Subspace clustering (SC) aims to cluster data lying in a union of low-dimensional subspaces. Usually, SC learns an affinity matrix and then performs spectral clustering. Both steps suffer from high time and space complexity, which leads to difficulty in clustering large datasets. This paper presents a method called k-Factorization Subspace Clustering (k-FSC) for large-scale subspace clustering. K-FSC directly factorizes the data into k groups via pursuing structured sparsity in the matrix factorization model. Thus, k-FSC avoids learning affinity matrix and performing eigenvalue decomposition, and hence has low time and space complexity on large datasets. An efficient algorithm is proposed to solve the optimization of k-FSC. In addition, k-FSC is able to handle noise, outliers, and missing data and applicable to arbitrarily large datasets and streaming data. Extensive experiments show that k-FSC outperforms state-of-the-art subspace clustering methods.

Predictive process analytics focuses on predicting the future states of running instances of a business process. While advanced machine learning techniques have been used to increase accuracy of predictions, the resulting predictive models lack transparency. Current explainable machine learning methods, such as LIME and SHAP, can be used to interpret black box models. However, it is unclear how fit for purpose these methods are in explaining process predictive models. In this paper, we draw on evaluation measures used in the field of explainable AI and propose functionally-grounded evaluation metrics for assessing explainable methods in predictive process analytics. We apply the proposed metrics to evaluate the performance of LIME and SHAP in interpreting process predictive models built on XGBoost, which has been shown to be relatively accurate in process predictions. We conduct the evaluation using three open source, real-world event logs and analyse the evaluation results to derive insights. The research contributes to understanding the trustworthiness of explainable methods for predictive process analytics as a fundamental and key step towards human user-oriented evaluation.

This paper considers parameter estimation for nonlinear state-space models, which is an important but challenging problem. We address this challenge by employing a variational inference (VI) approach, which is a principled method that has deep connections to maximum likelihood estimation. This VI approach ultimately provides estimates of the model as solutions to an optimisation problem, which is deterministic, tractable and can be solved using standard optimisation tools. A specialisation of this approach for systems with additive Gaussian noise is also detailed. The proposed method is examined numerically on a range of simulation and real examples with a focus on robustness to parameter initialisations; we additionally perform favourable comparisons against state-of-the-art alternatives.

Algorithms are now routinely used to make consequential decisions that affect human lives. Examples include college admissions, medical interventions or law enforcement. While algorithms empower us to harness all information hidden in vast amounts of data, they may inadvertently amplify existing biases in the available datasets. This concern has sparked increasing interest in fair machine learning, which aims to quantify and mitigate algorithmic discrimination. Indeed, machine learning models should undergo intensive tests to detect algorithmic biases before being deployed at scale. In this paper, we use ideas from the theory of optimal transport to propose a statistical hypothesis test for detecting unfair classifiers. Leveraging the geometry of the feature space, the test statistic quantifies the distance of the empirical distribution supported on the test samples to the manifold of distributions that render a pre-trained classifier fair. We develop a rigorous hypothesis testing mechanism for assessing the probabilistic fairness of any pre-trained logistic classifier, and we show both theoretically as well as empirically that the proposed test is asymptotically correct. In addition, the proposed framework offers interpretability by identifying the most favorable perturbation of the data so that the given classifier becomes fair.

Clustering of time series data exhibits a number of challenges not present in other settings, notably the problem of registration (alignment) of observed signals. Typical approaches include pre-registration to a user-specified template or time warping approaches which attempt to optimally align series with a minimum of distortion. For many signals obtained from recording or sensing devices, these methods may be unsuitable as a template signal is not available for pre-registration, while the distortion of warping approaches may obscure meaningful temporal information. We propose a new method for automatic time series alignment within a convex clustering problem. Our approach, Temporal Registration using Optimal Unitary Transformations (TROUT), is based on a novel distance metric between time series that is easy to compute and automatically identifies optimal alignment between pairs of time series. By embedding our new metric in a convex formulation, we retain well-known advantages of computational and statistical performance. We provide an efficient algorithm for TROUT-based clustering and demonstrate its superior performance over a range of competitors.

