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Boolean Matrix Factorization (BMF) aims to find an approximation of a given binary matrix as the Boolean product of two low-rank binary matrices. Binary data is ubiquitous in many fields, and representing data by binary matrices is common in medicine, natural language processing, bioinformatics, computer graphics, among many others. Unfortunately, BMF is computationally hard and heuristic algorithms are used to compute Boolean factorizations. Very recently, the theoretical breakthrough was obtained independently by two research groups. Ban et al. (SODA 2019) and Fomin et al. (Trans. Algorithms 2020) show that BMF admits an efficient polynomial-time approximation scheme (EPTAS). However, despite the theoretical importance, the high double-exponential dependence of the running times from the rank makes these algorithms unimplementable in practice. The primary research question motivating our work is whether the theoretical advances on BMF could lead to practical algorithms. The main conceptional contribution of our work is the following. While EPTAS for BMF is a purely theoretical advance, the general approach behind these algorithms could serve as the basis in designing better heuristics. We also use this strategy to develop new algorithms for related $\mathbb{F}_p$-Matrix Factorization. Here, given a matrix $A$ over a finite field GF($p$) where $p$ is a prime, and an integer $r$, our objective is to find a matrix $B$ over the same field with GF($p$)-rank at most $r$ minimizing some norm of $A-B$. Our empirical research on synthetic and real-world data demonstrates the advantage of the new algorithms over previous works on BMF and $\mathbb{F}_p$-Matrix Factorization.

It came to the attention of myself and the coauthors of (S., Rozowski, Silva, Rot, 2022) that a number of process calculi can be obtained by algebraically presenting the branching structure of the transition systems they specify. Labelled transition systems, for example, branch into sets of transitions, terms in the free semilattice generated by the transitions. Interpreting equational theories in the category of sets has undesirable limitations, and we would like to have more examples of presentations in other categories. In this brief article, I discuss monad presentations in the category of partially ordered sets and monotone maps. I focus on quantitative monads, namely free modules over ordered semirings, and give sufficient conditions for one of these to lift a monad on the category of sets. I also give a description of ordered semirings that are useful for specifying unguarded recursive calls. Examples include ordered probability theory and ordered semilattices.

Prototype Generation (PG) methods are typically considered for improving the efficiency of the $k$-Nearest Neighbour ($k$NN) classifier when tackling high-size corpora. Such approaches aim at generating a reduced version of the corpus without decreasing the classification performance when compared to the initial set. Despite their large application in multiclass scenarios, very few works have addressed the proposal of PG methods for the multilabel space. In this regard, this work presents the novel adaptation of four multiclass PG strategies to the multilabel case. These proposals are evaluated with three multilabel $k$NN-based classifiers, 12 corpora comprising a varied range of domains and corpus sizes, and different noise scenarios artificially induced in the data. The results obtained show that the proposed adaptations are capable of significantly improving -- both in terms of efficiency and classification performance -- the only reference multilabel PG work in the literature as well as the case in which no PG method is applied, also presenting a statistically superior robustness in noisy scenarios. Moreover, these novel PG strategies allow prioritising either the efficiency or efficacy criteria through its configuration depending on the target scenario, hence covering a wide area in the solution space not previously filled by other works.

We deal with Bayesian generative and discriminative classifiers. Given a model distribution $p(x, y)$, with the observation $y$ and the target $x$, one computes generative classifiers by firstly considering $p(x, y)$ and then using the Bayes rule to calculate $p(x | y)$. A discriminative model is directly given by $p(x | y)$, which is used to compute discriminative classifiers. However, recent works showed that the Bayesian Maximum Posterior classifier defined from the Naive Bayes (NB) or Hidden Markov Chain (HMC), both generative models, can also match the discriminative classifier definition. Thus, there are situations in which dividing classifiers into "generative" and "discriminative" is somewhat misleading. Indeed, such a distinction is rather related to the way of computing classifiers, not to the classifiers themselves. We present a general theoretical result specifying how a generative classifier induced from a generative model can also be computed in a discriminative way from the same model. Examples of NB and HMC are found again as particular cases, and we apply the general result to two original extensions of NB, and two extensions of HMC, one of which being original. Finally, we shortly illustrate the interest of the new discriminative way of computing classifiers in the Natural Language Processing (NLP) framework.

The ergodic decomposition theorem is a cornerstone result of dynamical systems and ergodic theory. It states that every invariant measure on a dynamical system is a mixture of ergodic ones. Here we formulate and prove the theorem in terms of string diagrams, using the formalism of Markov categories. We recover the usual measure-theoretic statement by instantiating our result in the category of stochastic kernels. Along the way we give a conceptual treatment of several concepts in the theory of deterministic and stochastic dynamical systems. In particular, - ergodic measures appear very naturally as particular cones of deterministic morphisms (in the sense of Markov categories); - the invariant $\sigma$-algebra of a dynamical system can be seen as a colimit in the category of Markov kernels. In line with other uses of category theory, once the necessary structures are in place, our proof of the main theorem is much simpler than traditional approaches. In particular, it does not use any quantitative limiting arguments, and it does not rely on the cardinality of the group or monoid indexing the dynamics. We hope that this result paves the way for further applications of category theory to dynamical systems, ergodic theory, and information theory.

