Large Convolutional Network models have recently demonstrated impressive classification performance on the ImageNet benchmark. However there is no clear understanding of why they perform so well, or how they might be improved. In this paper we address both issues. We introduce a novel visualization technique that gives insight into the function of intermediate feature layers and the operation of the classifier. We also perform an ablation study to discover the performance contribution from different model layers. This enables us to find model architectures that outperform Krizhevsky \etal on the ImageNet classification benchmark. We show our ImageNet model generalizes well to other datasets: when the softmax classifier is retrained, it convincingly beats the current state-of-the-art results on Caltech-101 and Caltech-256 datasets.
Microsoft Research（微软研究院）；Harbin Institute of Technology（哈尔滨工业大学）
Convolutional Neural Networks (CNNs) have achieved comparable error rates to well-trained human on ILSVRC2014 image classification task. To achieve better performance, the complexity of CNNs is continually increasing with deeper and bigger architectures. Though CNNs achieved promising external classification behavior, understanding of their internal work mechanism is still limited. In this work, we attempt to understand the internal work mechanism of CNNs by probing the internal representations in two comprehensive aspects, i.e., visualizing patches in the representation spaces constructed by …
Charité – Universitätsmedizin Berlin（柏林夏里特医学院）；Freie Universität Berlin（柏林自由大学）；Humboldt-Universität zu Berlin（柏林洪堡大学）；Berlin Institute of Health（柏林健康研究所）；Bernstein Center for Computational Neuroscience（伯恩斯坦计算神经科学中心）；Berlin Center for Advanced Neuroimaging（柏林高级神经影像中心）；Technical University Berlin（柏林工业大学）
Visualizing and interpreting convolutional neural networks (CNNs) is an important task to increase trust in automatic medical decision making systems. In this study, we train a 3D CNN to detect Alzheimer's disease based on structural MRI scans of the brain. Then, we apply four different gradient-based and occlusion-based visualization methods that explain the network's classification decisions by highlighting relevant areas in the input image. We compare the methods qualitatively and quantitatively. We find that all four methods focus on brain regions known to be involved in Alzheimer's diseas…
Image representations, from SIFT and bag of visual words to Convolutional Neural Networks (CNNs) are a crucial component of almost all computer vision systems. However, our understanding of them remains limited. In this paper we study several landmark representations, both shallow and deep, by a number of complementary visualization techniques. These visualizations are based on the concept of "natural pre-image", namely a natural-looking image whose representation has some notable property. We study in particular three such visualizations: inversion, in which the aim is to reconstruct an image…
Instituto Superior Tecnico, University of Lisbon（里斯本大学高等技术学院）
Fire localization in images and videos is an important step for an autonomous system to combat fire incidents. State-of-art image segmentation methods based on deep neural networks require a large number of pixel-annotated samples to train Convolutional Neural Networks (CNNs) in a fully-supervised manner. In this paper, we consider weakly supervised segmentation of fire in images, in which only image labels are used to train the network. We show that in the case of fire segmentation, which is a binary segmentation problem, the mean value of features in a mid-layer of classification CNN can per…
Because of their state-of-the-art performance in computer vision, CNNs are becoming increasingly popular in a variety of fields, including medicine. However, as neural networks are black box function approximators, it is difficult, if not impossible, for a medical expert to reason about their output. This could potentially result in the expert distrusting the network when he or she does not agree with its output. In such a case, explaining why the CNN makes a certain decision becomes valuable information. In this paper, we try to open the black box of the CNN by inspecting and visualizing the …
Deep learning models are now used in many different industries, while in certain domains safety is not a critical issue in the medical field it is a huge concern. Not only, we want the models to generalize well but we also want to know the models confidence respect to its decision and which features matter the most. Our team aims to develop a full pipeline in which not only displays the uncertainty of the models decision but also, the saliency map to show which sets of pixels of the input image contribute most to the predictions.
