Daily Papers

Given a K-vertex simplex in a d-dimensional space, suppose we measure n points on the simplex with noise (hence, some of the observed points fall outside the simplex). Vertex hunting is the problem of estimating the K vertices of the simplex. A popular vertex hunting algorithm is successive projection algorithm (SPA). However, SPA is observed to perform unsatisfactorily under strong noise or outliers. We propose pseudo-point SPA (pp-SPA). It uses a projection step and a denoise step to generate pseudo-points and feed them into SPA for vertex hunting. We derive error bounds for pp-SPA, leveraging on extreme value theory of (possibly) high-dimensional random vectors. The results suggest that pp-SPA has faster rates and better numerical performances than SPA. Our analysis includes an improved non-asymptotic bound for the original SPA, which is of independent interest.

We present a novel stochastic variational Gaussian process (GP) inference method, based on a posterior over a learnable set of weighted pseudo input-output points (coresets). Instead of a free-form variational family, the proposed coreset-based, variational tempered family for GPs (CVTGP) is defined in terms of the GP prior and the data-likelihood; hence, accommodating the modeling inductive biases. We derive CVTGP's lower bound for the log-marginal likelihood via marginalization of the proposed posterior over latent GP coreset variables, and show it is amenable to stochastic optimization. CVTGP reduces the learnable parameter size to O(M), enjoys numerical stability, and maintains O(M^3) time- and O(M^2) space-complexity, by leveraging a coreset-based tempered posterior that, in turn, provides sparse and explainable representations of the data. Results on simulated and real-world regression problems with Gaussian observation noise validate that CVTGP provides better evidence lower-bound estimates and predictive root mean squared error than alternative stochastic GP inference methods.

We have investigated the effects of strain on two-dimensional square lattices and examined the methods for inducing pseudo-magnetic fields. In both the columnar and staggered pi-flux square lattices, we have found that strain only modulates Fermi velocities rather than inducing pseudo-magnetic fields. However, spatially non-uniform on-site potentials (anisotropic hoppings) can create pseudo-magnetic fields in columnar (staggered) pi-flux square lattices. On the other hand, we demonstrate that strain does induce pseudo-magnetic fields in staggered zero-flux square lattices. By breaking a quarter of the bonds, we clarify that a staggered zero-flux square lattice is topologically equivalent to a honeycomb lattice and displays pseudo-vector potentials and pseudo-Landau levels at the Dirac points.

Prompting with natural language instructions has recently emerged as a popular method of harnessing the capabilities of large language models. Given the inherent ambiguity present in natural language, it is intuitive to consider the possible advantages of prompting with less ambiguous prompt styles, such as the use of pseudo-code. In this paper we explore if prompting via pseudo-code instructions helps improve the performance of pre-trained language models. We manually create a dataset of pseudo-code prompts for 132 different tasks spanning classification, QA and generative language tasks, sourced from the Super-NaturalInstructions dataset. Using these prompts along with their counterparts in natural language, we study their performance on two LLM families - BLOOM and CodeGen. Our experiments show that using pseudo-code instructions leads to better results, with an average increase (absolute) of 7-16 points in F1 scores for classification tasks and an improvement (relative) of 12-38% in aggregate ROUGE-L scores across all tasks. We include detailed ablation studies which indicate that code comments, docstrings, and the structural clues encoded in pseudo-code all contribute towards the improvement in performance. To the best of our knowledge our work is the first to demonstrate how pseudo-code prompts can be helpful in improving the performance of pre-trained LMs.

