The Incremental Multiresolution Matrix Factorization Algorithm

Multiresolution analysis and matrix factorization are foundational tools in computer vision. In this work, we study the interface between these two distinct topics and obtain techniques to uncover hierarchical block structure in symmetric matrices -- an important aspect in the success of many vision problems. Our new algorithm, the incremental multiresolution matrix factorization, uncovers such structure one feature at a time, and hence scales well to large matrices. We describe how this multiscale analysis goes much farther than what a direct global factorization of the data can identify. We evaluate the efficacy of the resulting factorizations for relative leveraging within regression tasks using medical imaging data. We also use the factorization on representations learned by popular deep networks, providing evidence of their ability to infer semantic relationships even when they are not explicitly trained to do so. We show that this algorithm can be used as an exploratory tool to improve the network architecture, and within numerous other settings in vision.Comment: Computer Vision and Pattern Recognition (CVPR) 2017, 10 page

Fast Temporal Wavelet Graph Neural Networks

Spatio-temporal signals forecasting plays an important role in numerous domains, especially in neuroscience and transportation. The task is challenging due to the highly intricate spatial structure, as well as the non-linear temporal dynamics of the network. To facilitate reliable and timely forecast for the human brain and traffic networks, we propose the Fast Temporal Wavelet Graph Neural Networks (FTWGNN) that is both time- and memory-efficient for learning tasks on timeseries data with the underlying graph structure, thanks to the theories of multiresolution analysis and wavelet theory on discrete spaces. We employ Multiresolution Matrix Factorization (MMF) (Kondor et al., 2014) to factorize the highly dense graph structure and compute the corresponding sparse wavelet basis that allows us to construct fast wavelet convolution as the backbone of our novel architecture. Experimental results on real-world PEMS-BAY, METR-LA traffic datasets and AJILE12 ECoG dataset show that FTWGNN is competitive with the state-of-the-arts while maintaining a low computational footprint. Our PyTorch implementation is publicly available at https://github.com/HySonLab/TWGNNComment: arXiv admin note: text overlap with arXiv:2111.0194

Thermography data fusion and non-negative matrix factorization for the evaluation of cultural heritage objects and buildings

The application of the thermal and infrared technology in different areas of research is considerably increasing. These applications involve nondestructive testing, medical analysis (computer aid diagnosis/detection—CAD), and arts and archeology, among many others. In the arts and archeology field, infrared technology provides significant contributions in terms of finding defects of possible impaired regions. This has been done through a wide range of different thermographic experiments and infrared methods. The proposed approach here focuses on application of some known factor analysis methods such as standard nonnegative matrix factorization (NMF) optimized by gradient-descent-based multiplicative rules (SNMF1) and standard NMF optimized by nonnegative least squares active-set algorithm (SNMF2) and eigen-decomposition approaches such as principal component analysis (PCA) in thermography, and candid covariance-free incremental principal component analysis in thermography to obtain the thermal features. On the one hand, these methods are usually applied as preprocessing before clustering for the purpose of segmentation of possible defects. On the other hand, a wavelet-based data fusion combines the data of each method with PCA to increase the accuracy of the algorithm. The quantitative assessment of these approaches indicates considerable segmentation along with the reasonable computational complexity. It shows the promising performance and demonstrated a confirmation for the outlined properties. In particular, a polychromatic wooden statue, a fresco, a painting on canvas, and a building were analyzed using the above-mentioned methods, and the accuracy of defect (or targeted) region segmentation up to 71.98%, 57.10%, 49.27%, and 68.53% was obtained, respectively

Tensor approximation in visualization and graphics

In this course, we will introduce the basic concepts of tensor approximation (TA) – a higher-order generalization of the SVD and PCA methods – as well as its applications to visual data representation, analysis and visualization, and bring the TA framework closer to visualization and computer graphics researchers and practitioners. The course will cover the theoretical background of TA methods, their properties and how to compute them, as well as practical applications of TA methods in visualization and computer graphics contexts. In a first theoretical part, the attendees will be instructed on the necessary mathematical background of TA methods to learn the basics skills of using and applying these new tools in the context of the representation of large multidimensional visual data. Specific and very noteworthy features of the TA framework are highlighted which can effectively be exploited for spatio-temporal multidimensional data representation and visualization purposes. In two application oriented sessions, compact TA data representation in scientific visualization and computer graphics as well as decomposition and reconstruction algorithms will be demonstrated. At the end of the course, the participants will have a good basic knowledge of TA methods along with a practical understanding of its potential application in visualization and graphics related projects