Higher School of Applied Sciences of Tlemcen（特莱姆森应用科学高等学院）；USTHB University（USTHB大学）；University of Mons（蒙斯大学）；University Hospital Center of Tlemcen（特莱姆森大学医院中心）；Tlemcen University（特莱姆森大学）
Computer aided detection and diagnosis systems based on deep learning have shown promising performance in breast cancer detection. However, there are cases where the obtained results lack justification. In this study, our objective is to highlight the regions of interest used by a convolutional neural network (CNN) for classifying histological images as benign or malignant. We compare these regions with the regions identified by pathologists. To achieve this, we employed the VGG19 architecture and tested three visualization methods: Gradient, LRP Z, and LRP Epsilon. Additionally, we experiment…
Scattering Transforms (or ScatterNets) introduced by Mallat are a promising start into creating a well-defined feature extractor to use for pattern recognition and image classification tasks. They are of particular interest due to their architectural similarity to Convolutional Neural Networks (CNNs), while requiring no parameter learning and still performing very well (particularly in constrained classification tasks). In this paper we visualize what the deeper layers of a ScatterNet are sensitive to using a 'DeScatterNet'. We show that the higher orders of ScatterNets are sensitive to comple…
University of Massachusetts, Amherst（马萨诸塞大学阿默斯特分校）
A number of recent approaches have used deep convolutional neural networks (CNNs) to build texture representations. Nevertheless, it is still unclear how these models represent texture and invariances to categorical variations. This work conducts a systematic evaluation of recent CNN-based texture descriptors for recognition and attempts to understand the nature of invariances captured by these representations. First we show that the recently proposed bilinear CNN model [25] is an excellent general-purpose texture descriptor and compares favorably to other CNN-based descriptors on various text…
Understanding and interpreting the decisions made by deep learning models is valuable in many domains. In computer vision, computing heatmaps from a deep network is a popular approach for visualizing and understanding deep networks. However, heatmaps that do not correlate with the network may mislead human, hence the performance of heatmaps in providing a faithful explanation to the underlying deep network is crucial. In this paper, we propose I-GOS, which optimizes for a heatmap so that the classification scores on the masked image would maximally decrease. The main novelty of the approach is…
Visual interpretability of Convolutional Neural Networks (CNNs) has gained significant popularity because of the great challenges that CNN complexity imposes to understanding their inner workings. Although many techniques have been proposed to visualize class features of CNNs, most of them do not provide a correspondence between inputs and the extracted features in specific layers. This prevents the discovery of stimuli that each layer responds better to. We propose an approach to visually interpret CNN features given a set of images by creating corresponding images that depict the most inform…
Guangxi Normal University（广西师范大学）；Huaqiao University（华侨大学）；JD Explore Academy（京东探索研究院）
Vision Transformer (ViT) has become a leading tool in various computer vision tasks, owing to its unique self-attention mechanism that learns visual representations explicitly through cross-patch information interactions. Despite having good success, the literature seldom explores the explainability of vision transformer, and there is no clear picture of how the attention mechanism with respect to the correlation across comprehensive patches will impact the performance and what is the further potential. In this work, we propose a novel explainable visualization approach to analyze and interpre…
Center for Research in Computer Vision（计算机视觉研究中心）；University of Exeter（埃克塞特大学）；University of Central Florida（中佛罗里达大学）
Deep convolutional neural networks have demonstrated high performances for fixation prediction in recent years. How they achieve this, however, is less explored and they remain to be black box models. Here, we attempt to shed light on the internal structure of deep saliency models and study what features they extract for fixation prediction. Specifically, we use a simple yet powerful architecture, consisting of only one CNN and a single resolution input, combined with a new loss function for pixel-wise fixation prediction during free viewing of natural scenes. We show that our simple method is…
We introduce a new, systematic framework for visualizing information flow in deep networks. Specifically, given any trained deep convolutional network model and a given test image, our method produces a compact support in the image domain that corresponds to a (high-resolution) feature that contributes to the given explanation. Our method is both computationally efficient as well as numerically robust. We present several preliminary numerical results that support the benefits of our framework over existing methods.
