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Solving PDEs with machine learning techniques has become a popular alternative to conventional methods. In this context, Neural networks (NNs) are among the most commonly used machine learning tools, and in those models, the choice of an appropriate loss function is critical. In general, the main goal is to guarantee that minimizing the loss during training translates to minimizing the error in the solution at the same rate. In this work, we focus on the time-harmonic Maxwell's equations, whose weak formulation takes H(curl) as the space of test functions. We propose a NN in which the loss function is a computable approximation of the dual norm of the weak-form PDE residual. To that end, we employ the Helmholtz decomposition of the space H(curl) and construct an orthonormal basis for this space in two and three spatial dimensions. Here, we use the Discrete Sine/Cosine Transform to accurately and efficiently compute the discrete version of our proposed loss function. Moreover, in the numerical examples we show a high correlation between the proposed loss function and the H(curl)-norm of the error, even in problems with low-regularity solutions.

In recent years, many semantic segmentation methods have been proposed to predict label of pixels in the scene. In general, we measure area prediction errors or boundary prediction errors for comparing methods. However, there is no intuitive evaluation metric that evaluates both aspects. In this work, we propose a new evaluation measure called weighted Intersection over Union (wIoU) for semantic segmentation. First, it build a weight map generated from a boundary distance map, allowing weighted evaluation for each pixel based on a boundary importance factor. The proposed wIoU can evaluate both contour and region by setting a boundary importance factor. We validated the effectiveness of wIoU on a dataset of 33 scenes and demonstrated its flexibility. Using the proposed metric, we expect more flexible and intuitive evaluation in semantic segmentation filed are possible.

Persistent homology is a central methodology in topological data analysis that has been successfully implemented in many fields and is becoming increasingly popular and relevant. The output of persistent homology is a persistence diagram -- a multiset of points supported on the upper half plane -- that is often used as a statistical summary of the topological features of data. In this paper, we study the random nature of persistent homology and estimate the density of expected persistence diagrams from observations using wavelets; we show that our wavelet-based estimator is optimal. Furthermore, we propose an adaptive estimator that offers a sparse representation of the expected persistence diagram that achieves near-optimality.

With the increasing adoption of AI-based systems across everyday life, the need to understand their decision-making mechanisms is correspondingly accelerating. The level at which we can trust the statistical inferences made from AI-based decision systems is an increasing concern, especially in high-risk systems such as criminal justice or medical diagnosis, where incorrect inferences may have tragic consequences. Despite their successes in providing solutions to problems involving real-world data, deep learning (DL) models cannot quantify the certainty of their predictions. And are frequently quite confident, even when their solutions are incorrect. This work presents a method to infer prominent features in two DL classification models trained on clinical and non-clinical text by employing techniques from topological and geometric data analysis. We create a graph of a model's prediction space and cluster the inputs into the graph's vertices by the similarity of features and prediction statistics. We then extract subgraphs demonstrating high-predictive accuracy for a given label. These subgraphs contain a wealth of information about features that the DL model has recognized as relevant to its decisions. We infer these features for a given label using a distance metric between probability measures, and demonstrate the stability of our method compared to the LIME interpretability method. This work demonstrates that we may gain insights into the decision mechanism of a DL model, which allows us to ascertain if the model is making its decisions based on information germane to the problem or identifies extraneous patterns within the data.

Lane marker detection is a crucial component of the autonomous driving and driver assistance systems. Modern deep lane detection methods with row-based lane representation exhibit excellent performance on lane detection benchmarks. Through preliminary oracle experiments, we firstly disentangle the lane representation components to determine the direction of our approach. We show that correct lane positions are already among the predictions of an existing row-based detector, and the confidence scores that accurately represent intersection-over-union (IoU) with ground truths are the most beneficial. Based on the finding, we propose LaneIoU that better correlates with the metric, by taking the local lane angles into consideration. We develop a novel detector coined CLRerNet featuring LaneIoU for the target assignment cost and loss functions aiming at the improved quality of confidence scores. Through careful and fair benchmark including cross validation, we demonstrate that CLRerNet outperforms the state-of-the-art by a large margin - enjoying F1 score of 81.43% compared with 80.47% of the existing method on CULane, and 86.47% compared with 86.10% on CurveLanes.

Ray tracing is an essential operation for realistic image synthesis. The acceleration of ray tracing has been studied for a long period of time because algorithms such as light transport simulations require a large amount of ray tracing. One of the major approaches to accelerate the intersections is to use bounding volumes for early pruning for primitives in the volume. The axis-aligned bounding box is a popular bounding volume for ray tracing because of its simplicity and efficiency. However, the conservative bounding volume may produce extra empty space in addition to its content. Especially, primitives that are thin and diagonal to the axis give false-positive hits on the box volume due to the extra space. Although more complex bounding volumes such as oriented bounding boxes may reduce more false-positive hits, they are computationally expensive. In this paper, we propose a novel culling approach to reduce false-positive hits for the bounding box by embedding a binary voxel data structure to the volume. As a ray is represented as a conservative voxel volume as well in our approach, the ray--voxel intersection is cheaply done by bitwise AND operations. Our method is applicable to hierarchical data structures such as bounding volume hierarchy (BVH). It reduces false-positive hits due to the ray--box test and reduces the number of intersections during the traversal of BVH in ray tracing. We evaluate the reduction of intersections with several scenes and show the possibility of performance improvement despite the culling overhead. We also introduce a compression approach with a lookup table for our voxel data. We show that our compressed voxel data achieves significant false-positive reductions with a small amount of memory.

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.

Text in natural images is of arbitrary orientations, requiring detection in terms of oriented bounding boxes. Normally, a multi-oriented text detector often involves two key tasks: 1) text presence detection, which is a classification problem disregarding text orientation; 2) oriented bounding box regression, which concerns about text orientation. Previous methods rely on shared features for both tasks, resulting in degraded performance due to the incompatibility of the two tasks. To address this issue, we propose to perform classification and regression on features of different characteristics, extracted by two network branches of different designs. Concretely, the regression branch extracts rotation-sensitive features by actively rotating the convolutional filters, while the classification branch extracts rotation-invariant features by pooling the rotation-sensitive features. The proposed method named Rotation-sensitive Regression Detector (RRD) achieves state-of-the-art performance on three oriented scene text benchmark datasets, including ICDAR 2015, MSRA-TD500, RCTW-17 and COCO-Text. Furthermore, RRD achieves a significant improvement on a ship collection dataset, demonstrating its generality on oriented object detection.

In this paper, we propose a conceptually simple and geometrically interpretable objective function, i.e. additive margin Softmax (AM-Softmax), for deep face verification. In general, the face verification task can be viewed as a metric learning problem, so learning large-margin face features whose intra-class variation is small and inter-class difference is large is of great importance in order to achieve good performance. Recently, Large-margin Softmax and Angular Softmax have been proposed to incorporate the angular margin in a multiplicative manner. In this work, we introduce a novel additive angular margin for the Softmax loss, which is intuitively appealing and more interpretable than the existing works. We also emphasize and discuss the importance of feature normalization in the paper. Most importantly, our experiments on LFW BLUFR and MegaFace show that our additive margin softmax loss consistently performs better than the current state-of-the-art methods using the same network architecture and training dataset. Our code has also been made available at //github.com/happynear/AMSoftmax