Domain Adaptation (DA) is important for deep learning-based medical image segmentation models to deal with testing images from a new target domain. As the source-domain data are usually unavailable when a trained model is deployed at a new center, Source-Free Domain Adaptation (SFDA) is appealing for data and annotation-efficient adaptation to the target domain. However, existing SFDA methods have a limited performance due to lack of sufficient supervision with source-domain images unavailable and target-domain images unlabeled. We propose a novel Uncertainty-aware Pseudo Label guided (UPL) SFDA method for medical image segmentation. Specifically, we propose Target Domain Growing (TDG) to enhance the diversity of predictions in the target domain by duplicating the pre-trained model's prediction head multiple times with perturbations. The different predictions in these duplicated heads are used to obtain pseudo labels for unlabeled target-domain images and their uncertainty to identify reliable pseudo labels. We also propose a Twice Forward pass Supervision (TFS) strategy that uses reliable pseudo labels obtained in one forward pass to supervise predictions in the next forward pass. The adaptation is further regularized by a mean prediction-based entropy minimization term that encourages confident and consistent results in different prediction heads. UPL-SFDA was validated with a multi-site heart MRI segmentation dataset, a cross-modality fetal brain segmentation dataset, and a 3D fetal tissue segmentation dataset. It improved the average Dice by 5.54, 5.01 and 6.89 percentage points for the three tasks compared with the baseline, respectively, and outperformed several state-of-the-art SFDA methods.

Most state-of-the-art point trackers are trained on synthetic data due to the difficulty of annotating real videos for this task. However, this can result in suboptimal performance due to the statistical gap between synthetic and real videos. In order to understand these issues better, we introduce CoTracker3, comprising a new tracking model and a new semi-supervised training recipe. This allows real videos without annotations to be used during training by generating pseudo-labels using off-the-shelf teachers. The new model eliminates or simplifies components from previous trackers, resulting in a simpler and often smaller architecture. This training scheme is much simpler than prior work and achieves better results using 1,000 times less data. We further study the scaling behaviour to understand the impact of using more real unsupervised data in point tracking. The model is available in online and offline variants and reliably tracks visible and occluded points.

Deep learning has achieved impressive results in nuclei segmentation, but the massive requirement for pixel-wise labels remains a significant challenge. To alleviate the annotation burden, existing methods generate pseudo masks for model training using point labels. However, the generated masks are inevitably different from the ground truth, and these dissimilarities are not handled reasonably during the network training, resulting in the subpar performance of the segmentation model. To tackle this issue, we propose a framework named DoNuSeg, enabling Dynamic pseudo label Optimization in point-supervised Nuclei Segmentation. Specifically, DoNuSeg takes advantage of class activation maps (CAMs) to adaptively capture regions with semantics similar to annotated points. To leverage semantic diversity in the hierarchical feature levels, we design a dynamic selection module to choose the optimal one among CAMs from different encoder blocks as pseudo masks. Meanwhile, a CAM-guided contrastive module is proposed to further enhance the accuracy of pseudo masks. In addition to exploiting the semantic information provided by CAMs, we consider location priors inherent to point labels, developing a task-decoupled structure for effectively differentiating nuclei. Extensive experiments demonstrate that DoNuSeg outperforms state-of-the-art point-supervised methods. The code is available at https://github.com/shinning0821/MICCAI24-DoNuSeg.

Pseudo-labeling is significant for semi-supervised instance segmentation, which generates instance masks and classes from unannotated images for subsequent training. However, in existing pipelines, pseudo-labels that contain valuable information may be directly filtered out due to mismatches in class and mask quality. To address this issue, we propose a novel framework, called pseudo-label aligning instance segmentation (PAIS), in this paper. In PAIS, we devise a dynamic aligning loss (DALoss) that adjusts the weights of semi-supervised loss terms with varying class and mask score pairs. Through extensive experiments conducted on the COCO and Cityscapes datasets, we demonstrate that PAIS is a promising framework for semi-supervised instance segmentation, particularly in cases where labeled data is severely limited. Notably, with just 1\% labeled data, PAIS achieves 21.2 mAP (based on Mask-RCNN) and 19.9 mAP (based on K-Net) on the COCO dataset, outperforming the current state-of-the-art model, \ie, NoisyBoundary with 7.7 mAP, by a margin of over 12 points. Code is available at: https://github.com/hujiecpp/PAIS.