George Washington University（乔治·华盛顿大学）；Temple University（天普大学）
For convolutional neural network models that optimize an image embedding, we propose a method to highlight the regions of images that contribute most to pairwise similarity. This work is a corollary to the visualization tools developed for classification networks, but applicable to the problem domains better suited to similarity learning. The visualization shows how similarity networks that are fine-tuned learn to focus on different features. We also generalize our approach to embedding networks that use different pooling strategies and provide a simple mechanism to support image similarity se…
Neural network compression and acceleration are widely demanded currently due to the resource constraints on most deployment targets. In this paper, through analyzing the filter activation, gradients, and visualizing the filters' functionality in convolutional neural networks, we show that the filters in higher layers learn extremely task-specific features, which are exclusive for only a small subset of the overall tasks, or even a single class. Based on such findings, we reveal the critical paths of information flow for different classes. And by their intrinsic property of exclusiveness, we p…
Residual networks are the current state of the art on ImageNet. Similar work in the direction of utilizing shortcut connections has been done extremely recently with derivatives of residual networks and with highway networks. This work potentially challenges our understanding that CNNs learn layers of local features that are followed by increasingly global features. Through qualitative visualization and empirical analysis, we explore the purpose that residual skip connections serve. Our assessments show that the residual shortcut connections force layers to refine features, as expected. We als…
Polish-Japanese Academy of Information Technology（波日信息技术学院）；Tooploox（Tooploox公司）；Warsaw University of Technology（华沙理工大学）
Classification of human emotions remains an important and challenging task for many computer vision algorithms, especially in the era of humanoid robots which coexist with humans in their everyday life. Currently proposed methods for emotion recognition solve this task using multi-layered convolutional networks that do not explicitly infer any facial features in the classification phase. In this work, we postulate a fundamentally different approach to solve emotion recognition task that relies on incorporating facial landmarks as a part of the classification loss function. To that end, we exte…
Hong Kong University of Science and Technology（香港科技大学）；Microsoft Research Asia（微软亚洲研究院）
Deep convolutional neural networks (CNNs) have achieved remarkable success in various fields. However, training an excellent CNN is practically a trial-and-error process that consumes a tremendous amount of time and computer resources. To accelerate the training process and reduce the number of trials, experts need to understand what has occurred in the training process and why the resulting CNN behaves as such. However, current popular training platforms, such as TensorFlow, only provide very little and general information, such as training/validation errors, which is far from enough to serve…
We present Smooth Grad-CAM++, a technique which combines two recent techniques: SMOOTHGRAD and Grad-CAM++. Smooth Grad-CAM++ has the capability of either visualizing a layer, subset of feature maps, or subset of neurons within a feature map at each instance. We experimented with few images, and we discovered that Smooth Grad-CAM++ produced more visually sharp maps with larger number of salient pixels highlighted in the given input images when compared with other methods. Smooth Grad-CAM++ will give insight into what our deep CNN models (including models trained on medical scan or imagery) lear…
Deep neural network (DNN) models have recently obtained state-of-the-art prediction accuracy for the transcription factor binding (TFBS) site classification task. However, it remains unclear how these approaches identify meaningful DNA sequence signals and give insights as to why TFs bind to certain locations. In this paper, we propose a toolkit called the Deep Motif Dashboard (DeMo Dashboard) which provides a suite of visualization strategies to extract motifs, or sequence patterns from deep neural network models for TFBS classification. We demonstrate how to visualize and understand three im…
How the human vision system addresses the object identity-preserving recognition problem is largely unknown. Here, we use a vision recognition-reconstruction network (RRN) to investigate the development, recognition, learning and forgetting mechanisms, and achieve similar characteristics to electrophysiological measurements in monkeys. First, in network development study, the RRN also experiences critical developmental stages characterized by specificities in neuron types, synapse and activation patterns, and visual task performance from the early stage of coarse salience map recognition to ma…
