach to causal inference from a non-experimental time series $X$, often referred to as "(linear) Granger causal analysis", is to regress present on past and interpret the regression matrix $\hat{B}$ causally. However, if there is an unmeasured time series $Z$ that influences $X$, then this approach can lead to wrong causal conclusions, i.e., distinct from those one would draw if one had additional information such as $Z$. In this paper we take a different approach: We assume that $X$ together with some hidden $Z$ forms a first order vector autoregressive (VAR) process with transition matrix $A$, and argue why it is more valid to interpret $A$ causally instead of $\hat{B}$. Then we examine under which conditions the most important parts of $A$ are identifiable or almost identifiable from only $X$. Essentially, sufficient conditions are (1) non-Gaussian, independent noise or (2) no influence from $X$ to $Z$. We present two estimation algorithms that are tailored towards conditions (1) and (2), respectively, and evaluate them on synthetic and real-world data. We discuss how to check the model using $X$.

stat.ML↗

arXiv2015

Causal and anti-causal learning in pattern recognition for neuroimaging

Pattern recognition in neuroimaging distinguishes between two types of models: encoding- and decoding models. This distinction is based on the insight that brain state features, that are found to be relevant in an experimental paradigm, carry a different meaning in encoding- than in decoding models. In this paper, we argue that this distinction is not sufficient: Relevant features in encoding- and decoding models carry a different meaning depending on whether they represent causal- or anti-causal relations. We provide a theoretical justification for this argument and conclude that causal inference is essential for interpretation in neuroimaging.

stat.ML↗

arXiv2015

Decoding index finger position from EEG using random forests

While invasively recorded brain activity is known to provide detailed information on motor commands, it is an open question at what level of detail information about positions of body parts can be decoded from non-invasively acquired signals. In this work it is shown that index finger positions can be differentiated from non-invasive electroencephalographic (EEG) recordings in healthy human subjects. Using a leave-one-subject-out cross-validation procedure, a random forest distinguished different index finger positions on a numerical keyboard above chance-level accuracy. Among the different spectral features investigated, high $β$-power (20-30 Hz) over contralateral sensorimotor cortex carried most information about finger position. Thus, these findings indicate that finger position is in principle decodable from non-invasive features of brain activity that generalize across individuals.

stat.ML↗