Bridging Convex and Nonconvex Optimization in Robust PCA: Noise, Outliers, and Missing Data

This paper delivers improved theoretical guarantees for the convex programming approach in low-rank matrix estimation, in the presence of (1) random noise, (2) gross sparse outliers, and (3) missing data. This problem, often dubbed as robust principal component analysis (robust PCA), finds applications in various domains. Despite the wide applicability of convex relaxation, the available statistical support (particularly the stability analysis vis-a-vis random noise) remains highly suboptimal, which we strengthen in this paper. When the unknown matrix is well-conditioned, incoherent, and of constant rank, we demonstrate that a principled convex program achieves near-optimal statistical accuracy, in terms of both the Euclidean loss and the $\ell_{\infty}$ loss. All of this happens even when nearly a constant fraction of observations are corrupted by outliers with arbitrary magnitudes. The key analysis idea lies in bridging the convex program in use and an auxiliary nonconvex optimization algorithm, and hence the title of this paper

Inference and Uncertainty Quantification for Noisy Matrix Completion

Noisy matrix completion aims at estimating a low-rank matrix given only partial and corrupted entries. Despite substantial progress in designing efficient estimation algorithms, it remains largely unclear how to assess the uncertainty of the obtained estimates and how to perform statistical inference on the unknown matrix (e.g.~constructing a valid and short confidence interval for an unseen entry). This paper takes a step towards inference and uncertainty quantification for noisy matrix completion. We develop a simple procedure to compensate for the bias of the widely used convex and nonconvex estimators. The resulting de-biased estimators admit nearly precise non-asymptotic distributional characterizations, which in turn enable optimal construction of confidence intervals\,/\,regions for, say, the missing entries and the low-rank factors. Our inferential procedures do not rely on sample splitting, thus avoiding unnecessary loss of data efficiency. As a byproduct, we obtain a sharp characterization of the estimation accuracy of our de-biased estimators, which, to the best of our knowledge, are the first tractable algorithms that provably achieve full statistical efficiency (including the preconstant). The analysis herein is built upon the intimate link between convex and nonconvex optimization --- an appealing feature recently discovered by \cite{chen2019noisy}.Comment: published at Proceedings of the National Academy of Sciences Nov 2019, 116 (46) 22931-2293

MetaViewer: Towards A Unified Multi-View Representation

Existing multi-view representation learning methods typically follow a specific-to-uniform pipeline, extracting latent features from each view and then fusing or aligning them to obtain the unified object representation. However, the manually pre-specify fusion functions and view-private redundant information mixed in features potentially degrade the quality of the derived representation. To overcome them, we propose a novel bi-level-optimization-based multi-view learning framework, where the representation is learned in a uniform-to-specific manner. Specifically, we train a meta-learner, namely MetaViewer, to learn fusion and model the view-shared meta representation in outer-level optimization. Start with this meta representation, view-specific base-learners are then required to rapidly reconstruct the corresponding view in inner-level. MetaViewer eventually updates by observing reconstruction processes from uniform to specific over all views, and learns an optimal fusion scheme that separates and filters out view-private information. Extensive experimental results in downstream tasks such as classification and clustering demonstrate the effectiveness of our method.Comment: 8 pages, 5 figures, conferenc

Tri-MipRF: Tri-Mip Representation for Efficient Anti-Aliasing Neural Radiance Fields

Despite the tremendous progress in neural radiance fields (NeRF), we still face a dilemma of the trade-off between quality and efficiency, e.g., MipNeRF presents fine-detailed and anti-aliased renderings but takes days for training, while Instant-ngp can accomplish the reconstruction in a few minutes but suffers from blurring or aliasing when rendering at various distances or resolutions due to ignoring the sampling area. To this end, we propose a novel Tri-Mip encoding that enables both instant reconstruction and anti-aliased high-fidelity rendering for neural radiance fields. The key is to factorize the pre-filtered 3D feature spaces in three orthogonal mipmaps. In this way, we can efficiently perform 3D area sampling by taking advantage of 2D pre-filtered feature maps, which significantly elevates the rendering quality without sacrificing efficiency. To cope with the novel Tri-Mip representation, we propose a cone-casting rendering technique to efficiently sample anti-aliased 3D features with the Tri-Mip encoding considering both pixel imaging and observing distance. Extensive experiments on both synthetic and real-world datasets demonstrate our method achieves state-of-the-art rendering quality and reconstruction speed while maintaining a compact representation that reduces 25% model size compared against Instant-ngp.Comment: Accepted to ICCV 2023 Project page: https://wbhu.github.io/projects/Tri-MipR

Do We Fully Understand Students' Knowledge States? Identifying and Mitigating Answer Bias in Knowledge Tracing