Judgments about personality based on facial appearance are strong effectors in social decision making, and are known to have impact on areas from presidential elections to jury decisions. Recent work has shown that it is possible to predict perception of memorability, trustworthiness, intelligence and other attributes in human face images. The most successful of these approaches require face images expertly annotated with key facial landmarks. We demonstrate a Convolutional Neural Network (CNN) model that is able to perform the same task without the need for landmark features, thereby greatly …
Picasso is a free open-source (Eclipse Public License) web application written in Python for rendering standard visualizations useful for analyzing convolutional neural networks. Picasso ships with occlusion maps and saliency maps, two visualizations which help reveal issues that evaluation metrics like loss and accuracy might hide: for example, learning a proxy classification task. Picasso works with the Tensorflow deep learning framework, and Keras (when the model can be loaded into the Tensorflow backend). Picasso can be used with minimal configuration by deep learning researchers and engin…
Breast cancer has the highest incidence and second highest mortality rate for women in the US. Our study aims to utilize deep learning for benign/malignant classification of mammogram tumors using a subset of cases from the Digital Database of Screening Mammography (DDSM). Though it was a small dataset from the view of Deep Learning (about 1000 patients), we show that currently state of the art architectures of deep learning can find a robust signal, even when trained from scratch. Using convolutional neural networks (CNNs), we are able to achieve an accuracy of 85% and an ROC AUC of 0.91, whi…
University of Exeter（埃克塞特大学）；Aalto University（阿尔托大学）；MarkableAI（MarkableAI公司）
Recently, data-driven deep saliency models have achieved high performance and have outperformed classical saliency models, as demonstrated by results on datasets such as the MIT300 and SALICON. Yet, there remains a large gap between the performance of these models and the inter-human baseline. Some outstanding questions include what have these models learned, how and where they fail, and how they can be improved. This article attempts to answer these questions by analyzing the representations learned by individual neurons located at the intermediate layers of deep saliency models. To this end,…
Grinnell College（格林内尔学院）；Oregon State University（俄勒冈州立大学）
Recently, several networks that operate directly on point clouds have been proposed. There is significant utility in understanding their mechanisms to classify point clouds, which can potentially help diagnosing these networks and designing better architectures. In this paper, we propose a novel approach to visualize features important to the point cloud classifiers. Our approach is based on smoothing curved areas on a point cloud. After prominent features were smoothed, the resulting point cloud can be evaluated on the network to assess whether the feature is important to the classifier. A te…
University of Electronic Science and Technology of China（电子科技大学）
Many works have concentrated on visualizing and understanding the inner mechanism of convolutional neural networks (CNNs) by generating images that activate some specific neurons, which is called deep visualization. However, it is still unclear what the filters extract from images intuitively. In this paper, we propose a modified code inversion algorithm, called feature map inversion, to understand the function of filter of interest in CNNs. We reveal that every filter extracts a specific texture. The texture from higher layer contains more colours and more intricate structures. We also demons…
Visualizing key features in X-ray images of epoxy resins for improved material classification using singular value decomposition of deep learning features
WPI-Advanced Institute for Materials Research (AIMR), Tohoku University（东北大学WPI先进材料研究所（AIMR））；Research Institute for Electronic Science, Hokkaido University（北海道大学电子科学研究所）；MathAM-OIL, Tohoku University（东北大学MathAM-OIL）；AIST（产业技术综合研究所）
University of Toronto（多伦多大学）；LG AI Research（LG人工智能研究院）
Visualizing the features captured by Convolutional Neural Networks (CNNs) is one of the conventional approaches to interpret the predictions made by these models in numerous image recognition applications. Grad-CAM is a popular solution that provides such a visualization by combining the activation maps obtained from the model. However, the average gradient-based terms deployed in this method underestimates the contribution of the representations discovered by the model to its predictions. Addressing this problem, we introduce a solution to tackle this issue by computing the path integral of t…
Melanoma is considered to be the most aggressive form of skin cancer. Due to the similar shape of malignant and benign cancerous lesions, doctors spend considerably more time when diagnosing these findings. At present, the evaluation of malignancy is performed primarily by invasive histological examination of the suspicious lesion. Developing an accurate classifier for early and efficient detection can minimize and monitor the harmful effects of skin cancer and increase patient survival rates. This paper proposes a multi-class classification task using the CoAtNet architecture, a hybrid model …
Technical University of Denmark（丹麦技术大学）；Terma A/S（Terma公司）
