Daily Papers

Using deep learning, 3D autonomous driving semantic segmentation has become a well-studied subject, with methods that can reach very high performance. Nonetheless, because of the limited size of the training datasets, these models cannot see every type of object and scene found in real-world applications. The ability to be reliable in these various unknown environments is called domain generalization. Despite its importance, domain generalization is relatively unexplored in the case of 3D autonomous driving semantic segmentation. To fill this gap, this paper presents the first benchmark for this application by testing state-of-the-art methods and discussing the difficulty of tackling Laser Imaging Detection and Ranging (LiDAR) domain shifts. We also propose the first method designed to address this domain generalization, which we call 3DLabelProp. This method relies on leveraging the geometry and sequentiality of the LiDAR data to enhance its generalization performances by working on partially accumulated point clouds. It reaches a mean Intersection over Union (mIoU) of 50.4% on SemanticPOSS and of 55.2% on PandaSet solid-state LiDAR while being trained only on SemanticKITTI, making it the state-of-the-art method for generalization (+5% and +33% better, respectively, than the second best method). The code for this method is available on GitHub: https://github.com/JulesSanchez/3DLabelProp.

Reliably identifying reinforced concrete defects (RCDs)plays a crucial role in assessing the structural integrity, traffic safety, and long-term durability of concrete bridges, which represent the most common bridge type worldwide. Nevertheless, available datasets for the recognition of RCDs are small in terms of size and class variety, which questions their usability in real-world scenarios and their role as a benchmark. Our contribution to this problem is "dacl10k", an exceptionally diverse RCD dataset for multi-label semantic segmentation comprising 9,920 images deriving from real-world bridge inspections. dacl10k distinguishes 12 damage classes as well as 6 bridge components that play a key role in the building assessment and recommending actions, such as restoration works, traffic load limitations or bridge closures. In addition, we examine baseline models for dacl10k which are subsequently evaluated. The best model achieves a mean intersection-over-union of 0.42 on the test set. dacl10k, along with our baselines, will be openly accessible to researchers and practitioners, representing the currently biggest dataset regarding number of images and class diversity for semantic segmentation in the bridge inspection domain.

This paper proposes a new framework to detect, segment, and estimate the localization of the eyes from a periocular Near-Infra-Red iris image under alcohol consumption. The purpose of the system is to measure the fitness for duty. Fitness systems allow us to determine whether a person is physically or psychologically able to perform their tasks. Our framework is based on an object detector trained from scratch to detect both eyes from a single image. Then, two efficient networks were used for semantic segmentation; a Criss-Cross attention network and DenseNet10, with only 122,514 and 210,732 parameters, respectively. These networks can find the pupil, iris, and sclera. In the end, the binary output eye mask is used for pupil and iris diameter estimation with high precision. Five state-of-the-art algorithms were used for this purpose. A mixed proposal reached the best results. A second contribution is establishing an alcohol behavior curve to detect the alcohol presence utilizing a stream of images captured from an iris instance. Also, a manually labeled database with more than 20k images was created. Our best method obtains a mean Intersection-over-Union of 94.54% with DenseNet10 with only 210,732 parameters and an error of only 1-pixel on average.

Foundation models have the potential to transform the landscape of remote sensing (RS) data analysis by enabling large computer vision models to be pre-trained on vast amounts of remote sensing data. These models can then be fine-tuned with small amounts of labeled training and applied to a variety of applications. Most existing foundation models are designed for high spatial resolution, cloud-free satellite imagery or photos, limiting their applicability in scenarios that require frequent temporal monitoring or broad spectral profiles. As a result, foundation models trained solely on cloud-free images have limited utility for applications that involve atmospheric variables or require atmospheric corrections. We introduce SatVision-TOA, a novel foundation model pre-trained on 14-band MODIS L1B Top-Of-Atmosphere (TOA) radiance imagery, addressing the need for models pre-trained to handle moderate- and coarse-resolution all-sky remote sensing data. The SatVision-TOA model is pre-trained using a Masked-Image-Modeling (MIM) framework and the SwinV2 architecture, and learns detailed contextual representations through self-supervised learning without the need for labels. It is a 3 billion parameter model that is trained on 100 million images. To our knowledge this is the largest foundation model trained solely on satellite RS imagery. Results show that SatVision-TOA achieves superior performance over baseline methods on downstream tasks such as 3D cloud retrieval. Notably, the model achieves a mean intersection over union (mIOU) of 0.46, a substantial improvement over the baseline mIOU of 0.22. Additionally, the rate of false negative results in the fine-tuning task were reduced by over 50% compared to the baseline. Our work advances pre-trained vision modeling for multispectral RS by learning from a variety of atmospheric and aerosol conditions to improve cloud and land surface monitoring.

