We study the Kronecker product regression problem, in which the design matrix is a Kronecker product of two or more matrices. Given $A_i \in \mathbb{R}^{n_i \times d_i}$ for $i=1,2,\dots,q$ where $n_i \gg d_i$ for each $i$, and $b \in \mathbb{R}^{n_1 n_2 \cdots n_q}$, let $\mathcal{A} = A_1 \otimes A_2 \otimes \cdots \otimes A_q$. Then for $p \in [1,2]$, the goal is to find $x \in \mathbb{R}^{d_1 \cdots d_q}$ that approximately minimizes $\|\mathcal{A}x - b\|_p$. Recently, Diao, Song, Sun, and Woodruff (AISTATS, 2018) gave an algorithm which is faster than forming the Kronecker product $\mathcal{A}$ Specifically, for $p=2$ their running time is $O(\sum_{i=1}^q \text{nnz}(A_i) + \text{nnz}(b))$, where nnz$(A_i)$ is the number of non-zero entries in $A_i$. Note that nnz$(b)$ can be as large as $n_1 \cdots n_q$. For $p=1,$ $q=2$ and $n_1 = n_2$, they achieve a worse bound of $O(n_1^{3/2} \text{poly}(d_1d_2) + \text{nnz}(b))$. In this work, we provide significantly faster algorithms. For $p=2$, our running time is $O(\sum_{i=1}^q \text{nnz}(A_i) )$, which has no dependence on nnz$(b)$. For $p<2$, our running time is $O(\sum_{i=1}^q \text{nnz}(A_i) + \text{nnz}(b))$, which matches the prior best running time for $p=2$. We also consider the related all-pairs regression problem, where given $A \in \mathbb{R}^{n \times d}, b \in \mathbb{R}^n$, we want to solve $\min_{x} \|\bar{A}x - \bar{b}\|_p$, where $\bar{A} \in \mathbb{R}^{n^2 \times d}, \bar{b} \in \mathbb{R}^{n^2}$ consist of all pairwise differences of the rows of $A,b$. We give an $O(\text{nnz}(A))$ time algorithm for $p \in[1,2]$, improving the $\Omega(n^2)$ time needed to form $\bar{A}$. Finally, we initiate the study of Kronecker product low rank and low $t$-rank approximation. For input $\mathcal{A}$ as above, we give $O(\sum_{i=1}^q \text{nnz}(A_i))$ time algorithms, which is much faster than computing $\mathcal{A}$.

Accurately forecasting the future movements of surrounding vehicles is essential for safe and efficient operations of autonomous driving cars. This task is difficult because a vehicle's moving trajectory is greatly determined by its driver's intention, which is often hard to estimate. By leveraging attention mechanisms along with long short-term memory (LSTM) networks, this work learns the relation between a driver's intention and the vehicle's changing positions relative to road infrastructures, and uses it to guide the prediction. Different from other state-of-the-art solutions, our work treats the on-road lanes as non-Euclidean structures, unfolds the vehicle's moving history to form a spatio-temporal graph, and uses methods from Graph Neural Networks to solve the problem. Not only is our approach a pioneering attempt in using non-Euclidean methods to process static environmental features around a predicted object, our model also outperforms other state-of-the-art models in several metrics. The practicability and interpretability analysis of the model shows great potential for large-scale deployment in various autonomous driving systems in addition to our own.

Working with any gradient-based machine learning algorithm involves the tedious task of tuning the optimizer's hyperparameters, such as the learning rate. There exist many techniques for automated hyperparameter optimization, but they typically introduce even more hyperparameters to control the hyperparameter optimization process. We propose to instead learn the hyperparameters themselves by gradient descent, and furthermore to learn the hyper-hyperparameters by gradient descent as well, and so on ad infinitum. As these towers of gradient-based optimizers grow, they become significantly less sensitive to the choice of top-level hyperparameters, hence decreasing the burden on the user to search for optimal values.

Deep neural networks (DNNs) can outperform human brains in specific tasks, but due to their lack of transparency in decision-making processes, it remains uncertain whether we can completely rely on DNNs when it comes to high-risk problems requiring rigorous decisions. As DNNs' operations rely on a massive number of linear/nonlinear computations in both parallel and sequential, it is impossible to distinguish every factor influencing their decision. Further, DNNs cannot be empirically tested in every potential condition. Therefore, we should consider the possibility that DNNs' decisions can be corrupted by unexpected factors; for instance, adversarial attacks can deceive DNNs quite effectively. Before DNNs are deployed to solve high-stakes problems, such vulnerability must be overcome. In our study, we propose an algorithm to provide better insights into DNNs' decision-making processes. Our experiments suggest that this algorithm can effectively trace DNNs' decision processes on a layer-by-layer basis and be used to detect adversarial attacks.

