Information diagram and the I-measure are useful mnemonics where random variables are treated as sets, and entropy and mutual information are treated as a signed measure. Although the I-measure has been successful in machine proofs of entropy inequalities, the theoretical underpinning of the ``random variables as sets'' analogy has been unclear until the recent works on mappings from random variables to sets by Ellerman (recovering order-$2$ Tsallis entropy over general probability space), and Down and Mediano (recovering Shannon entropy over discrete probability space). We generalize these constructions by designing a mapping which recovers the Shannon entropy (and the information density) over general probability space. Moreover, it has an intuitive interpretation based on the arrival time in a Poisson process, allowing us to understand the union, intersection and difference between (sets corresponding to) random variables and events. Cross entropy, KL divergence, and conditional entropy given an event, can be obtained as set intersections. We propose a generalization of the information diagram that also includes events, and demonstrate its usage by a diagrammatic proof of Fano's inequality.
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We study stochastic Cubic Newton methods for solving general possibly non-convex minimization problems. We propose a new framework, which we call the helper framework, that provides a unified view of the stochastic and variance-reduced second-order algorithms equipped with global complexity guarantees. It can also be applied to learning with auxiliary information. Our helper framework offers the algorithm designer high flexibility for constructing and analyzing the stochastic Cubic Newton methods, allowing arbitrary size batches, and the use of noisy and possibly biased estimates of the gradients and Hessians, incorporating both the variance reduction and the lazy Hessian updates. We recover the best-known complexities for the stochastic and variance-reduced Cubic Newton, under weak assumptions on the noise. A direct consequence of our theory is the new lazy stochastic second-order method, which significantly improves the arithmetic complexity for large dimension problems. We also establish complexity bounds for the classes of gradient-dominated objectives, that include convex and strongly convex problems. For Auxiliary Learning, we show that using a helper (auxiliary function) can outperform training alone if a given similarity measure is small.
The aim of latent variable disentanglement is to infer the multiple informative latent representations that lie behind a data generation process and is a key factor in controllable data generation. In this paper, we propose a deep neural network-based self-supervised learning method to infer the disentangled rhythmic and harmonic representations behind music audio generation. We train a variational autoencoder that generates an audio mel-spectrogram from two latent features representing the rhythmic and harmonic content. In the training phase, the variational autoencoder is trained to reconstruct the input mel-spectrogram given its pitch-shifted version. At each forward computation in the training phase, a vector rotation operation is applied to one of the latent features, assuming that the dimensions of the feature vectors are related to pitch intervals. Therefore, in the trained variational autoencoder, the rotated latent feature represents the pitch-related information of the mel-spectrogram, and the unrotated latent feature represents the pitch-invariant information, i.e., the rhythmic content. The proposed method was evaluated using a predictor-based disentanglement metric on the learned features. Furthermore, we demonstrate its application to the automatic generation of music remixes.
We investigate the equational theory of Kleene algebra terms with variable complements -- (language) complement where it applies only to variables -- w.r.t. languages. While the equational theory w.r.t. languages coincides with the language equivalence (under the standard language valuation) for Kleene algebra terms, this coincidence is broken if we extend the terms with complements. In this paper, we prove the decidability of some fragments of the equational theory: the universality problem is coNP-complete, and the inequational theory t <= s is coNP-complete when t does not contain Kleene-star. To this end, we introduce words-to-letters valuations; they are sufficient valuations for the equational theory and ease us in investigating the equational theory w.r.t. languages. Additionally, we prove that for words with variable complements, the equational theory coincides with the word equivalence.
We investigate the complexity of several manipulation and control problems under numerous prevalent approval-based multiwinner voting rules. Particularly, the rules we study include approval voting (AV), satisfaction approval voting (SAV), net-satisfaction approval voting (NSAV), proportional approval voting (PAV), approval-based Chamberlin-Courant voting (ABCCV), minimax approval voting (MAV), etc. We show that these rules generally resist the strategic types scrutinized in the paper, with only a few exceptions. In addition, we also obtain many fixed-parameter tractability results for these problems with respect to several natural parameters, and derive polynomial-time algorithms for certain special cases.
The distributed data analytic system -- Spark is a common choice for processing massive volumes of heterogeneous data, while it is challenging to tune its parameters to achieve high performance. Recent studies try to employ auto-tuning techniques to solve this problem but suffer from three issues: limited functionality, high overhead, and inefficient search. In this paper, we present a general and efficient Spark tuning framework that can deal with the three issues simultaneously. First, we introduce a generalized tuning formulation, which can support multiple tuning goals and constraints conveniently, and a Bayesian optimization (BO) based solution to solve this generalized optimization problem. Second, to avoid high overhead from additional offline evaluations in existing methods, we propose to tune parameters along with the actual periodic executions of each job (i.e., online evaluations). To ensure safety during online job executions, we design a safe configuration acquisition method that models the safe region. Finally, three innovative techniques are leveraged to further accelerate the search process: adaptive sub-space generation, approximate gradient descent, and meta-learning method. We have implemented this framework as an independent cloud service, and applied it to the data platform in Tencent. The empirical results on both public benchmarks and large-scale production tasks demonstrate its superiority in terms of practicality, generality, and efficiency. Notably, this service saves an average of 57.00% memory cost and 34.93% CPU cost on 25K in-production tasks within 20 iterations, respectively.
