Florian Kalinke

Tobias Fuchs, Florian Kalinke, Nadja Klein

Learning generative models in settings where the source and target distributions are only specified through unpaired samples is gaining in importance. Here, one frequently-used model are Schrödinger bridges (SB), which represent the most likely evolution between both endpoint distributions. To accelerate training, simulation-free SBs avoid the path simulation of the original SB models. However, learning simulation-free SBs requires paired data; a coupling of the source and target samples is obtained as the solution of the entropic optimal transport (OT) problem. As obtaining the optimal global coupling is infeasible in many practical cases, the entropic OT problem is iteratively solved on minibatches instead. Still, the repeated cost remains substantial and the locality can distort the global transport geometry. We propose quantized diffusion Schrödinger bridges (QDSB), which compute the endpoint coupling on anchor-quantized endpoint distributions and lift the resulting plan back to original data points through cell-wise sampling. We show that the regularized optimal coupling is stable w.r.t. anchor quantization, with an error controlled by the quality of the anchor approximation. In real-world experiments, QDSB matches the sample quality of existing baselines, requiring substantially less time. Code and data are available at github.com/mathefuchs/qdsb.

Florian Kalinke, Zoltán Szabó, Bharath K. Sriperumbudur

Kernel Stein discrepancy (KSD) is among the most popular goodness-of-fit (GoF) measures on general domains with a large number of successful deployments. One of the main applications of KSD is in constructing powerful GoF tests. However, tests relying on the classical U-/V-statistic-based KSD estimators have two major drawbacks. (i) Their runtime scales quadratically in the number of samples. (ii) Their asymptotic null distribution is computationally intractable in most cases, typically handled by bootstrapping. While it is known that the Nyström method permits accelerating KSD estimation with no loss of statistical accuracy under mild conditions, to the best of our knowledge, the fundamental question of its impact on bootstrap-based GoF testing is open; resolving this question is the focus of the current paper. In particular, we prove that the key properties of the quadratic-time bootstrapped KSD-based GoF test (asymptotic level and local consistency) are preserved by its Nyström acceleration. We numerically demonstrate the efficiency of the accelerated KSD estimator and bootstrap in the context of GoF testing of spherical and functional data. Our numerical results show that the Nyström-accelerated method performs statistically on-par with the quadratic-time approach, while requiring substantially smaller runtime.

Florian Kalinke, Zoltán Szabó

Kernel techniques are among the most popular and powerful approaches of data science. Among the key features that make kernels ubiquitous are (i) the number of domains they have been designed for, (ii) the Hilbert structure of the function class associated to kernels facilitating their statistical analysis, and (iii) their ability to represent probability distributions without loss of information. These properties give rise to the immense success of Hilbert-Schmidt independence criterion (HSIC) which is able to capture joint independence of random variables under mild conditions, and permits closed-form estimators with quadratic computational complexity (w.r.t. the sample size). In order to alleviate the quadratic computational bottleneck in large-scale applications, multiple HSIC approximations have been proposed, however these estimators are restricted to $M=2$ random variables, do not extend naturally to the $M\ge 2$ case, and lack theoretical guarantees. In this work, we propose an alternative Nyström-based HSIC estimator which handles the $M\ge 2$ case, prove its consistency, and demonstrate its applicability in multiple contexts, including synthetic examples, dependency testing of media annotations, and causal discovery.

Marco Heyden, Edouard Fouché, Vadim Arzamasov et al.

Change detection is of fundamental importance when analyzing data streams. Detecting changes both quickly and accurately enables monitoring and prediction systems to react, e.g., by issuing an alarm or by updating a learning algorithm. However, detecting changes is challenging when observations are high-dimensional. In high-dimensional data, change detectors should not only be able to identify when changes happen, but also in which subspace they occur. Ideally, one should also quantify how severe they are. Our approach, ABCD, has these properties. ABCD learns an encoder-decoder model and monitors its accuracy over a window of adaptive size. ABCD derives a change score based on Bernstein's inequality to detect deviations in terms of accuracy, which indicate changes. Our experiments demonstrate that ABCD outperforms its best competitor by up to 20% in F1-score on average. It can also accurately estimate changes' subspace, together with a severity measure that correlates with the ground truth.

Florian Kalinke, Marco Heyden, Georg Gntuni et al.