Convolutional Neural Networks (Convnets) have achieved good results in a range of computer vision tasks the recent years. Though given a lot of attention, visualizing the learned representations to interpret Convnets, still remains a challenging task. The high dimensionality of internal representations and the high abstractions of deep layers are the main challenges when visualizing Convnet functionality. We present in this paper a technique based on clustering internal Convnet representations with a Dirichlet Process Gaussian Mixture Model, for visualization of learned representations in Conv…
University of New Mexico（新墨西哥大学）；Kent State University（肯特州立大学）；US Army Research Laboratory（美国陆军研究实验室）；Los Alamos National Laboratory（洛斯阿拉莫斯国家实验室）
Convolutional Neural Networks(CNNs) are complex systems. They are trained so they can adapt their internal connections to recognize images, texts and more. It is both interesting and helpful to visualize the dynamics within such deep artificial neural networks so that people can understand how these artificial networks are learning and making predictions. In the field of scientific simulations, visualization tools like Paraview have long been utilized to provide insights and understandings. We present in situ TensorView to visualize the training and functioning of CNNs as if they are systems o…
Carnegie Mellon University（卡内基梅隆大学）；University of Michigan（密歇根大学）；Georgia Tech（佐治亚理工学院）；HKUST（香港科技大学）
The promise of multimodal models for real-world applications has inspired research in visualizing and understanding their internal mechanics with the end goal of empowering stakeholders to visualize model behavior, perform model debugging, and promote trust in machine learning models. However, modern multimodal models are typically black-box neural networks, which makes it challenging to understand their internal mechanics. How can we visualize the internal modeling of multimodal interactions in these models? Our paper aims to fill this gap by proposing MultiViz, a method for analyzing the beh…
Massachusetts Institute of Technology（麻省理工学院）；MIT-IBM Watson AI Lab（MIT-IBM沃森人工智能实验室）；IBM Research（IBM研究院）；The Chinese University of Hong Kong（香港中文大学）
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-…
Institute of Fundamental Technological Research, \ Academy of Sciences, Warsaw, Poland Corresponding author, e-mail
Attenuation coefficient (AC) is a fundamental measure of tissue acoustical properties, which can be used in medical diagnostics. In this work, we investigate the feasibility of using convolutional neural networks (CNNs) to directly estimate AC from radio-frequency (RF) ultrasound signals. To develop the CNNs we used RF signals collected from tissue mimicking numerical phantoms for the AC values in a range from 0.1 to 1.5 dB/(MHz*cm). The models were trained based on 1-D patches of RF data. We obtained mean absolute AC estimation errors of 0.08, 0.12, 0.20, 0.25 for the patch lengths: 10 mm, 5 …
Karlsruhe Institute of Technology（卡尔斯鲁厄理工学院）；University of Mannheim（曼海姆大学）
Convolutional neural networks (CNNs) have achieved astonishing performance on various image classification tasks, but it is difficult for humans to understand how a classification comes about. Recent literature proposes methods to explain the classification process to humans. These focus mostly on visualizing feature maps and filter weights, which are not very intuitive for non-experts in analyzing a CNN classification. In this paper, we propose FilTag, an approach to effectively explain CNNs even to non-experts. The idea is that when images of a class frequently activate a convolutional filte…
Montreal Institute for Learning Algorithms（蒙特利尔学习算法研究所）；ConscientAI Labs（ConscientAI实验室）；Microsoft Research New England（微软新英格兰研究院）
We present a project that aims to generate images that depict accurate, vivid, and personalized outcomes of climate change using Cycle-Consistent Adversarial Networks (CycleGANs). By training our CycleGAN model on street-view images of houses before and after extreme weather events (e.g. floods, forest fires, etc.), we learn a mapping that can then be applied to images of locations that have not yet experienced these events. This visual transformation is paired with climate model predictions to assess likelihood and type of climate-related events in the long term (50 years) in order to bring t…
Supercomputer Education and Research Centre（超级计算机教育与研究中心）；Indian Institute of Science（印度科学学院）
With a view to provide a user-friendly interface for designing, training and developing deep learning frameworks, we have developed Expresso, a GUI tool written in Python. Expresso is built atop Caffe, the open-source, prize-winning framework popularly used to develop Convolutional Neural Networks. Expresso provides a convenient wizard-like graphical interface which guides the user through various common scenarios -- data import, construction and training of deep networks, performing various experiments, analyzing and visualizing the results of these experiments. The multi-threaded nature of E…
Most modern convolutional neural networks (CNNs) used for object recognition are built using the same principles: Alternating convolution and max-pooling layers followed by a small number of fully connected layers. We re-evaluate the state of the art for object recognition from small images with convolutional networks, questioning the necessity of different components in the pipeline. We find that max-pooling can simply be replaced by a convolutional layer with increased stride without loss in accuracy on several image recognition benchmarks. Following this finding -- and building on other rec…