Interactive Design and Optics-Based Visualization of Arbitrary Non-Euclidean Kaleidoscopic Orbifolds
Orbifolds are a modern mathematical concept that arises in the research of hyperbolic geometry with applications in computer graphics and visualization. In this paper, we make use of rooms with mirrors as the visual metaphor for orbifolds. Given any arbitrary two-dimensional kaleidoscopic orbifold, we provide an algorithm to construct a Euclidean, spherical, or hyperbolic polygon to match the orbifold. This polygon is then used to create a room for which the polygon serves as the floor and the ceiling. With our system that implements M\"obius transformations, the user can interactively edit the scene and see the reflections of the edited objects. To correctly visualize non-Euclidean orbifolds, we adapt the rendering algorithms to account for the geodesics in these spaces, which light rays follow. Our interactive orbifold design system allows the user to create arbitrary two-dimensional kaleidoscopic orbifolds. In addition, our mirror-based orbifold visualization approach has the potential of helping our users gain insight on the orbifold, including its orbifold notation as well as its universal cover, which can also be the spherical space and the hyperbolic space.
Mesh degeneration is a bottleneck for fluid-structure interaction (FSI) simulations and for shape optimization via the method of mappings. In both cases, an appropriate mesh motion technique is required. The choice is typically based on heuristics, e.g., the solution operators of partial differential equations (PDE), such as the Laplace or biharmonic equation. Especially the latter, which shows good numerical performance for large displacements, is expensive. Moreover, from a continuous perspective, choosing the mesh motion technique is to a certain extent arbitrary and has no influence on the physically relevant quantities. Therefore, we consider approaches inspired by machine learning. We present a hybrid PDE-NN approach, where the neural network (NN) serves as parameterization of a coefficient in a second order nonlinear PDE. We ensure existence of solutions for the nonlinear PDE by the choice of the neural network architecture. Moreover, we present an approach where a neural network corrects the harmonic extension such that the boundary displacement is not changed. In order to avoid technical difficulties in coupling finite element and machine learning software, we work with a splitting of the monolithic FSI system into three smaller subsystems. This allows to solve the mesh motion equation in a separate step. We assess the quality of the learned mesh motion technique by applying it to a FSI benchmark problem.
Wireless communication systems to date primarily rely on the orthogonality of resources to facilitate the design and implementation, from user access to data transmission. Emerging applications and scenarios in the sixth generation (6G) wireless systems will require massive connectivity and transmission of a deluge of data, which calls for more flexibility in the design concept that goes beyond orthogonality. Furthermore, recent advances in signal processing and learning have attracted considerable attention, as they provide promising approaches to various complex and previously intractable problems of signal processing in many fields. This article provides an overview of research efforts to date in the field of signal processing and learning for next-generation multiple access, with an emphasis on massive random access and non-orthogonal multiple access. The promising interplay with new technologies and the challenges in learning-based NGMA are discussed.
We propose a novel methodology to define assistance systems that rely on information fusion to combine different sources of information while providing an assessment. The main contribution of this paper is providing a general framework for the fusion of n number of information sources using the evidence theory. The fusion provides a more robust prediction and an associated uncertainty that can be used to assess the prediction likeliness. Moreover, we provide a methodology for the information fusion of two primary sources: an ensemble classifier based on machine data and an expert-centered model. We demonstrate the information fusion approach using data from an industrial setup, which rounds up the application part of this research. Furthermore, we address the problem of data drift by proposing a methodology to update the data-based models using an evidence theory approach. We validate the approach using the Benchmark Tennessee Eastman while doing an ablation study of the model update parameters.
Object detection typically assumes that training and test data are drawn from an identical distribution, which, however, does not always hold in practice. Such a distribution mismatch will lead to a significant performance drop. In this work, we aim to improve the cross-domain robustness of object detection. We tackle the domain shift on two levels: 1) the image-level shift, such as image style, illumination, etc, and 2) the instance-level shift, such as object appearance, size, etc. We build our approach based on the recent state-of-the-art Faster R-CNN model, and design two domain adaptation components, on image level and instance level, to reduce the domain discrepancy. The two domain adaptation components are based on H-divergence theory, and are implemented by learning a domain classifier in adversarial training manner. The domain classifiers on different levels are further reinforced with a consistency regularization to learn a domain-invariant region proposal network (RPN) in the Faster R-CNN model. We evaluate our newly proposed approach using multiple datasets including Cityscapes, KITTI, SIM10K, etc. The results demonstrate the effectiveness of our proposed approach for robust object detection in various domain shift scenarios.
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