Detecting changes is of fundamental importance when analyzing data streams and has many applications, e.g., in predictive maintenance, fraud detection, or medicine. A principled approach to detect changes is to compare the distributions of observations within the stream to each other via hypothesis testing. Maximum mean discrepancy (MMD), a (semi-)metric on the space of probability distributions, provides powerful non-parametric two-sample tests on kernel-enriched domains. In particular, MMD is able to detect any disparity between distributions under mild conditions. However, classical MMD estimators suffer from a quadratic runtime complexity, which renders their direct use for change detection in data streams impractical. In this article, we propose a new change detection algorithm, called Maximum Mean Discrepancy on Exponential Windows (MMDEW), that combines the benefits of MMD with an efficient computation based on exponential windows. We prove that MMDEW enjoys polylogarithmic runtime and logarithmic memory complexity and show empirically that it outperforms the state of the art on benchmark data streams.

Pouya Roudaki, Shakeel Gavioli-Akilagun, Florian Kalinke et al.

We introduce kernel integrated $R^2$, a new measure of statistical dependence that combines the local normalization principle of the recently introduced integrated $R^2$ with the flexibility of reproducing kernel Hilbert spaces (RKHSs). The proposed measure extends integrated $R^2$ from scalar responses to responses taking values on general spaces equipped with a characteristic kernel, allowing to measure dependence of multivariate, functional, and structured data, while remaining sensitive to tail behaviour and oscillatory dependence structures. We establish that (i) this new measure takes values in $[0,1]$, (ii) equals zero if and only if independence holds, and (iii) equals one if and only if the response is almost surely a measurable function of the covariates. Two estimators are proposed: a graph-based method using $K$-nearest neighbours and an RKHS-based method built on conditional mean embeddings. We prove consistency and derive convergence rates for the graph-based estimator, showing its adaptation to intrinsic dimensionality. Numerical experiments on simulated data and a real data experiment in the context of dependency testing for media annotations demonstrate competitive power against state-of-the-art dependence measures, particularly in settings involving non-linear and structured relationships.

Tobias Fuchs, Florian Kalinke, Klemens Böhm

In real-world applications, one often encounters ambiguously labeled data, where different annotators assign conflicting class labels. Partial-label learning allows training classifiers in this weakly supervised setting, where state-of-the-art methods already show good predictive performance. However, even the best algorithms give incorrect predictions, which can have severe consequences when they impact actions or decisions. We propose a novel risk-consistent nearest-neighbor-based partial-label learning algorithm with a reject option, that is, the algorithm can reject unsure predictions. Extensive experiments on artificial and real-world datasets show that our method provides the best trade-off between the number and accuracy of non-rejected predictions when compared to our competitors, which use confidence thresholds for rejecting unsure predictions. When evaluated without the reject option, our nearest-neighbor-based approach also achieves competitive prediction performance.

Tobias Fuchs, Florian Kalinke

Real-world data is often ambiguous; for example, human annotation produces instances with multiple conflicting class labels. Partial-label learning (PLL) aims at training a classifier in this challenging setting, where each instance is associated with a set of candidate labels and one correct, but unknown, class label. A multitude of algorithms targeting this setting exists and, to enhance their prediction quality, several extensions that are applicable across a wide range of PLL methods have been introduced. While many of these extensions rely on heuristics, this article proposes a novel enhancing method that incrementally prunes candidate sets using conformal prediction. To work around the missing labeled validation set, which is typically required for conformal prediction, we propose a strategy that alternates between training a PLL classifier to label the validation set, leveraging these predicted class labels for calibration, and pruning candidate labels that are not part of the resulting conformal sets. In this sense, our method alternates between empirical risk minimization and candidate set pruning. We establish that our pruning method preserves the conformal validity with respect to the unknown ground truth. Our extensive experiments on artificial and real-world data show that the proposed approach significantly improves the test set accuracies of several state-of-the-art PLL classifiers.

Tobias Fuchs, Florian Kalinke

Real-world training data is often noisy; for example, human annotators assign conflicting class labels to the same instances. Partial-label learning (PLL) is a weakly supervised learning paradigm that allows training classifiers in this context without manual data cleaning. While state-of-the-art methods have good predictive performance, their predictions are sensitive to high noise levels, out-of-distribution data, and adversarial perturbations. We propose a novel PLL method based on subjective logic, which explicitly represents uncertainty by leveraging the magnitudes of the underlying neural network's class activation values. Thereby, we effectively incorporate prior knowledge about the class labels by using a novel label weight re-distribution strategy that we prove to be optimal. We empirically show that our method yields more robust predictions in terms of predictive performance under high PLL noise levels, handling out-of-distribution examples, and handling adversarial perturbations on the test instances.