Neural networks have been proposed recently for positioning and channel charting of user equipments (UEs) in wireless systems. Both of these approaches process channel state information (CSI) that is acquired at a multi-antenna base-station in order to learn a function that maps CSI to location information. CSI-based positioning using deep neural networks requires a dataset that contains both CSI and associated location information. Channel charting (CC) only requires CSI information to extract relative position information. Since CC builds on dimensionality reduction, it can be implemented using autoencoders. In this paper, we propose a unified architecture based on Siamese networks that can be used for supervised UE positioning and unsupervised channel charting. In addition, our framework enables semisupervised positioning, where only a small set of location information is available during training. We use simulations to demonstrate that Siamese networks achieve similar or better performance than existing positioning and CC approaches with a single, unified neural network architecture.

In some misspecified settings, the posterior distribution in Bayesian statistics may lead to inconsistent estimates. To fix this issue, it has been suggested to replace the likelihood by a pseudo-likelihood, that is the exponential of a loss function enjoying suitable robustness properties. In this paper, we build a pseudo-likelihood based on the Maximum Mean Discrepancy, defined via an embedding of probability distributions into a reproducing kernel Hilbert space. We show that this MMD-Bayes posterior is consistent and robust to model misspecification. As the posterior obtained in this way might be intractable, we also prove that reasonable variational approximations of this posterior enjoy the same properties. We provide details on a stochastic gradient algorithm to compute these variational approximations. Numerical simulations indeed suggest that our estimator is more robust to misspecification than the ones based on the likelihood. Keywords: Maximum Mean Discrepancy, Robust estimation, Variational inference.

Generative processes in biology and other fields often produce data that can be regarded as resulting from a composition of basic features. Here we present an unsupervised method based on autoencoders for inferring these basic features of data. The main novelty in our approach is that the training is based on the optimization of the `local entropy' rather than the standard loss, resulting in a more robust inference, and enhancing the performance on this type of data considerably. Algorithmically, this is realized by training an interacting system of replicated autoencoders. We apply this method to synthetic and protein sequence data, and show that it is able to infer a hidden representation that correlates well with the underlying generative process, without requiring any prior knowledge.

We provide a rigorous mathematical treatment to the crowding issue in data visualization when high dimensional data sets are projected down to low dimensions for visualization. By properly adjusting the capacity of high dimensional balls, our method makes right enough room to prepare for the embedding. A key component of the proposed method is an estimation of the correlation dimension at various scales which reflects the data density variation. The proposed adjustment to the capacity applies to any distance (Euclidean, geodesic, diffusion) and can potentially be used in many existing methods to mitigate the crowding during the dimension reduction. We demonstrate the effectiveness of the new method using synthetic and real datasets.

We present the first application of adaptive machine learning to the identification of anomalies in a data set of non-periodic astronomical light curves. The method follows an active learning strategy where highly informative objects are selected to be labelled. This new information is subsequently used to improve the machine learning model, allowing its accuracy to evolve with the addition of every new classification. For the case of anomaly detection, the algorithm aims to maximize the number of real anomalies presented to the expert by slightly modifying the decision boundary of a traditional isolation forest in each iteration. As a proof of concept, we apply the Active Anomaly Discovery (AAD) algorithm to light curves from the Open Supernova Catalog and compare its results to those of a static Isolation Forest (IF). For both methods, we visually inspected objects within 2% highest anomaly scores. We show that AAD was able to identify 80% more true anomalies than IF. This result is the first evidence that AAD algorithms can play a central role in the search for new physics in the era of large scale sky surveys.

Predicting the next position of movable objects has been a problem for at least the last three decades, referred to as trajectory prediction. In our days, the vast amounts of data being continuously produced add the big data dimension to the trajectory prediction problem, which we are trying to tackle by creating a {\lambda}-Architecture based analytics platform. This platform performs both batch and stream analytics tasks and then combines them to perform analytical tasks that cannot be performed by analyzing any of these layers by itself. The biggest benefit of this platform is its context agnostic trait, which allows us to use it for any use case, as long as a time-stamped geolocation stream is provided. The experimental results presented prove that each part of the {\lambda}-Architecture performs well at certain targets, making a combination of these parts a necessity in order to improve the overall accuracy and performance of the platform.