Fast Evaluation of Generalized Todd Polynomials: Applications to MacMahon's Partition Analysis and Integer Programming

The Todd polynomials $td_k=td_k(b_1,b_2,\dots,b_m)$ are defined by their generating functions $$\sum_{k\ge 0} td_k s^k = \prod_{i=1}^m \frac{b_i s}{e^{b_i s}-1}.$$ It appears as a basic block in Todd class of a toric variety, which is important in the theory of lattice polytopes and in number theory. We find generalized Todd polynomials arise naturally in MacMahon's partition analysis, especially in Erhart series computation.We give fast evaluation of generalized Todd polynomials for numerical $b_i$'s. In order to do so, we develop fast operations in the quotient ring $\mathbb{Z}_p[[x]]$ modulo $s^d$ for large prime $p$. As applications, i) we recompute the Ehrhart series of magic squares of order 6, which was first solved by the first named author. The running time is reduced from 70 days to about 1 day; ii) we give a polynomial time algorithm for Integer Linear Programming when the dimension is fixed, with a good performance.Comment: 2 table

Algebraic Volume for Polytope Arise from Ehrhart Theory

Volume computation for $d$-polytopes $\mathcal{P}$ is fundamental in mathematics. There are known volume computation algorithms, mostly based on triangulation or signed-decomposition of $\mathcal{P}$. We consider $\mathrm{cone}(\mathcal{P})$ as a lift of $\mathcal{P}$ in view of Ehrhart theory. By using technique from algebraic combinatorics, we obtain a volume algorithm using only signed simplicial cone decompositions of $\mathrm{cone}(\P)$. Each cone is associated with a simple algebraic volume formula. Summing them gives the volume of the polytope. Our volume formula applies to various kind of cases. In particular, we use it to explain the traditional triangulation method and Lawrence's signed decomposition method. Moreover, we give a completely new primal-dual method for volume computation. This solves the traditional problem in this area: All existing methods are hopelessly impractical for either the class of simple polytopes or the class of simplicial polytopes. Our method has a good performance in computer experiments.Comment: 26 pages, 4 figure

The Value of Backers’ Word-of-Mouth in Screening Crowdfunding Projects: An Empirical Investigation

Reward-based crowdfunding is an emerging financing channel for entrepreneurs to raise money for their innovative projects. How to screen the crowdfunding projects is critical for crowdfunding platform, project founder, and potential backers. This study aims to investigate whether backers’ word-of-mouth (WOM) is a valuable input to generate collective intelligence for project screening. Specially, we answer three questions. First, is backers’ WOM an effective signal for implementation performance of crowdfunding projects? Second, how do the WOM help screen projects during the fund-raising process? Third, which kind of comments (positive or negative) is more effective in screening crowdfunding projects? Research hypotheses were developed based on theories of collective intelligence and WOM communication. Using a cross section dataset and a panel dataset, we get the following findings. First, backers’ negative WOM can effectively predict project implementation performance, however positive WOM does not have that prediction power. The prediction power of positive and negative WOM differs significantly. One possible reason is that negative WOM does contain more information of project quality. Second, project with more accumulative negative WOM tend to attract fewer subsequent backers. However, accumulative positive WOM is not helpful for attracting more potential backers. We conclude that negative WOM is useful for project screening project, because it is a signal of project quality, and meanwhile it could prevent backers make subsequent investments

Disentangled Representation Learning

Disentangled Representation Learning (DRL) aims to learn a model capable of identifying and disentangling the underlying factors hidden in the observable data in representation form. The process of separating underlying factors of variation into variables with semantic meaning benefits in learning explainable representations of data, which imitates the meaningful understanding process of humans when observing an object or relation. As a general learning strategy, DRL has demonstrated its power in improving the model explainability, controlability, robustness, as well as generalization capacity in a wide range of scenarios such as computer vision, natural language processing, data mining etc. In this article, we comprehensively review DRL from various aspects including motivations, definitions, methodologies, evaluations, applications and model designs. We discuss works on DRL based on two well-recognized definitions, i.e., Intuitive Definition and Group Theory Definition. We further categorize the methodologies for DRL into four groups, i.e., Traditional Statistical Approaches, Variational Auto-encoder Based Approaches, Generative Adversarial Networks Based Approaches, Hierarchical Approaches and Other Approaches. We also analyze principles to design different DRL models that may benefit different tasks in practical applications. Finally, we point out challenges in DRL as well as potential research directions deserving future investigations. We believe this work may provide insights for promoting the DRL research in the community.Comment: 22 pages,9 figure