End-to-end Network has become increasingly important in multi-tasking. One prominent example of this is the growing significance of a driving perception system in autonomous driving. This paper systematically studies an end-to-end perception network for multi-tasking and proposes several key optimizations to improve accuracy. First, the paper proposes efficient segmentation head and box/class prediction networks based on weighted bidirectional feature network. Second, the paper proposes automatically customized anchor for each level in the weighted bidirectional feature network. Third, the paper proposes an efficient training loss function and training strategy to balance and optimize network. Based on these optimizations, we have developed an end-to-end perception network to perform multi-tasking, including traffic object detection, drivable area segmentation and lane detection simultaneously, called HybridNets, which achieves better accuracy than prior art. In particular, HybridNets achieves 77.3 mean Average Precision on Berkeley DeepDrive Dataset, outperforms lane detection with 31.6 mean Intersection Over Union with 12.83 million parameters and 15.6 billion floating-point operations. In addition, it can perform visual perception tasks in real-time and thus is a practical and accurate solution to the multi-tasking problem. Code is available at https://github.com/datvuthanh/HybridNets.

Application of deep learning on histopathological whole slide images (WSIs) holds promise of improving diagnostic efficiency and reproducibility but is largely dependent on the ability to write computer code or purchase commercial solutions. We present a code-free pipeline utilizing free-to-use, open-source software (QuPath, DeepMIB, and FastPathology) for creating and deploying deep learning-based segmentation models for computational pathology. We demonstrate the pipeline on a use case of separating epithelium from stroma in colonic mucosa. A dataset of 251 annotated WSIs, comprising 140 hematoxylin-eosin (HE)-stained and 111 CD3 immunostained colon biopsy WSIs, were developed through active learning using the pipeline. On a hold-out test set of 36 HE and 21 CD3-stained WSIs a mean intersection over union score of 96.6% and 95.3% was achieved on epithelium segmentation. We demonstrate pathologist-level segmentation accuracy and clinical acceptable runtime performance and show that pathologists without programming experience can create near state-of-the-art segmentation solutions for histopathological WSIs using only free-to-use software. The study further demonstrates the strength of open-source solutions in its ability to create generalizable, open pipelines, of which trained models and predictions can seamlessly be exported in open formats and thereby used in external solutions. All scripts, trained models, a video tutorial, and the full dataset of 251 WSIs with ~31k epithelium annotations are made openly available at https://github.com/andreped/NoCodeSeg to accelerate research in the field.

There has been tremendous progress in multimodal Large Language Models (LLMs). Recent works have extended these models to video input with promising instruction following capabilities. However, an important missing piece is temporal localization. These models cannot accurately answer the "When?" questions. We identify three key aspects that limit their temporal localization capabilities: (i) time representation, (ii) architecture, and (iii) data. We address these shortcomings by proposing Language Instructed Temporal-Localization Assistant (LITA) with the following features: (1) We introduce time tokens that encode timestamps relative to the video length to better represent time in videos. (2) We introduce SlowFast tokens in the architecture to capture temporal information at fine temporal resolution. (3) We emphasize temporal localization data for LITA. In addition to leveraging existing video datasets with timestamps, we propose a new task, Reasoning Temporal Localization (RTL), along with the dataset, ActivityNet-RTL, for learning and evaluating this task. Reasoning temporal localization requires both the reasoning and temporal localization of Video LLMs. LITA demonstrates strong performance on this challenging task, nearly doubling the temporal mean intersection-over-union (mIoU) of baselines. In addition, we show that our emphasis on temporal localization also substantially improves video-based text generation compared to existing Video LLMs, including a 36% relative improvement of Temporal Understanding. Code is available at: https://github.com/NVlabs/LITA