Dense retrieval approaches can overcome the lexical gap and lead to significantly improved search results. However, they require large amounts of training data which is not available for most domains. As shown in previous work (Thakur et al., 2021b), the performance of dense retrievers severely degrades under a domain shift. This limits the usage of dense retrieval approaches to only a few domains with large training datasets. In this paper, we propose the novel unsupervised domain adaptation method Generative Pseudo Labeling (GPL), which combines a query generator with pseudo labeling from a cross-encoder. On six representative domain-specialized datasets, we find the proposed GPL can outperform an out-of-the-box state-of-the-art dense retrieval approach by up to 9.3 points nDCG@10. GPL requires less (unlabeled) data from the target domain and is more robust in its training than previous methods. We further investigate the role of six recent pre-training methods in the scenario of domain adaptation for retrieval tasks, where only three could yield improved results. The best approach, TSDAE (Wang et al., 2021) can be combined with GPL, yielding another average improvement of 1.4 points nDCG@10 across the six tasks. The code and the models are available at https://github.com/UKPLab/gpl.

Breast dynamic contrast-enhanced magnetic resonance imaging (DCE-MRI) plays an important role in the screening and prognosis assessment of high-risk breast cancer. The segmentation of cancerous regions is essential useful for the subsequent analysis of breast MRI. To alleviate the annotation effort to train the segmentation networks, we propose a weakly-supervised strategy using extreme points as annotations for breast cancer segmentation. Without using any bells and whistles, our strategy focuses on fully exploiting the learning capability of the routine training procedure, i.e., the train - fine-tune - retrain process. The network first utilizes the pseudo-masks generated using the extreme points to train itself, by minimizing a contrastive loss, which encourages the network to learn more representative features for cancerous voxels. Then the trained network fine-tunes itself by using a similarity-aware propagation learning (SimPLe) strategy, which leverages feature similarity between unlabeled and positive voxels to propagate labels. Finally the network retrains itself by employing the pseudo-masks generated using previous fine-tuned network. The proposed method is evaluated on our collected DCE-MRI dataset containing 206 patients with biopsy-proven breast cancers. Experimental results demonstrate our method effectively fine-tunes the network by using the SimPLe strategy, and achieves a mean Dice value of 81%.

We investigate a new threat to neural sequence-to-sequence (seq2seq) models: training-time attacks that cause models to "spin" their outputs so as to support an adversary-chosen sentiment or point of view -- but only when the input contains adversary-chosen trigger words. For example, a spinned summarization model outputs positive summaries of any text that mentions the name of some individual or organization. Model spinning introduces a "meta-backdoor" into a model. Whereas conventional backdoors cause models to produce incorrect outputs on inputs with the trigger, outputs of spinned models preserve context and maintain standard accuracy metrics, yet also satisfy a meta-task chosen by the adversary. Model spinning enables propaganda-as-a-service, where propaganda is defined as biased speech. An adversary can create customized language models that produce desired spins for chosen triggers, then deploy these models to generate disinformation (a platform attack), or else inject them into ML training pipelines (a supply-chain attack), transferring malicious functionality to downstream models trained by victims. To demonstrate the feasibility of model spinning, we develop a new backdooring technique. It stacks an adversarial meta-task onto a seq2seq model, backpropagates the desired meta-task output to points in the word-embedding space we call "pseudo-words," and uses pseudo-words to shift the entire output distribution of the seq2seq model. We evaluate this attack on language generation, summarization, and translation models with different triggers and meta-tasks such as sentiment, toxicity, and entailment. Spinned models largely maintain their accuracy metrics (ROUGE and BLEU) while shifting their outputs to satisfy the adversary's meta-task. We also show that, in the case of a supply-chain attack, the spin functionality transfers to downstream models.

