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This work considers supervised learning to count from images and their corresponding point annotations. Where density-based counting methods typically use the point annotations only to create Gaussian-density maps, which act as the supervision signal, the starting point of this work is that point annotations have counting potential beyond density map generation. We introduce two methods that repurpose the available point annotations to enhance counting performance. The first is a counting-specific augmentation that leverages point annotations to simulate occluded objects in both input and density images to enhance the network's robustness to occlusions. The second method, foreground distillation, generates foreground masks from the point annotations, from which we train an auxiliary network on images with blacked-out backgrounds. By doing so, it learns to extract foreground counting knowledge without interference from the background. These methods can be seamlessly integrated with existing counting advances and are adaptable to different loss functions. We demonstrate complementary effects of the approaches, allowing us to achieve robust counting results even in challenging scenarios such as background clutter, occlusion, and varying crowd densities. Our proposed approach achieves strong counting results on multiple datasets, including ShanghaiTech Part\_A and Part\_B, UCF\_QNRF, JHU-Crowd++, and NWPU-Crowd.

Graph clustering is a fundamental task in network analysis where the goal is to detect sets of nodes that are well-connected to each other but sparsely connected to the rest of the graph. We present faster approximation algorithms for an NP-hard parameterized clustering framework called LambdaCC, which is governed by a tunable resolution parameter and generalizes many other clustering objectives such as modularity, sparsest cut, and cluster deletion. Previous LambdaCC algorithms are either heuristics with no approximation guarantees, or computationally expensive approximation algorithms. We provide fast new approximation algorithms that can be made purely combinatorial. These rely on a new parameterized edge labeling problem we introduce that generalizes previous edge labeling problems that are based on the principle of strong triadic closure and are of independent interest in social network analysis. Our methods are orders of magnitude more scalable than previous approximation algorithms and our lower bounds allow us to obtain a posteriori approximation guarantees for previous heuristics that have no approximation guarantees of their own.

We address univariate root isolation when the polynomial's coefficients are in a multiple field extension. We consider a polynomial $F \in L[Y]$, where $L$ is a multiple algebraic extension of $\mathbb{Q}$. We provide aggregate bounds for $F$ and algorithmic and bit-complexity results for the problem of isolating its roots. For the latter problem we follow a common approach based on univariate root isolation algorithms. For the particular case where $F$ does not have multiple roots, we achieve a bit-complexity in $\tilde{\mathcal{O}}_B(n d^{2n+2}(d+n\tau))$, where $d$ is the total degree and $\tau$ is the bitsize of the involved polynomials.In the general case we need to enhance our algorithm with a preprocessing step that determines the number of distinct roots of $F$. We follow a numerical, yet certified, approach that has bit-complexity $\tilde{\mathcal{O}}_B(n^2d^{3n+3}\tau + n^3 d^{2n+4}\tau)$.

We introduce Probabilistic Coordinate Fields (PCFs), a novel geometric-invariant coordinate representation for image correspondence problems. In contrast to standard Cartesian coordinates, PCFs encode coordinates in correspondence-specific barycentric coordinate systems (BCS) with affine invariance. To know \textit{when and where to trust} the encoded coordinates, we implement PCFs in a probabilistic network termed PCF-Net, which parameterizes the distribution of coordinate fields as Gaussian mixture models. By jointly optimizing coordinate fields and their confidence conditioned on dense flows, PCF-Net can work with various feature descriptors when quantifying the reliability of PCFs by confidence maps. An interesting observation of this work is that the learned confidence map converges to geometrically coherent and semantically consistent regions, which facilitates robust coordinate representation. By delivering the confident coordinates to keypoint/feature descriptors, we show that PCF-Net can be used as a plug-in to existing correspondence-dependent approaches. Extensive experiments on both indoor and outdoor datasets suggest that accurate geometric invariant coordinates help to achieve the state of the art in several correspondence problems, such as sparse feature matching, dense image registration, camera pose estimation, and consistency filtering. Further, the interpretable confidence map predicted by PCF-Net can also be leveraged to other novel applications from texture transfer to multi-homography classification.

Spatial point patterns are a commonly recorded form of data in ecology, medicine, astronomy, criminology, epidemiology and many other application fields. One way to understand their second order dependence structure is via their spectral density function. However, unlike time series analysis, for point processes such approaches are currently underutilized. In part, this is because the interpretation of the spectral representation of point patterns is challenging. In this paper, we demonstrate how to band-pass filter point patterns, thus enabling us to explore the spectral representation of point patterns in space by isolating the signal corresponding to certain sets of wavenumbers.