Discussions of minimum parking requirement policies often include maps of parking lots, which are time consuming to construct manually. Open source datasets for such parking lots are scarce, particularly for US cities. This paper introduces the idea of using Near-Infrared (NIR) channels as input and several post-processing techniques to improve the prediction of off-street surface parking lots using satellite imagery. We constructed two datasets with 12,617 image-mask pairs each: one with 3-channel (RGB) and another with 4-channel (RGB + NIR). The datasets were used to train five deep learning models (OneFormer, Mask2Former, SegFormer, DeepLabV3, and FCN) for semantic segmentation, classifying images to differentiate between parking and non-parking pixels. Our results demonstrate that the NIR channel improved accuracy because parking lots are often surrounded by grass, even though the NIR channel needed to be upsampled from a lower resolution. Post-processing including eliminating erroneous holes, simplifying edges, and removing road and building footprints further improved the accuracy. Best model, OneFormer trained on 4-channel input and paired with post-processing techniques achieves a mean Intersection over Union (mIoU) of 84.9 percent and a pixel-wise accuracy of 96.3 percent.

Monitoring of land cover and land use is crucial in natural resources management. Automatic visual mapping can carry enormous economic value for agriculture, forestry, or public administration. Satellite or aerial images combined with computer vision and deep learning enable precise assessment and can significantly speed up change detection. Aerial imagery usually provides images with much higher pixel resolution than satellite data allowing more detailed mapping. However, there is still a lack of aerial datasets made for the segmentation, covering rural areas with a resolution of tens centimeters per pixel, manual fine labels, and highly publicly important environmental instances like buildings, woods, water, or roads. Here we introduce LandCover.ai (Land Cover from Aerial Imagery) dataset for semantic segmentation. We collected images of 216.27 sq. km rural areas across Poland, a country in Central Europe, 39.51 sq. km with resolution 50 cm per pixel and 176.76 sq. km with resolution 25 cm per pixel and manually fine annotated four following classes of objects: buildings, woodlands, water, and roads. Additionally, we report simple benchmark results, achieving 85.56% of mean intersection over union on the test set. It proves that the automatic mapping of land cover is possible with a relatively small, cost-efficient, RGB-only dataset. The dataset is publicly available at https://landcover.ai.linuxpolska.com/

The escalating global concern over extensive food wastage necessitates innovative solutions to foster a net-zero lifestyle and reduce emissions. The LILA home composter presents a convenient means of recycling kitchen scraps and daily food waste into nutrient-rich, high-quality compost. To capture the nutritional information of the produced compost, we have created and annotated a large high-resolution image dataset of kitchen food waste with segmentation masks of 19 nutrition-rich categories. Leveraging this dataset, we benchmarked four state-of-the-art semantic segmentation models on food waste segmentation, contributing to the assessment of compost quality of Nitrogen, Phosphorus, or Potassium. The experiments demonstrate promising results of using segmentation models to discern food waste produced in our daily lives. Based on the experiments, SegFormer, utilizing MIT-B5 backbone, yields the best performance with a mean Intersection over Union (mIoU) of 67.09. Class-based results are also provided to facilitate further analysis of different food waste classes.

Autonomous agents often require accurate methods for detecting and localizing changes in their environment, particularly when observations are captured from unconstrained and inconsistent viewpoints. We propose a novel label-free, pose-agnostic change detection method that integrates information from multiple viewpoints to construct a change-aware 3D Gaussian Splatting (3DGS) representation of the scene. With as few as 5 images of the post-change scene, our approach can learn an additional change channel in a 3DGS and produce change masks that outperform single-view techniques. Our change-aware 3D scene representation additionally enables the generation of accurate change masks for unseen viewpoints. Experimental results demonstrate state-of-the-art performance in complex multi-object scenes, achieving a 1.7x and 1.5x improvement in Mean Intersection Over Union and F1 score respectively over other baselines. We also contribute a new real-world dataset to benchmark change detection in diverse challenging scenes in the presence of lighting variations.

Accurately segmenting different organs from medical images is a critical prerequisite for computer-assisted diagnosis and intervention planning. This study proposes a deep learning-based approach for segmenting various organs from CT and MRI scans and classifying diseases. Our study introduces a novel technique integrating momentum within residual blocks for enhanced training dynamics in medical image analysis. We applied our method in two distinct tasks: segmenting liver, lung, & colon data and classifying abdominal pelvic CT and MRI scans. The proposed approach has shown promising results, outperforming state-of-the-art methods on publicly available benchmarking datasets. For instance, in the lung segmentation dataset, our approach yielded significant enhancements over the TransNetR model, including a 5.72% increase in dice score, a 5.04% improvement in mean Intersection over Union (mIoU), an 8.02% improvement in recall, and a 4.42% improvement in precision. Hence, incorporating momentum led to state-of-the-art performance in both segmentation and classification tasks, representing a significant advancement in the field of medical imaging.