As a significant step for human face modeling, editing, and generation, face landmarking aims at extracting facial keypoints from images. A generalizable face landmarker is required in practice because real-world facial images, e.g., the avatars in animations and games, are often stylized in various ways. However, achieving generalizable face landmarking is challenging due to the diversity of facial styles and the scarcity of labeled stylized faces. In this study, we propose a simple but effective paradigm to learn a generalizable face landmarker based on labeled real human faces and unlabeled stylized faces. Our method learns the face landmarker as the key module of a conditional face warper. Given a pair of real and stylized facial images, the conditional face warper predicts a warping field from the real face to the stylized one, in which the face landmarker predicts the ending points of the warping field and provides us with high-quality pseudo landmarks for the corresponding stylized facial images. Applying an alternating optimization strategy, we learn the face landmarker to minimize i) the discrepancy between the stylized faces and the warped real ones and ii) the prediction errors of both real and pseudo landmarks. Experiments on various datasets show that our method outperforms existing state-of-the-art domain adaptation methods in face landmarking tasks, leading to a face landmarker with better generalizability. Code is available at https://plustwo0.github.io/project-face-landmarker}{https://plustwo0.github.io/project-face-landmarker.

Mesh deformation plays a pivotal role in many 3D vision tasks including dynamic simulations, rendering, and reconstruction. However, defining an efficient discrepancy between predicted and target meshes remains an open problem. A prevalent approach in current deep learning is the set-based approach which measures the discrepancy between two surfaces by comparing two randomly sampled point-clouds from the two meshes with Chamfer pseudo-distance. Nevertheless, the set-based approach still has limitations such as lacking a theoretical guarantee for choosing the number of points in sampled point-clouds, and the pseudo-metricity and the quadratic complexity of the Chamfer divergence. To address these issues, we propose a novel metric for learning mesh deformation. The metric is defined by sliced Wasserstein distance on meshes represented as probability measures that generalize the set-based approach. By leveraging probability measure space, we gain flexibility in encoding meshes using diverse forms of probability measures, such as continuous, empirical, and discrete measures via varifold representation. After having encoded probability measures, we can compare meshes by using the sliced Wasserstein distance which is an effective optimal transport distance with linear computational complexity and can provide a fast statistical rate for approximating the surface of meshes. To the end, we employ a neural ordinary differential equation (ODE) to deform the input surface into the target shape by modeling the trajectories of the points on the surface. Our experiments on cortical surface reconstruction demonstrate that our approach surpasses other competing methods in multiple datasets and metrics.

Training high-accuracy 3D detectors necessitates massive labeled 3D annotations with 7 degree-of-freedom, which is laborious and time-consuming. Therefore, the form of point annotations is proposed to offer significant prospects for practical applications in 3D detection, which is not only more accessible and less expensive but also provides strong spatial information for object localization. In this paper, we empirically discover that it is non-trivial to merely adapt Point-DETR to its 3D form, encountering two main bottlenecks: 1) it fails to encode strong 3D prior into the model, and 2) it generates low-quality pseudo labels in distant regions due to the extreme sparsity of LiDAR points. To overcome these challenges, we introduce Point-DETR3D, a teacher-student framework for weakly semi-supervised 3D detection, designed to fully capitalize on point-wise supervision within a constrained instance-wise annotation budget.Different from Point-DETR which encodes 3D positional information solely through a point encoder, we propose an explicit positional query initialization strategy to enhance the positional prior. Considering the low quality of pseudo labels at distant regions produced by the teacher model, we enhance the detector's perception by incorporating dense imagery data through a novel Cross-Modal Deformable RoI Fusion (D-RoI).Moreover, an innovative point-guided self-supervised learning technique is proposed to allow for fully exploiting point priors, even in student models.Extensive experiments on representative nuScenes dataset demonstrate our Point-DETR3D obtains significant improvements compared to previous works. Notably, with only 5% of labeled data, Point-DETR3D achieves over 90% performance of its fully supervised counterpart.