Florian Kalinke, Zoltan Szabo

Kernel techniques are among the most influential approaches in data science and statistics. Under mild conditions, the reproducing kernel Hilbert space associated to a kernel is capable of encoding the independence of $M\ge 2$ random variables. Probably the most widespread independence measure relying on kernels is the so-called Hilbert-Schmidt independence criterion (HSIC; also referred to as distance covariance in the statistics literature). Despite various existing HSIC estimators designed since its introduction close to two decades ago, the fundamental question of the rate at which HSIC can be estimated is still open. In this work, we prove that the minimax optimal rate of HSIC estimation on $\mathbb R^d$ for Borel measures containing the Gaussians with continuous bounded translation-invariant characteristic kernels is $\mathcal O\!\left(n^{-1/2}\right)$. Specifically, our result implies the optimality in the minimax sense of many of the most-frequently used estimators (including the U-statistic, the V-statistic, and the Nyström-based one) on $\mathbb R^d$.

Jose Cribeiro-Ramallo, Agnideep Aich, Florian Kalinke et al.

Kernel Stein discrepancies (KSDs) have emerged as a powerful tool for quantifying goodness-of-fit over the last decade, featuring numerous successful applications. To the best of our knowledge, all existing KSD estimators with known rate achieve $\sqrt n$-convergence. In this work, we present two complementary results (with different proof strategies), establishing that the minimax lower bound of KSD estimation is $n^{-1/2}$ and settling the optimality of these estimators. Our first result focuses on KSD estimation on $\mathbb R^d$ with the Langevin-Stein operator; our explicit constant for the Gaussian kernel indicates that the difficulty of KSD estimation may increase exponentially with the dimensionality $d$. Our second result settles the minimax lower bound for KSD estimation on general domains.

Florian Kalinke, Shakeel Gavioli-Akilagun

This article studies the problem of online non-parametric change point detection in multivariate data streams. We approach the problem through the lens of kernel-based two-sample testing and introduce a sequential testing procedure based on random Fourier features, running with logarithmic time complexity per observation and with overall logarithmic space complexity. The algorithm has two advantages compared to the state of the art. First, our approach is genuinely online, and no access to training data known to be from the pre-change distribution is necessary. Second, the algorithm does not require the user to specify a window parameter over which local tests are to be calculated. We prove strong theoretical guarantees on the algorithm's performance, including information-theoretic bounds demonstrating that the detection delay is optimal in the minimax sense. Numerical studies on real and synthetic data show that our algorithm is competitive with respect to the state of the art.

Florian Kalinke, Zoltan Szabo, Bharath K. Sriperumbudur

Kernel methods underpin many of the most successful approaches in data science and statistics, and they allow representing probability measures as elements of a reproducing kernel Hilbert space without loss of information. Recently, the kernel Stein discrepancy (KSD), which combines Stein's method with the flexibility of kernel techniques, gained considerable attention. Through the Stein operator, KSD allows the construction of powerful goodness-of-fit tests where it is sufficient to know the target distribution up to a multiplicative constant. However, the typical U- and V-statistic-based KSD estimators suffer from a quadratic runtime complexity, which hinders their application in large-scale settings. In this work, we propose a Nyström-based KSD acceleration -- with runtime $\mathcal O\left(mn+m^3\right)$ for $n$ samples and $m\ll n$ Nyström points -- , show its $\sqrt{n}$-consistency with a classical sub-Gaussian assumption, and demonstrate its applicability for goodness-of-fit testing on a suite of benchmarks. We also show the $\sqrt n$-consistency of the quadratic-time KSD estimator.

Edouard Fouché, Florian Kalinke, Klemens Böhm

In the real world, data streams are ubiquitous -- think of network traffic or sensor data. Mining patterns, e.g., outliers or clusters, from such data must take place in real time. This is challenging because (1) streams often have high dimensionality, and (2) the data characteristics may change over time. Existing approaches tend to focus on only one aspect, either high dimensionality or the specifics of the streaming setting. For static data, a common approach to deal with high dimensionality -- known as subspace search -- extracts low-dimensional, `interesting' projections (subspaces), in which patterns are easier to find. In this paper, we address both Challenge (1) and (2) by generalising subspace search to data streams. Our approach, Streaming Greedy Maximum Random Deviation (SGMRD), monitors interesting subspaces in high-dimensional data streams. It leverages novel multivariate dependency estimators and monitoring techniques based on bandit theory. We show that the benefits of SGMRD are twofold: (i) It monitors subspaces efficiently, and (ii) this improves the results of downstream data mining tasks, such as outlier detection. Our experiments, performed against synthetic and real-world data, demonstrate that SGMRD outperforms its competitors by a large margin.