There is a long-standing problem of repeated patterns in correspondence problems, where mismatches frequently occur because of inherent ambiguity. The unique position information associated with repeated patterns makes coordinate representations a useful supplement to appearance representations for improving feature correspondences. However, the issue of appropriate coordinate representation has remained unresolved. In this study, we demonstrate that geometric-invariant coordinate representations, such as barycentric coordinates, can significantly reduce mismatches between features. The first step is to establish a theoretical foundation for geometrically invariant coordinates. We present a seed matching and filtering network (SMFNet) that combines feature matching and consistency filtering with a coarse-to-fine matching strategy in order to acquire reliable sparse correspondences. We then introduce DEGREE, a novel anchor-to-barycentric (A2B) coordinate encoding approach, which generates multiple affine-invariant correspondence coordinates from paired images. DEGREE can be used as a plug-in with standard descriptors, feature matchers, and consistency filters to improve the matching quality. Extensive experiments in synthesized indoor and outdoor datasets demonstrate that DEGREE alleviates the problem of repeated patterns and helps achieve state-of-the-art performance. Furthermore, DEGREE also reports competitive performance in the third Image Matching Challenge at CVPR 2021. This approach offers a new perspective to alleviate the problem of repeated patterns and emphasizes the importance of choosing coordinate representations for feature correspondences.

Agents navigating in 3D environments require some form of memory, which should hold a compact and actionable representation of the history of observations useful for decision taking and planning. In most end-to-end learning approaches the representation is latent and usually does not have a clearly defined interpretation, whereas classical robotics addresses this with scene reconstruction resulting in some form of map, usually estimated with geometry and sensor models and/or learning. In this work we propose to learn an actionable representation of the scene independently of the targeted downstream task and without explicitly optimizing reconstruction. The learned representation is optimized by a blind auxiliary agent trained to navigate with it on multiple short sub episodes branching out from a waypoint and, most importantly, without any direct visual observation. We argue and show that the blindness property is important and forces the (trained) latent representation to be the only means for planning. With probing experiments we show that the learned representation optimizes navigability and not reconstruction. On downstream tasks we show that it is robust to changes in distribution, in particular the sim2real gap, which we evaluate with a real physical robot in a real office building, significantly improving performance.

Key Point Analysis (KPA) has been recently proposed for deriving fine-grained insights from collections of textual comments. KPA extracts the main points in the data as a list of concise sentences or phrases, termed key points, and quantifies their prevalence. While key points are more expressive than word clouds and key phrases, making sense of a long, flat list of key points, which often express related ideas in varying levels of granularity, may still be challenging. To address this limitation of KPA, we introduce the task of organizing a given set of key points into a hierarchy, according to their specificity. Such hierarchies may be viewed as a novel type of Textual Entailment Graph. We develop ThinkP, a high quality benchmark dataset of key point hierarchies for business and product reviews, obtained by consolidating multiple annotations. We compare different methods for predicting pairwise relations between key points, and for inferring a hierarchy from these pairwise predictions. In particular, for the task of computing pairwise key point relations, we achieve significant gains over existing strong baselines by applying directional distributional similarity methods to a novel distributional representation of key points, and further boost performance via weak supervision.

Persistent homology (PH) provides topological descriptors for geometric data, such as weighted graphs, which are interpretable, stable to perturbations, and invariant under, e.g., relabeling. Most applications of PH focus on the one-parameter case -- where the descriptors summarize the changes in topology of data as it is filtered by a single quantity of interest -- and there is now a wide array of methods enabling the use of one-parameter PH descriptors in data science, which rely on the stable vectorization of these descriptors as elements of a Hilbert space. Although the multiparameter PH (MPH) of data that is filtered by several quantities of interest encodes much richer information than its one-parameter counterpart, the scarceness of stability results for MPH descriptors has so far limited the available options for the stable vectorization of MPH. In this paper, we aim to bring together the best of both worlds by showing how the interpretation of signed barcodes -- a recent family of MPH descriptors -- as signed measures leads to natural extensions of vectorization strategies from one parameter to multiple parameters. The resulting feature vectors are easy to define and to compute, and provably stable. While, as a proof of concept, we focus on simple choices of signed barcodes and vectorizations, we already see notable performance improvements when comparing our feature vectors to state-of-the-art topology-based methods on various types of data.

Graph convolution networks (GCN) are increasingly popular in many applications, yet remain notoriously hard to train over large graph datasets. They need to compute node representations recursively from their neighbors. Current GCN training algorithms suffer from either high computational costs that grow exponentially with the number of layers, or high memory usage for loading the entire graph and node embeddings. In this paper, we propose a novel efficient layer-wise training framework for GCN (L-GCN), that disentangles feature aggregation and feature transformation during training, hence greatly reducing time and memory complexities. We present theoretical analysis for L-GCN under the graph isomorphism framework, that L-GCN leads to as powerful GCNs as the more costly conventional training algorithm does, under mild conditions. We further propose L^2-GCN, which learns a controller for each layer that can automatically adjust the training epochs per layer in L-GCN. Experiments show that L-GCN is faster than state-of-the-arts by at least an order of magnitude, with a consistent of memory usage not dependent on dataset size, while maintaining comparable prediction performance. With the learned controller, L^2-GCN can further cut the training time in half. Our codes are available at //github.com/Shen-Lab/L2-GCN.