Point annotations are considerably more time-efficient than bounding box annotations. However, how to use cheap point annotations to boost the performance of semi-supervised object detection remains largely unsolved. In this work, we present Point-Teaching, a weakly semi-supervised object detection framework to fully exploit the point annotations. Specifically, we propose a Hungarian-based point matching method to generate pseudo labels for point annotated images. We further propose multiple instance learning (MIL) approaches at the level of images and points to supervise the object detector with point annotations. Finally, we propose a simple-yet-effective data augmentation, termed point-guided copy-paste, to reduce the impact of the unmatched points. Experiments demonstrate the effectiveness of our method on a few datasets and various data regimes.

We study the joint image of lines incident to points, meaning the set of image tuples obtained from fixed cameras observing a varying 3D point-line incidence. We prove a formula for the number of complex critical points of the triangulation problem that aims to compute a 3D point-line incidence from noisy images. Our formula works for an arbitrary number of images and measures the intrinsic difficulty of this triangulation. Additionally, we conduct numerical experiments using homotopy continuation methods, comparing different approaches of triangulation of such incidences. In our setup, exploiting the incidence relations gives both a faster point reconstruction and in three views more accurate.

Few prior works study deep learning on point sets. PointNet by Qi et al. is a pioneer in this direction. However, by design PointNet does not capture local structures induced by the metric space points live in, limiting its ability to recognize fine-grained patterns and generalizability to complex scenes. In this work, we introduce a hierarchical neural network that applies PointNet recursively on a nested partitioning of the input point set. By exploiting metric space distances, our network is able to learn local features with increasing contextual scales. With further observation that point sets are usually sampled with varying densities, which results in greatly decreased performance for networks trained on uniform densities, we propose novel set learning layers to adaptively combine features from multiple scales. Experiments show that our network called PointNet++ is able to learn deep point set features efficiently and robustly. In particular, results significantly better than state-of-the-art have been obtained on challenging benchmarks of 3D point clouds.

Singular regular points often arise in differential equations describing physical phenomena such as fluid dynamics, electromagnetism, and gravitation. Traditional numerical techniques often fail or become unstable near these points, requiring the use of semi-analytical tools, such as series expansions and perturbative methods, in combination with numerical algorithms; or to invoke more sophisticated methods. In this work, we take an alternative route and leverage the power of machine learning to exploit Physics Informed Neural Networks (PINNs) as a modern approach to solving ordinary differential equations with singular points. PINNs utilize deep learning architectures to approximate solutions by embedding the differential equations into the loss function of the neural network. We discuss the advantages of PINNs in handling singularities, particularly their ability to bypass traditional grid-based methods and provide smooth approximations across irregular regions. Techniques for enhancing the accuracy of PINNs near singular points, such as adaptive loss weighting, are used in order to achieve high efficiency in the training of the network. We exemplify our results by studying four differential equations of interest in mathematics and gravitation -- the Legendre equation, the hypergeometric equation, the solution for black hole space-times in theories of Lorentz violating gravity, and the spherical accretion of a perfect fluid in a Schwarzschild geometry.

Recent approaches representing 3D objects and scenes using Gaussian splats show increased rendering speed across a variety of platforms and devices. While rendering such representations is indeed extremely efficient, storing and transmitting them is often prohibitively expensive. To represent large-scale scenes, one often needs to store millions of 3D Gaussians, occupying gigabytes of disk space. This poses a very practical limitation, prohibiting widespread adoption.Several solutions have been proposed to strike a balance between disk size and rendering quality, noticeably reducing the visual quality. In this work, we propose a new representation that dramatically reduces the hard drive footprint while featuring similar or improved quality when compared to the standard 3D Gaussian splats. When compared to other compact solutions, ours offers higher quality renderings with significantly reduced storage, being able to efficiently run on a mobile device in real-time. Our key observation is that nearby points in the scene can share similar representations. Hence, only a small ratio of 3D points needs to be stored. We introduce an approach to identify such points which are called parent points. The discarded points called children points along with attributes can be efficiently predicted by tiny MLPs.