Land cover (LC) segmentation plays a critical role in various applications, including environmental analysis and natural disaster management. However, generating accurate LC maps is a complex and time-consuming task that requires the expertise of multiple annotators and regular updates to account for environmental changes. In this work, we introduce SPADA, a framework for fuel map delineation that addresses the challenges associated with LC segmentation using sparse annotations and domain adaptation techniques for semantic segmentation. Performance evaluations using reliable ground truths, such as LUCAS and Urban Atlas, demonstrate the technique's effectiveness. SPADA outperforms state-of-the-art semantic segmentation approaches as well as third-party products, achieving a mean Intersection over Union (IoU) score of 42.86 and an F1 score of 67.93 on Urban Atlas and LUCAS, respectively.

Self-attention is of vital importance in semantic segmentation as it enables modeling of long-range context, which translates into improved performance. We argue that it is equally important to model short-range context, especially to tackle cases where not only the regions of interest are small and ambiguous, but also when there exists an imbalance between the semantic classes. To this end, we propose Masked Supervised Learning (MaskSup), an effective single-stage learning paradigm that models both short- and long-range context, capturing the contextual relationships between pixels via random masking. Experimental results demonstrate the competitive performance of MaskSup against strong baselines in both binary and multi-class segmentation tasks on three standard benchmark datasets, particularly at handling ambiguous regions and retaining better segmentation of minority classes with no added inference cost. In addition to segmenting target regions even when large portions of the input are masked, MaskSup is also generic and can be easily integrated into a variety of semantic segmentation methods. We also show that the proposed method is computationally efficient, yielding an improved performance by 10\% on the mean intersection-over-union (mIoU) while requiring 3times less learnable parameters.

Accurate computer-aided polyp detection and segmentation during colonoscopy examinations can help endoscopists resect abnormal tissue and thereby decrease chances of polyps growing into cancer. Towards developing a fully automated model for pixel-wise polyp segmentation, we propose ResUNet++, which is an improved ResUNet architecture for colonoscopic image segmentation. Our experimental evaluations show that the suggested architecture produces good segmentation results on publicly available datasets. Furthermore, ResUNet++ significantly outperforms U-Net and ResUNet, two key state-of-the-art deep learning architectures, by achieving high evaluation scores with a dice coefficient of 81.33%, and a mean Intersection over Union (mIoU) of 79.27% for the Kvasir-SEG dataset and a dice coefficient of 79.55%, and a mIoU of 79.62% with CVC-612 dataset.

Purpose: The capacity to isolate and recognize individual characters from facsimile images of papyrus manuscripts yields rich opportunities for digital analysis. For this reason the `ICDAR 2023 Competition on Detection and Recognition of Greek Letters on Papyri' was held as part of the 17th International Conference on Document Analysis and Recognition. This paper discusses our submission to the competition. Methods: We used an ensemble of YOLOv8 models to detect and classify individual characters and employed two different approaches for refining the character predictions, including a transformer based DeiT approach and a ResNet-50 model trained on a large corpus of unlabelled data using SimCLR, a self-supervised learning method. Results: Our submission won the recognition challenge with a mAP of 42.2%, and was runner-up in the detection challenge with a mean average precision (mAP) of 51.4%. At the more relaxed intersection over union threshold of 0.5, we achieved the highest mean average precision and mean average recall results for both detection and classification. Conclusion: The results demonstrate the potential for these techniques for automated character recognition on historical manuscripts. We ran the prediction pipeline on more than 4,500 images from the Oxyrhynchus Papyri to illustrate the utility of our approach, and we release the results publicly in multiple formats.