Universidad de Cantabria（坎塔布里亚大学）；Instituto de Investigación Valdecilla (IDIVAL)（瓦尔迪西利亚研究所（IDIVAL））；CIBER-BBN（生物工程、生物材料与纳米医学生物医学研究网络中心）；Dartmouth College（达特茅斯学院）
The following work presents how autoencoding all the possible hidden activations of a network for a given problem can provide insight about its structure, behavior, and vulnerabilities. The method, termed self-introspection, can show that a trained model showcases similar activation patterns (albeit randomly distributed due to initialization) when shown data belonging to the same category, and classification errors occur in fringe areas where the activations are not as clearly defined, suggesting some form of random, slowly varying, implicit encoding occurring within deep networks, that can be…
Bosch Center for Artificial Intelligence (BCAI)（博世人工智能中心）；University of Bonn（波恩大学）
To verify and validate networks, it is essential to gain insight into their decisions, limitations as well as possible shortcomings of training data. In this work, we propose a post-hoc, optimization based visual explanation method, which highlights the evidence in the input image for a specific prediction. Our approach is based on a novel technique to defend against adversarial evidence (i.e. faulty evidence due to artefacts) by filtering gradients during optimization. The defense does not depend on human-tuned parameters. It enables explanations which are both fine-grained and preserve the c…
Tsinghua University（清华大学）；Beijing National Research Center for Information Science and Technology（北京信息科学与技术国家研究中心）
Explaining deep convolutional neural networks has been recently drawing increasing attention since it helps to understand the networks' internal operations and why they make certain decisions. Saliency maps, which emphasize salient regions largely connected to the network's decision-making, are one of the most common ways for visualizing and analyzing deep networks in the computer vision community. However, saliency maps generated by existing methods cannot represent authentic information in images due to the unproven proposals about the weights of activation maps which lack solid theoretical …
In recent days, Convolutional Neural Networks (CNN) have demonstrated impressive performance in medical image analysis. However, there is a lack of clear understanding of why and how the Convolutional Neural Network performs so well for image analysis task. How CNN analyzes an image and discriminates among samples of different classes are usually considered as non-transparent. As a result, it becomes difficult to apply CNN based approaches in clinical procedures and automated disease diagnosis systems. In this paper, we consider this issue and work on visualizing and understanding the decision…
Cornell University（康奈尔大学）；University of Wyoming（怀俄明大学）；Jet Propulsion Laboratory, California Institute of Technology（加州理工学院喷气推进实验室）
Recent years have produced great advances in training large, deep neural networks (DNNs), including notable successes in training convolutional neural networks (convnets) to recognize natural images. However, our understanding of how these models work, especially what computations they perform at intermediate layers, has lagged behind. Progress in the field will be further accelerated by the development of better tools for visualizing and interpreting neural nets. We introduce two such tools here. The first is a tool that visualizes the activations produced on each layer of a trained convnet a…
Clouds frequently cover the Earth's surface and pose an omnipresent challenge to optical Earth observation methods. The vast majority of remote sensing approaches either selectively choose single cloud-free observations or employ a pre-classification strategy to identify and mask cloudy pixels. We follow a different strategy and treat cloud coverage as noise that is inherent to the observed satellite data. In prior work, we directly employed a straightforward \emph{convolutional long short-term memory} network for vegetation classification without explicit cloud filtering and achieved state-of…
Southwestern University of Finance and Economics（西南财经大学）；Sichuan University（四川大学）
To collectively forecast the demand for ride-sourcing services in all regions of a city, the deep learning approaches have been applied with commendable results. However, the local statistical differences throughout the geographical layout of the city make the spatial stationarity assumption of the convolution invalid, which limits the performance of CNNs on the demand forecasting task. In this paper, we propose a novel deep learning framework called LC-ST-FCN (locally connected spatiotemporal fully-convolutional neural network) to address the unique challenges of the region-level demand forec…
Deep convolutional networks provide state of the art classifications and regressions results over many high-dimensional problems. We review their architecture, which scatters data with a cascade of linear filter weights and non-linearities. A mathematical framework is introduced to analyze their properties. Computations of invariants involve multiscale contractions, the linearization of hierarchical symmetries, and sparse separations. Applications are discussed.