Training with sparse annotations is known to reduce the performance of object detectors. Previous methods have focused on proxies for missing ground truth annotations in the form of pseudo-labels for unlabeled boxes. We observe that existing methods suffer at higher levels of sparsity in the data due to noisy pseudo-labels. To prevent this, we propose an end-to-end system that learns to separate the proposals into labeled and unlabeled regions using Pseudo-positive mining. While the labeled regions are processed as usual, self-supervised learning is used to process the unlabeled regions thereby preventing the negative effects of noisy pseudo-labels. This novel approach has multiple advantages such as improved robustness to higher sparsity when compared to existing methods. We conduct exhaustive experiments on five splits on the PASCAL-VOC and COCO datasets achieving state-of-the-art performance. We also unify various splits used across literature for this task and present a standardized benchmark. On average, we improve by 2.6, 3.9 and 9.6 mAP over previous state-of-the-art methods on three splits of increasing sparsity on COCO. Our project is publicly available at https://www.cs.umd.edu/~sakshams/SparseDet.

We present Meta Pseudo Labels, a semi-supervised learning method that achieves a new state-of-the-art top-1 accuracy of 90.2% on ImageNet, which is 1.6% better than the existing state-of-the-art. Like Pseudo Labels, Meta Pseudo Labels has a teacher network to generate pseudo labels on unlabeled data to teach a student network. However, unlike Pseudo Labels where the teacher is fixed, the teacher in Meta Pseudo Labels is constantly adapted by the feedback of the student's performance on the labeled dataset. As a result, the teacher generates better pseudo labels to teach the student. Our code will be available at https://github.com/google-research/google-research/tree/master/meta_pseudo_labels.

To characterize the location (mean, median) of a set of graphs, one needs a notion of centrality that is adapted to metric spaces, since graph sets are not Euclidean spaces. A standard approach is to consider the Frechet mean. In this work, we equip a set of graphs with the pseudometric defined by the norm between the eigenvalues of their respective adjacency matrix. Unlike the edit distance, this pseudometric reveals structural changes at multiple scales, and is well adapted to studying various statistical problems for graph-valued data. We describe an algorithm to compute an approximation to the sample Frechet mean of a set of undirected unweighted graphs with a fixed size using this pseudometric.

Recent advancements in self-supervised learning in the point cloud domain have demonstrated significant potential. However, these methods often suffer from drawbacks, including lengthy pre-training time, the necessity of reconstruction in the input space, or the necessity of additional modalities. In order to address these issues, we introduce Point-JEPA, a joint embedding predictive architecture designed specifically for point cloud data. To this end, we introduce a sequencer that orders point cloud patch embeddings to efficiently compute and utilize their proximity based on the indices during target and context selection. The sequencer also allows shared computations of the patch embeddings' proximity between context and target selection, further improving the efficiency. Experimentally, our method achieves competitive results with state-of-the-art methods while avoiding the reconstruction in the input space or additional modality.

In this paper, we introduce a novel geometry-aware self-training framework for room layout estimation models on unseen scenes with unlabeled data. Our approach utilizes a ray-casting formulation to aggregate multiple estimates from different viewing positions, enabling the computation of reliable pseudo-labels for self-training. In particular, our ray-casting approach enforces multi-view consistency along all ray directions and prioritizes spatial proximity to the camera view for geometry reasoning. As a result, our geometry-aware pseudo-labels effectively handle complex room geometries and occluded walls without relying on assumptions such as Manhattan World or planar room walls. Evaluation on publicly available datasets, including synthetic and real-world scenarios, demonstrates significant improvements in current state-of-the-art layout models without using any human annotation.