This study systematically performed an extensive real-world evaluation of the performances of all configurations of YOLOv8, YOLOv9, YOLOv10, YOLO11( or YOLOv11), and YOLOv12 object detection algorithms in terms of precision, recall, mean Average Precision at 50\% Intersection over Union (mAP@50), and computational speeds including pre-processing, inference, and post-processing times immature green apple (or fruitlet) detection in commercial orchards. Additionally, this research performed and validated in-field counting of the fruitlets using an iPhone and machine vision sensors. Among the configurations, YOLOv12l recorded the highest recall rate at 0.90, compared to all other configurations of YOLO models. Likewise, YOLOv10x achieved the highest precision score of 0.908, while YOLOv9 Gelan-c attained a precision of 0.903. Analysis of [email protected] revealed that YOLOv9 Gelan-base and YOLOv9 Gelan-e reached peak scores of 0.935, with YOLO11s and YOLOv12l following closely at 0.933 and 0.931, respectively. For counting validation using images captured with an iPhone 14 Pro, the YOLO11n configuration demonstrated outstanding accuracy, recording RMSE values of 4.51 for Honeycrisp, 4.59 for Cosmic Crisp, 4.83 for Scilate, and 4.96 for Scifresh; corresponding MAE values were 4.07, 3.98, 7.73, and 3.85. Similar performance trends were observed with RGB-D sensor data. Moreover, sensor-specific training on Intel Realsense data significantly enhanced model performance. YOLOv11n achieved highest inference speed of 2.4 ms, outperforming YOLOv8n (4.1 ms), YOLOv9 Gelan-s (11.5 ms), YOLOv10n (5.5 ms), and YOLOv12n (4.6 ms), underscoring its suitability for real-time object detection applications. (YOLOv12 architecture, YOLOv11 Architecture, YOLOv12 object detection, YOLOv11 object detecion, YOLOv12 segmentation)

Intersection over Union (IoU) is the most popular evaluation metric used in the object detection benchmarks. However, there is a gap between optimizing the commonly used distance losses for regressing the parameters of a bounding box and maximizing this metric value. The optimal objective for a metric is the metric itself. In the case of axis-aligned 2D bounding boxes, it can be shown that IoU can be directly used as a regression loss. However, IoU has a plateau making it infeasible to optimize in the case of non-overlapping bounding boxes. In this paper, we address the weaknesses of IoU by introducing a generalized version as both a new loss and a new metric. By incorporating this generalized IoU (GIoU) as a loss into the state-of-the art object detection frameworks, we show a consistent improvement on their performance using both the standard, IoU based, and new, GIoU based, performance measures on popular object detection benchmarks such as PASCAL VOC and MS COCO.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids clustering. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not exist for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains and applications. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm that achieve an O(k)-fold speedup in the second ("SWAP") phase of the algorithm, but will still find the same results as the original PAM algorithm. If we relax the choice of swaps performed (while retaining comparable quality), we can further accelerate the algorithm by eagerly performing additional swaps in each iteration. With the substantially faster SWAP, we can now explore faster initialization strategies, because (i) the classic ("BUILD") initialization now becomes the bottleneck, and (ii) our swap is fast enough to compensate for worse starting conditions. We also show how the CLARA and CLARANS algorithms benefit from the proposed modifications. While we do not study the parallelization of our approach in this work, it can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100,200, we observed a 458x respectively 1191x speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets, and in particular to higher k.

Clustering non-Euclidean data is difficult, and one of the most used algorithms besides hierarchical clustering is the popular algorithm Partitioning Around Medoids (PAM), also simply referred to as k-medoids. In Euclidean geometry the mean-as used in k-means-is a good estimator for the cluster center, but this does not hold for arbitrary dissimilarities. PAM uses the medoid instead, the object with the smallest dissimilarity to all others in the cluster. This notion of centrality can be used with any (dis-)similarity, and thus is of high relevance to many domains such as biology that require the use of Jaccard, Gower, or more complex distances. A key issue with PAM is its high run time cost. We propose modifications to the PAM algorithm to achieve an O(k)-fold speedup in the second SWAP phase of the algorithm, but will still find the same results as the original PAM algorithm. If we slightly relax the choice of swaps performed (at comparable quality), we can further accelerate the algorithm by performing up to k swaps in each iteration. With the substantially faster SWAP, we can now also explore alternative strategies for choosing the initial medoids. We also show how the CLARA and CLARANS algorithms benefit from these modifications. It can easily be combined with earlier approaches to use PAM and CLARA on big data (some of which use PAM as a subroutine, hence can immediately benefit from these improvements), where the performance with high k becomes increasingly important. In experiments on real data with k=100, we observed a 200-fold speedup compared to the original PAM SWAP algorithm, making PAM applicable to larger data sets as long as we can afford to compute a distance matrix, and in particular to higher k (at k=2, the new SWAP was only 1.5 times faster, as the speedup is expected to increase with k).