Making Queries Tractable on Big Data with Preprocessing

A query class is traditionally considered tractable if there exists a polynomial-time (PTIME) algorithm to answer its queries. When it comes to big data, however, PTIME al-gorithms often become infeasible in practice. A traditional and effective approach to coping with this is to preprocess data off-line, so that queries in the class can be subsequently evaluated on the data efficiently. This paper aims to pro-vide a formal foundation for this approach in terms of com-putational complexity. (1) We propose a set of Π-tractable queries, denoted by ΠT0Q, to characterize classes of queries that can be answered in parallel poly-logarithmic time (NC) after PTIME preprocessing. (2) We show that several natu-ral query classes are Π-tractable and are feasible on big data. (3) We also study a set ΠTQ of query classes that can be ef-fectively converted to Π-tractable queries by re-factorizing its data and queries for preprocessing. We introduce a form of NC reductions to characterize such conversions. (4) We show that a natural query class is complete for ΠTQ. (5) We also show that ΠT0Q ⊂ P unless P = NC, i.e., the set ΠT0Q of all Π-tractable queries is properly contained in the set P of all PTIME queries. Nonetheless, ΠTQ = P, i.e., all PTIME query classes can be made Π-tractable via proper re-factorizations. This work is a step towards understanding the tractability of queries in the context of big data. 1

RISE: An Incremental Trust-Region Method for Robust Online Sparse Least-Squares Estimation

Many point estimation problems in robotics, computer vision, and machine learning can be formulated as instances of the general problem of minimizing a sparse nonlinear sum-of-squares objective function. For inference problems of this type, each input datum gives rise to a summand in the objective function, and therefore performing online inference corresponds to solving a sequence of sparse nonlinear least-squares minimization problems in which additional summands are added to the objective function over time. In this paper, we present Robust Incremental least-Squares Estimation (RISE), an incrementalized version of the Powell's Dog-Leg numerical optimization method suitable for use in online sequential sparse least-squares minimization. As a trust-region method, RISE is naturally robust to objective function nonlinearity and numerical ill-conditioning and is provably globally convergent for a broad class of inferential cost functions (twice-continuously differentiable functions with bounded sublevel sets). Consequently, RISE maintains the speed of current state-of-the-art online sparse least-squares methods while providing superior reliability.United States. Office of Naval Research (Grant N00014-12-1-0093)United States. Office of Naval Research (Grant N00014-11-1-0688)United States. Office of Naval Research (Grant N00014-06-1-0043)United States. Office of Naval Research (Grant N00014-10-1-0936)United States. Air Force Research Laboratory (Contract FA8650-11-C-7137

Fluid flow estimation with multiscale ensemble filters based on motion measurements under location uncertainty

International audienceThis paper proposes a novel multi-scale fluid flow data assimilation approach, which integrates and complements the advantages of a Bayesian sequential assimilation technique, the Weighted Ensemble Kalman filter (WEnKF). The data assimilation proposed in this work incorporates measurement brought by an efficient multiscale stochastic formulation of the well-known Lucas-Kanade (LK) estimator. This estimator has the great advantage to provide uncertainties associated to the motion measurements at different scales. The proposed assimilation scheme benefits from this multiscale uncertainty information and enables to enforce a physically plausible dynamical consistency of the estimated motion fields along the image sequence. Experimental evaluations are presented on synthetic and real fluid flow sequences

Multipole Graph Neural Operator for Parametric Partial Differential Equations

One of the main challenges in using deep learning-based methods for simulating physical systems and solving partial differential equations (PDEs) is formulating physics-based data in the desired structure for neural networks. Graph neural networks (GNNs) have gained popularity in this area since graphs offer a natural way of modeling particle interactions and provide a clear way of discretizing the continuum models. However, the graphs constructed for approximating such tasks usually ignore long-range interactions due to unfavorable scaling of the computational complexity with respect to the number of nodes. The errors due to these approximations scale with the discretization of the system, thereby not allowing for generalization under mesh-refinement. Inspired by the classical multipole methods, we purpose a novel multi-level graph neural network framework that captures interaction at all ranges with only linear complexity. Our multi-level formulation is equivalent to recursively adding inducing points to the kernel matrix, unifying GNNs with multi-resolution matrix factorization of the kernel. Experiments confirm our multi-graph network learns discretization-invariant solution operators to PDEs and can be evaluated in linear time

Application of the Progressive Wavelet Correlation in Image Retrieving

In the paper we apply progressive wavelet correlation along with Fourier methods for searching and retrieving images stored in a database. The searching consists of three incremental steps, each of which quadruples the number of correlation points. The process can be halted at any stage if the intermediate results indicate that the correlation will not result in a match. We perform a series of image search experiments that cover the following scenarios: (A) the given image is present in the database; (B) copies of the given image are present but with different names; (C) similar (but not identical) images are present; and (D) the given image is not present. Experiments are performed with data bases up to 1000 images, using the Oracle database and the Matlab component Database Toolbox for operations with databases