A key step in any scanning-based asset creation workflow is to convert unordered point clouds to a surface. Classical methods (e.g., Poisson reconstruction) start to degrade in the presence of noisy and partial scans. Hence, deep learning based methods have recently been proposed to produce complete surfaces, even from partial scans. However, such data-driven methods struggle to generalize to new shapes with large geometric and topological variations. We present Points2Surf, a novel patch-based learning framework that produces accurate surfaces directly from raw scans without normals. Learning a prior over a combination of detailed local patches and coarse global information improves generalization performance and reconstruction accuracy. Our extensive comparison on both synthetic and real data demonstrates a clear advantage of our method over state-of-the-art alternatives on previously unseen classes (on average, Points2Surf brings down reconstruction error by 30\% over SPR and by 270\%+ over deep learning based SotA methods) at the cost of longer computation times and a slight increase in small-scale topological noise in some cases. Our source code, pre-trained model, and dataset are available on: https://github.com/ErlerPhilipp/points2surf

A key contributor to recent progress in 3D detection from single images is monocular depth estimation. Existing methods focus on how to leverage depth explicitly, by generating pseudo-pointclouds or providing attention cues for image features. More recent works leverage depth prediction as a pretraining task and fine-tune the depth representation while training it for 3D detection. However, the adaptation is insufficient and is limited in scale by manual labels. In this work, we propose to further align depth representation with the target domain in unsupervised fashions. Our methods leverage commonly available LiDAR or RGB videos during training time to fine-tune the depth representation, which leads to improved 3D detectors. Especially when using RGB videos, we show that our two-stage training by first generating pseudo-depth labels is critical because of the inconsistency in loss distribution between the two tasks. With either type of reference data, our multi-task learning approach improves over the state of the art on both KITTI and NuScenes, while matching the test-time complexity of its single task sub-network.

High-fidelity human 3D models can now be learned directly from videos, typically by combining a template-based surface model with neural representations. However, obtaining a template surface requires expensive multi-view capture systems, laser scans, or strictly controlled conditions. Previous methods avoid using a template but rely on a costly or ill-posed mapping from observation to canonical space. We propose a hybrid point-based representation for reconstructing animatable characters that does not require an explicit surface model, while being generalizable to novel poses. For a given video, our method automatically produces an explicit set of 3D points representing approximate canonical geometry, and learns an articulated deformation model that produces pose-dependent point transformations. The points serve both as a scaffold for high-frequency neural features and an anchor for efficiently mapping between observation and canonical space. We demonstrate on established benchmarks that our representation overcomes limitations of prior work operating in either canonical or in observation space. Moreover, our automatic point extraction approach enables learning models of human and animal characters alike, matching the performance of the methods using rigged surface templates despite being more general. Project website: https://lemonatsu.github.io/npc/

Recent years have seen a surprising connection between the physics of scattering amplitudes and a class of mathematical objects--the positive Grassmannian, positive loop Grassmannians, tree and loop Amplituhedra--which have been loosely referred to as "positive geometries". The connection between the geometry and physics is provided by a unique differential form canonically determined by the property of having logarithmic singularities (only) on all the boundaries of the space, with residues on each boundary given by the canonical form on that boundary. In this paper we initiate an exploration of "positive geometries" and "canonical forms" as objects of study in their own right in a more general mathematical setting. We give a precise definition of positive geometries and canonical forms, introduce general methods for finding forms for more complicated positive geometries from simpler ones, and present numerous examples of positive geometries in projective spaces, Grassmannians, and toric, cluster and flag varieties. We also illustrate a number of strategies for computing canonical forms which yield interesting representations for the forms associated with wide classes of positive geometries, ranging from the simplest Amplituhedra to new expressions for the volume of arbitrary convex polytopes.

This paper gives three formulas for the pseudoinverse of a matrix product A = CR. The first is sometimes correct, the second is always correct, and the third is almost never correct. But that third randomized pseudoinverse A^+_r may be very useful when A is a very large matrix. 1. A^+ = R^+C^+ when A = CR and C has independent columns and R has independent rows. 2. A^+ = (C^+CR)^+(CRR^+)^+ is always correct. 3. A^+_r = (P^TCR)^+P^TCRQ(CRQ)^+ = A^+ only when rank(P^TA) = rank(AQ) = rank(A) with A = CR.