In object detection, an intersection over union (IoU) threshold is required to define positives and negatives. An object detector, trained with low IoU threshold, e.g. 0.5, usually produces noisy detections. However, detection performance tends to degrade with increasing the IoU thresholds. Two main factors are responsible for this: 1) overfitting during training, due to exponentially vanishing positive samples, and 2) inference-time mismatch between the IoUs for which the detector is optimal and those of the input hypotheses. A multi-stage object detection architecture, the Cascade R-CNN, is proposed to address these problems. It consists of a sequence of detectors trained with increasing IoU thresholds, to be sequentially more selective against close false positives. The detectors are trained stage by stage, leveraging the observation that the output of a detector is a good distribution for training the next higher quality detector. The resampling of progressively improved hypotheses guarantees that all detectors have a positive set of examples of equivalent size, reducing the overfitting problem. The same cascade procedure is applied at inference, enabling a closer match between the hypotheses and the detector quality of each stage. A simple implementation of the Cascade R-CNN is shown to surpass all single-model object detectors on the challenging COCO dataset. Experiments also show that the Cascade R-CNN is widely applicable across detector architectures, achieving consistent gains independently of the baseline detector strength. The code will be made available at https://github.com/zhaoweicai/cascade-rcnn.

In statistics, independent, identically distributed random samples do not carry a natural ordering, and their statistics are typically invariant with respect to permutations of their order. Thus, an n-sample in a space M can be considered as an element of the quotient space of M^n modulo the permutation group. The present paper takes this definition of sample space and the related concept of orbit types as a starting point for developing a geometric perspective on statistics. We aim at deriving a general mathematical setting for studying the behavior of empirical and population means in spaces ranging from smooth Riemannian manifolds to general stratified spaces. We fully describe the orbifold and path-metric structure of the sample space when M is a manifold or path-metric space, respectively. These results are non-trivial even when M is Euclidean. We show that the infinite sample space exists in a Gromov-Hausdorff type sense and coincides with the Wasserstein space of probability distributions on M. We exhibit Fr\'echet means and k-means as metric projections onto 1-skeleta or k-skeleta in Wasserstein space, and we define a new and more general notion of polymeans. This geometric characterization via metric projections applies equally to sample and population means, and we use it to establish asymptotic properties of polymeans such as consistency and asymptotic normality.

This paper presents a new, provably-convergent algorithm for computing the flag-mean and flag-median of a set of points on a flag manifold under the chordal metric. The flag manifold is a mathematical space consisting of flags, which are sequences of nested subspaces of a vector space that increase in dimension. The flag manifold is a superset of a wide range of known matrix spaces, including Stiefel and Grassmanians, making it a general object that is useful in a wide variety computer vision problems. To tackle the challenge of computing first order flag statistics, we first transform the problem into one that involves auxiliary variables constrained to the Stiefel manifold. The Stiefel manifold is a space of orthogonal frames, and leveraging the numerical stability and efficiency of Stiefel-manifold optimization enables us to compute the flag-mean effectively. Through a series of experiments, we show the competence of our method in Grassmann and rotation averaging, as well as principal component analysis. We release our source code under https://github.com/nmank/FlagAveraging.

This paper studies the role of over-parametrization in solving non-convex optimization problems. The focus is on the important class of low-rank matrix sensing, where we propose an infinite hierarchy of non-convex problems via the lifting technique and the Burer-Monteiro factorization. This contrasts with the existing over-parametrization technique where the search rank is limited by the dimension of the matrix and it does not allow a rich over-parametrization of an arbitrary degree. We show that although the spurious solutions of the problem remain stationary points through the hierarchy, they will be transformed into strict saddle points (under some technical conditions) and can be escaped via local search methods. This is the first result in the literature showing that over-parametrization creates a negative curvature for escaping spurious solutions. We also derive a bound on how much over-parametrization is requited to enable the elimination of spurious solutions.

Let F_{k,d}(n) be the maximal size of a set {A}subseteq [n] such that the equation \[a_1a_2\dots a_k=x^d, \; a_1<a_2<\ldots<a_k\] has no solution with a_1,a_2,ldots,a_kA and integer x. Erdos, S\'ark\"ozy and T. S\'os studied F_{k,2}, and gave bounds when k=2,3,4,6 and also in the general case. We study the problem for d=3, and provide bounds for k=2,3,4,6 and 9, furthermore, in the general case, as well. In particular, we refute an 18 years old conjecture of Verstra\"ete. We also introduce another function f_{k,d} closely related to F_{k,d}: While the original problem requires a_1, ldots , a_k to all be distinct, we can relax this and only require that the multiset of the a_i's cannot be partitioned into d-tuples where each d-tuple consists of d copies of the same number.