Predicting the dynamics of neural network parameters during training is one of the key challenges in building a theoretical foundation for deep learning. A central obstacle is that the motion of a network in high-dimensional parameter space undergoes discrete finite steps along complex stochastic gradients derived from real-world datasets. We circumvent this obstacle through a unifying theoretical framework based on intrinsic symmetries embedded in a network's architecture that are present for any dataset. We show that any such symmetry imposes stringent geometric constraints on gradients and Hessians, leading to an associated conservation law in the continuous-time limit of stochastic gradient descent (SGD), akin to Noether's theorem in physics. We further show that finite learning rates used in practice can actually break these symmetry induced conservation laws. We apply tools from finite difference methods to derive modified gradient flow, a differential equation that better approximates the numerical trajectory taken by SGD at finite learning rates. We combine modified gradient flow with our framework of symmetries to derive exact integral expressions for the dynamics of certain parameter combinations. We empirically validate our analytic predictions for learning dynamics on VGG-16 trained on Tiny ImageNet. Overall, by exploiting symmetry, our work demonstrates that we can analytically describe the learning dynamics of various parameter combinations at finite learning rates and batch sizes for state of the art architectures trained on any dataset. Just like the fundamental laws of classical and quantum mechanics taught us how to control and optimize the physical world for engineering purposes, a better understanding of the laws governing neural network learning dynamics can have a profound impact on the optimization of artificial neural networks.

Accurate 3D object detection (3DOD) is crucial for safe navigation of complex environments by autonomous robots. Regressing accurate 3D bounding boxes in cluttered environments based on sparse LiDAR data is however a highly challenging problem. We address this task by exploring recent advances in conditional energy-based models (EBMs) for probabilistic regression. While methods employing EBMs for regression have demonstrated impressive performance on 2D object detection in images, these techniques are not directly applicable to 3D bounding boxes. In this work, we therefore design a differentiable pooling operator for 3D bounding boxes, serving as the core module of our EBM network. We further integrate this general approach into the state-of-the-art 3D object detector SA-SSD. On the KITTI dataset, our proposed approach consistently outperforms the SA-SSD baseline across all 3DOD metrics, demonstrating the potential of EBM-based regression for highly accurate 3DOD. Code is available at https://github.com/fregu856/ebms_3dod.

Analyzing data from the Large Hadron Collider (LHC) present a hyper challenge. A given collision event may result in hundreds of outgoing particles, each with many features (momentum, electric charge, etc.). This hyper variate phase space is then observed by complex multi-channel detectors that are essentially hyperspectral cameras. The LHC detectors have millions of readout channels and dimensionality reduction is essential for data analysis. One natural and nearly lossless reduction is the reconstruction of charged particle trajectories ('tracks'). The innermost layers of the detectors at the LHC are constructed to register the passage of charged particles without significantly altering the particle energy or direction. In the ATLAS and CMS detectors, this is achieved using silicon sensors that are finely segmented in one or two directions and are called strips and pixels, respectively. We will focus on pixels, although our methodology applies more generally. Typically, the first step in a tracking algorithm is the construction of seeds, which are sets of three or more hit pixel clusters that can be used to fit charged-particle trajectories (see e.g.

Neural Tangent Kernel (NTK) theory is widely used to study the dynamics of infinitely-wide deep neural networks (DNNs) under gradient descent. But do the results for infinitely-wide networks give us hints about the behaviour of real finite-width ones? In this paper we study empirically when NTK theory is valid in practice for fully-connected ReLu and sigmoid networks. We find out that whether a network is in the NTK regime depends on the hyperparameters of random initialization and network's depth. In particular, NTK theory does not explain behaviour of sufficiently deep networks initialized so that their gradients explode: the kernel is random at initialization and changes significantly during training, contrary to NTK theory. On the other hand, in case of vanishing gradients DNNs are in the NTK regime but become untrainable rapidly with depth. We also describe a framework to study generalization properties of DNNs by means of NTK theory and discuss its limits.

Dimension reduction (DR) techniques such as t-SNE, UMAP, and TriMAP have demonstrated impressive visualization performance on many real world datasets. One tension that has always faced these methods is the trade-off between preservation of global structure and preservation of local structure: these methods can either handle one or the other, but not both. In this work, our main goal is to understand what aspects of DR methods are important for preserving both local and global structure: it is difficult to design a better method without a true understanding of the choices we make in our algorithms and their empirical impact on the lower-dimensional embeddings they produce. Towards the goal of local structure preservation, we provide several useful design principles for DR loss functions based on our new understanding of the mechanisms behind successful DR methods. Towards the goal of global structure preservation, our analysis illuminates that the choice of which components to preserve is important. We leverage these insights to design a new algorithm for DR, called Pairwise Controlled Manifold Approximation Projection (PaCMAP), which preserves both local and global structure. Our work provides several unexpected insights into what design choices both to make and avoid when constructing DR algorithms.