We address the problem of regressing 3D human pose and shape from a single image, with a focus on 3D accuracy. The current best methods leverage large datasets of 3D pseudo-ground-truth (p-GT) and 2D keypoints, leading to robust performance. With such methods, we observe a paradoxical decline in 3D pose accuracy with increasing 2D accuracy. This is caused by biases in the p-GT and the use of an approximate camera projection model. We quantify the error induced by current camera models and show that fitting 2D keypoints and p-GT accurately causes incorrect 3D poses. Our analysis defines the invalid distances within which minimizing 2D and p-GT losses is detrimental. We use this to formulate a new loss Threshold-Adaptive Loss Scaling (TALS) that penalizes gross 2D and p-GT losses but not smaller ones. With such a loss, there are many 3D poses that could equally explain the 2D evidence. To reduce this ambiguity we need a prior over valid human poses but such priors can introduce unwanted bias. To address this, we exploit a tokenized representation of human pose and reformulate the problem as token prediction. This restricts the estimated poses to the space of valid poses, effectively providing a uniform prior. Extensive experiments on the EMDB and 3DPW datasets show that our reformulated keypoint loss and tokenization allows us to train on in-the-wild data while improving 3D accuracy over the state-of-the-art. Our models and code are available for research at https://tokenhmr.is.tue.mpg.de.

Hyperbolic space has proven to be well-suited for capturing hierarchical relations in data, such as trees and directed acyclic graphs. Prior work introduced the concept of entailment cones, which uses partial orders defined by nested cones in the Poincar\'e ball to model hierarchies. Here, we introduce the ``shadow cones" framework, a physics-inspired entailment cone construction. Specifically, we model partial orders as subset relations between shadows formed by a light source and opaque objects in hyperbolic space. The shadow cones framework generalizes entailment cones to a broad class of formulations and hyperbolic space models beyond the Poincar\'e ball. This results in clear advantages over existing constructions: for example, shadow cones possess better optimization properties over constructions limited to the Poincar\'e ball. Our experiments on datasets of various sizes and hierarchical structures show that shadow cones consistently and significantly outperform existing entailment cone constructions. These results indicate that shadow cones are an effective way to model partial orders in hyperbolic space, offering physically intuitive and novel insights about the nature of such structures.

We consider the problem of approximating a function from L^2 by an element of a given m-dimensional space V_m, associated with some feature map varphi, using evaluations of the function at random points x_1,dots,x_n. After recalling some results on optimal weighted least-squares using independent and identically distributed points, we consider weighted least-squares using projection determinantal point processes (DPP) or volume sampling. These distributions introduce dependence between the points that promotes diversity in the selected features varphi(x_i). We first provide a generalized version of volume-rescaled sampling yielding quasi-optimality results in expectation with a number of samples n = O(mlog(m)), that means that the expected L^2 error is bounded by a constant times the best approximation error in L^2. Also, further assuming that the function is in some normed vector space H continuously embedded in L^2, we further prove that the approximation is almost surely bounded by the best approximation error measured in the H-norm. This includes the cases of functions from L^infty or reproducing kernel Hilbert spaces. Finally, we present an alternative strategy consisting in using independent repetitions of projection DPP (or volume sampling), yielding similar error bounds as with i.i.d. or volume sampling, but in practice with a much lower number of samples. Numerical experiments illustrate the performance of the different strategies.

Illuminating the surface of a convex body with parallel beams of light in a given direction generates a shadow region. We prove sharp regularity results for the boundary of this shadow in every direction of illumination. Moreover, techniques are developed for investigating the regularity of the region generated by orthogonally projecting a convex set onto another. As an application we study the geometry and Hausdorff dimension of the singular set corresponding to a Monge-Ampere equation.