Deep learning techniques have led to remarkable breakthroughs in the field of generic object detection and have spawned a lot of scene-understanding tasks in recent years. Scene graph has been the focus of research because of its powerful semantic representation and applications to scene understanding. Scene Graph Generation (SGG) refers to the task of automatically mapping an image into a semantic structural scene graph, which requires the correct labeling of detected objects and their relationships. Although this is a challenging task, the community has proposed a lot of SGG approaches and achieved good results. In this paper, we provide a comprehensive survey of recent achievements in this field brought about by deep learning techniques. We review 138 representative works that cover different input modalities, and systematically summarize existing methods of image-based SGG from the perspective of feature extraction and fusion. We attempt to connect and systematize the existing visual relationship detection methods, to summarize, and interpret the mechanisms and the strategies of SGG in a comprehensive way. Finally, we finish this survey with deep discussions about current existing problems and future research directions. This survey will help readers to develop a better understanding of the current research status and ideas.

Generative adversarial networks (GANs) have been extensively studied in the past few years. Arguably their most significant impact has been in the area of computer vision where great advances have been made in challenges such as plausible image generation, image-to-image translation, facial attribute manipulation and similar domains. Despite the significant successes achieved to date, applying GANs to real-world problems still poses significant challenges, three of which we focus on here. These are: (1) the generation of high quality images, (2) diversity of image generation, and (3) stable training. Focusing on the degree to which popular GAN technologies have made progress against these challenges, we provide a detailed review of the state of the art in GAN-related research in the published scientific literature. We further structure this review through a convenient taxonomy we have adopted based on variations in GAN architectures and loss functions. While several reviews for GANs have been presented to date, none have considered the status of this field based on their progress towards addressing practical challenges relevant to computer vision. Accordingly, we review and critically discuss the most popular architecture-variant, and loss-variant GANs, for tackling these challenges. Our objective is to provide an overview as well as a critical analysis of the status of GAN research in terms of relevant progress towards important computer vision application requirements. As we do this we also discuss the most compelling applications in computer vision in which GANs have demonstrated considerable success along with some suggestions for future research directions. Code related to GAN-variants studied in this work is summarized on //github.com/sheqi/GAN_Review.

Deep Learning algorithms have achieved the state-of-the-art performance for Image Classification and have been used even in security-critical applications, such as biometric recognition systems and self-driving cars. However, recent works have shown those algorithms, which can even surpass the human capabilities, are vulnerable to adversarial examples. In Computer Vision, adversarial examples are images containing subtle perturbations generated by malicious optimization algorithms in order to fool classifiers. As an attempt to mitigate these vulnerabilities, numerous countermeasures have been constantly proposed in literature. Nevertheless, devising an efficient defense mechanism has proven to be a difficult task, since many approaches have already shown to be ineffective to adaptive attackers. Thus, this self-containing paper aims to provide all readerships with a review of the latest research progress on Adversarial Machine Learning in Image Classification, however with a defender's perspective. Here, novel taxonomies for categorizing adversarial attacks and defenses are introduced and discussions about the existence of adversarial examples are provided. Further, in contrast to exisiting surveys, it is also given relevant guidance that should be taken into consideration by researchers when devising and evaluating defenses. Finally, based on the reviewed literature, it is discussed some promising paths for future research.

Substantial progress has been made recently on developing provably accurate and efficient algorithms for low-rank matrix factorization via nonconvex optimization. While conventional wisdom often takes a dim view of nonconvex optimization algorithms due to their susceptibility to spurious local minima, simple iterative methods such as gradient descent have been remarkably successful in practice. The theoretical footings, however, had been largely lacking until recently. In this tutorial-style overview, we highlight the important role of statistical models in enabling efficient nonconvex optimization with performance guarantees. We review two contrasting approaches: (1) two-stage algorithms, which consist of a tailored initialization step followed by successive refinement; and (2) global landscape analysis and initialization-free algorithms. Several canonical matrix factorization problems are discussed, including but not limited to matrix sensing, phase retrieval, matrix completion, blind deconvolution, robust principal component analysis, phase synchronization, and joint alignment. Special care is taken to illustrate the key technical insights underlying their analyses. This article serves as a testament that the integrated consideration of optimization and statistics leads to fruitful research findings.

Generative Adversarial Networks (GANs) have recently achieved impressive results for many real-world applications, and many GAN variants have emerged with improvements in sample quality and training stability. However, they have not been well visualized or understood. How does a GAN represent our visual world internally? What causes the artifacts in GAN results? How do architectural choices affect GAN learning? Answering such questions could enable us to develop new insights and better models. In this work, we present an analytic framework to visualize and understand GANs at the unit-, object-, and scene-level. We first identify a group of interpretable units that are closely related to object concepts using a segmentation-based network dissection method. Then, we quantify the causal effect of interpretable units by measuring the ability of interventions to control objects in the output. We examine the contextual relationship between these units and their surroundings by inserting the discovered object concepts into new images. We show several practical applications enabled by our framework, from comparing internal representations across different layers, models, and datasets, to improving GANs by locating and removing artifact-causing units, to interactively manipulating objects in a scene. We provide open source interpretation tools to help researchers and practitioners better understand their GAN models.