Knowledge tracing (KT) aims to monitor students' evolving knowledge states through their learning interactions with concept-related questions, and can be indirectly evaluated by predicting how students will perform on future questions. In this paper, we observe that there is a common phenomenon of answer bias, i.e., a highly unbalanced distribution of correct and incorrect answers for each question. Existing models tend to memorize the answer bias as a shortcut for achieving high prediction performance in KT, thereby failing to fully understand students' knowledge states. To address this issue, we approach the KT task from a causality perspective. A causal graph of KT is first established, from which we identify that the impact of answer bias lies in the direct causal effect of questions on students' responses. A novel COunterfactual REasoning (CORE) framework for KT is further proposed, which separately captures the total causal effect and direct causal effect during training, and mitigates answer bias by subtracting the latter from the former in testing. The CORE framework is applicable to various existing KT models, and we implement it based on the prevailing DKT, DKVMN, and AKT models, respectively. Extensive experiments on three benchmark datasets demonstrate the effectiveness of CORE in making the debiased inference for KT.Comment: 13 page

(<b><i>α, β</i></b>) - Lock Resolution Method of Linguistic Truth-Valued Intuitionistic Fuzzy First-order Logic

Automated reasoning is an important research direction in artificial intelligence, and resolution method is an efficient logical reasoning tool, it deserves to be studied. This paper establishes linguistic truth-valued intuitionistic fuzzy first-order logic (LTV-IFFL) system. To improve the resolution efficiency, the (αβ)-lock resolution method is introduced into LTV-IFFL system and requires that the resolution literal is the literal with the smallest lock number. Its soundness and completeness are proved. Finally, we give the (αβ) -locked resolution of the LTV-IFFL and apply it to the example.</p

Synthesis and Electrochemical Performance of Polyacrylonitrile Carbon Nanostructure Microspheres for Supercapacitor Application

Polyacrylonitrile (PAN) carbon nanostructure microspheres (CNM) with the average particle size of 200 nm were prepared in the range of 500 to 800°C. The precursors of CNM were obtained through soap-free emulsion polymerization followed by freeze drying, oxidative stabilization, and half-carbonization. KOH was employed as the activation agent of the precursor material, and the ratio between KOH and the precursor was selected as 2 : 1. The element content, pore structure, nitrogen-containing functional groups, and microstructure characterization were characterized via elemental analysis, N2 adsorption at low temperature, X-ray photoelectron spectroscopy (XPS), scanning electron microscopy (SEM), and transmission electron microscopy (TEM), and the electrochemical properties were examined as well. The results revealed that the CNM displayed specific surface area as high as 2134 m2/g and the total pore volume could reach 2.01 cm3/g when the activation temperature was 700°C. Furthermore, its specific capacitance in 3 M KOH and 1 M organic electrolyte could reach 311 F/g and 179 F/g, respectively. And, also, abundant functional groups of N-5 and N-6 were rich in the surface of the material, which could cause Faraday reaction and got the increasing specific capacitance via improvement of the wettability of the electrode material

SIMULATION OF THE SEGMENT FILLING INSERTION FABRICS AT THE YARN LEVEL

Fabrics with segment filling insertion are finding application in several traditional luxurious textiles, clothing, and in the latest time as well for smart textiles. Segment filling allows the integration of conductive yarns for contacting areas, keeping the textile character of the structures. This work presents a method for 3D modeling woven structures with segment filling at the yarn level. The pattern image is analyzed by an image processing tool, written in Python, and used to create the initial weaving information. After that, the different regions are filled with suitable preselected weave types, such as plain, twill, or others. Finally, this data is used to compute the 3D coordinates of the weft and warp yarns, and saved in a suitable format. The 3D visualization is done by the TexMind Viewer, which allows its advanced version export in various formats for FEM, CFD, and other computations

The New Method and Application of Friction Torque for Extended Reach Well

The extended reach well has the characteristics of long horizontal displacement, big hold angle and long open hole. The forecast and control of friction torque is one of the key factors of successful drilling of extended reach well. Based on the characteristics of extended reach well, considering the effect of drill string stiffness and drill string buckling on the friction & torque of the tube, a modified 3D soft-string calculation model for friction & torque of the extended reach well is proposed. The corresponding software has been programmed, and the model is applied in well A. The calculation result shows that the new method’s calculation result is consistent with the experimental result, and the torque and hook load errors are within 10%. The method could satisfy the engineering requirement, which provide a good guidance for the friction & torque analysis in the process of profile optimal design and drilling operation for the extended reach well. Key words: Extended reach well; Friction＆torque; Drill string stiffness; Drill string buckling; Drilling operatio