GICI-LIB: A GNSS/INS/Camera Integrated Navigation Library

Accurate navigation is essential for autonomous robots and vehicles. In recent years, the integration of the Global Navigation Satellite System (GNSS), Inertial Navigation System (INS), and camera has garnered considerable attention due to its robustness and high accuracy in diverse environments. In such systems, fully utilizing the role of GNSS is cumbersome because of the diverse choices of formulations, error models, satellite constellations, signal frequencies, and service types, which lead to different precision, robustness, and usage dependencies. To clarify the capacity of GNSS algorithms and accelerate the development efficiency of employing GNSS in multi-sensor fusion algorithms, we open source the GNSS/INS/Camera Integration Library (GICI-LIB), together with detailed documentation and a comprehensive land vehicle dataset. A factor graph optimization-based multi-sensor fusion framework is established, which combines almost all GNSS measurement error sources by fully considering temporal and spatial correlations between measurements. The graph structure is designed for flexibility, making it easy to form any kind of integration algorithm. For illustration, four Real-Time Kinematic (RTK)-based algorithms from GICI-LIB are evaluated using our dataset. Results confirm the potential of the GICI system to provide continuous precise navigation solutions in a wide spectrum of urban environments.Comment: Open-source: https://github.com/chichengcn/gici-open. This work has been submitted to the IEEE for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessibl

Constraining Depth Map Geometry for Multi-View Stereo: A Dual-Depth Approach with Saddle-shaped Depth Cells

Learning-based multi-view stereo (MVS) methods deal with predicting accurate depth maps to achieve an accurate and complete 3D representation. Despite the excellent performance, existing methods ignore the fact that a suitable depth geometry is also critical in MVS. In this paper, we demonstrate that different depth geometries have significant performance gaps, even using the same depth prediction error. Therefore, we introduce an ideal depth geometry composed of Saddle-Shaped Cells, whose predicted depth map oscillates upward and downward around the ground-truth surface, rather than maintaining a continuous and smooth depth plane. To achieve it, we develop a coarse-to-fine framework called Dual-MVSNet (DMVSNet), which can produce an oscillating depth plane. Technically, we predict two depth values for each pixel (Dual-Depth), and propose a novel loss function and a checkerboard-shaped selecting strategy to constrain the predicted depth geometry. Compared to existing methods,DMVSNet achieves a high rank on the DTU benchmark and obtains the top performance on challenging scenes of Tanks and Temples, demonstrating its strong performance and generalization ability. Our method also points to a new research direction for considering depth geometry in MVS.Comment: Accepted by ICCV 202

Receptors that bind to PEDF and their therapeutic roles in retinal diseases

Retinal neovascular, neurodegenerative, and inflammatory diseases represented by diabetic retinopathy are the main types of blinding eye disorders that continually cause the increased burden worldwide. Pigment epithelium-derived factor (PEDF) is an endogenous factor with multiple effects including neurotrophic activity, anti-angiogenesis, anti-tumorigenesis, and anti-inflammatory activity. PEDF activity depends on the interaction with the proteins on the cell surface. At present, seven independent receptors, including adipose triglyceride lipase, laminin receptor, lipoprotein receptor-related protein, plexin domain-containing 1, plexin domain-containing 2, F1-ATP synthase, and vascular endothelial growth factor receptor 2, have been demonstrated and confirmed to be high affinity receptors for PEDF. Understanding the interactions between PEDF and PEDF receptors, their roles in normal cellular metabolism and the response the initiate in disease will be accommodating for elucidating the ways in which inflammation, angiogenesis, and neurodegeneration exacerbate disease pathology. In this review, we firstly introduce PEDF receptors comprehensively, focusing particularly on their expression pattern, ligands, related diseases, and signal transduction pathways, respectively. We also discuss the interactive ways of PEDF and receptors to expand the prospective understanding of PEDF receptors in the diagnosis and treatment of retinal diseases

MeLo: Low-rank Adaptation is Better than Fine-tuning for Medical Image Diagnosis

The common practice in developing computer-aided diagnosis (CAD) models based on transformer architectures usually involves fine-tuning from ImageNet pre-trained weights. However, with recent advances in large-scale pre-training and the practice of scaling laws, Vision Transformers (ViT) have become much larger and less accessible to medical imaging communities. Additionally, in real-world scenarios, the deployments of multiple CAD models can be troublesome due to problems such as limited storage space and time-consuming model switching. To address these challenges, we propose a new method MeLo (Medical image Low-rank adaptation), which enables the development of a single CAD model for multiple clinical tasks in a lightweight manner. It adopts low-rank adaptation instead of resource-demanding fine-tuning. By fixing the weight of ViT models and only adding small low-rank plug-ins, we achieve competitive results on various diagnosis tasks across different imaging modalities using only a few trainable parameters. Specifically, our proposed method achieves comparable performance to fully fine-tuned ViT models on four distinct medical imaging datasets using about 0.17% trainable parameters. Moreover, MeLo adds only about 0.5MB of storage space and allows for extremely fast model switching in deployment and inference. Our source code and pre-trained weights are available on our website (https://absterzhu.github.io/melo.github.io/).Comment: 5 pages, 3 figure