Multivariate Hawkes Processes for Large-Scale Inference

Low rank approximation of matrices is an important tool in machine learning. Given a data matrix, low rank approximation helps to find factors, patterns, and provides concise representations for the data. Research on low rank approximation usually focuses on real matrices. However, in many applications data are binary (categorical) rather than continuous. This leads to the problem of low rank approximation of binary matrices. Here we are given a d x n binary matrix A and a small integer k < d. The goal is to find two binary matrices U and V of sizes d x k and k x n respectively, so that the Frobenius norm of A - U V is minimized. There are two models of this problem, depending on the definition of the dot product of binary vectors: The GF(2) model and the Boolean semiring model. Unlike low rank approximation of a real matrix which can be efficiently solved by Singular Value Decomposition, we show that approximation of a binary matrix is NP-hard, even for k=1.\nIn this paper, our main concern is the problem of Column Subset Selection (CSS), in which the low rank matrix U must be formed by k columns of the data matrix, and we are interested in the approximation ratio achievable by CSS for binary matrices. For the GF(2) model, we show that CSS has approximation ratio bounded by k/2+1+k/(2(2^k-1)) and this is asymptotically tight. For the Boolean model, it turns out that CSS is no longer sufficient to obtain a bound. We then develop a Generalized CSS (GCSS) procedure in which the columns of U are generated from Boolean formulas operating bitwise on selected columns of the data matrix. We show that the approximation ratio achieved by GCSS is bounded by 2^(k-1)+1, and argue that an exponential dependency on k is seems inherent.

Acoustic modeling using mixtures of multivariate Gaussians is the prevalent approach for many speech processing problems. Computing likelihoods against a large set of Gaussians is required as a part of many speech processing systems and it is the computationally dominant phase for LVCSR systems. We express the likelihood computation as a multiplication of matrices representing augmented feature vectors and Gaussian parameters. The computational gain of this approach over traditional methods is by exploiting the structure of these matrices and efficient implementation of their multiplication. In particular, we explore direct low-rank approximation of the Gaussian parameter matrix and indirect derivation of low-rank factors of the Gaussian parameter matrix by optimum approximation of the likelihood matrix. We show that both the methods lead to similar speedups but the latter leads to far lesser impact on the recognition accuracy. Experiments on a 1138 word vocabulary RM1 task using Sphinx 3.7 system show that, for a typical case the matrix multiplication approach leads to overall speedup of 46%. Both the low-rank approximation methods increase the speedup to around 60%, with the former method increasing the word error rate (WER) from 3.2% to 6.6%, while the latter increases the WER from 3.2% to 3.5%.

Least Squares Support Vector Machine (LS-SVM) has been recently applied to non-parametric identification of Linear Parameter Varying (LPV) systems, described by the AutoRegressive with eXogenous input (ARX). However, the online application of LPV-ARX system in the LS-SVM setting requires high computational time, related to the number of training data used to compute the coefficients of the identified model, limiting the possibility to use the method to real-time applications. In this paper, the authors propose the Low-Rank (LR) matrix approximation and a pruning based approach to compute a sparse solution. In particular, the pruning algorithm is considered to compute off-line a sparse solution of Lagrangian multipliers and then speed up the testing stage, whereas the LR matrix approximation allows to speed up the training stage. The proposed approach has been tested by identifying a subsystem of a vehicle powertrain model by the input/output data collected from the simulation model. The proposed approach has been compared with respect to the standard approach based on LS-SVM. The methods are tested on the considered real-world problem and the proposed approach permits to reduce the execution time of about 77% on average in the considered identification problem, corresponding to a degradation of the identification result less than 0.2% with respect to the standard solution.

In this paper, we introduce a method for multivariate function approximation using function evaluations, Chebyshev polynomials, and tensor-based compression techniques via the Tucker format. We develop novel randomized techniques to accomplish the tensor compression, provide a detailed analysis of the computational costs, provide insight into the error of the resulting approximations, and discuss the benefits of the proposed approaches. We also apply the tensor-based function approximation to develop low-rank matrix approximations to kernel matrices that describe pairwise interactions between two sets of points; the resulting low-rank approximations are efficient to compute and store (the complexity is linear in the number of points). We present an adaptive version of the function and kernel approximation that determines an approximation that satisfies a user-specified relative error over a set of random points. We extend our approach to the case where the kernel requires repeated evaluations for many values of (hyper)parameters that govern the kernel. We give detailed numerical experiments on example problems involving multivariate function approximation, low-rank matrix approximations of kernel matrices involving well-separated clusters of sources and target points, and a global low-rank approximation of kernel matrices with an application to Gaussian processes. We observe speedups up to 18X over standard matrix-based approaches.

We describe a local low rank approximation method based on super-rays for light field compression. Super-rays can be seen as a set of super-pixels that are coherent across all light field views. A super-ray based disparity estimation method is proposed using a low rank prior, in order to be able to align all the super-pixels forming each super-ray. A dedicated super-ray construction method is described that constrains the super-pixels forming a given super-ray to be all of the same shape and size, dealing with occlusions. This constraint is needed so that the super-rays can be used as a support of angular dimensionality reduction based on low rank matrix approximation. A low rank matrix approximation is then computed on the disparity compensated super-rays using a singular value decomposition (SVD). A coding algorithm is then described for the different components of the resulting low rank approximation. Experimental results show performance gains compared with two reference light field coding schemes (HEVC-based scheme and JPEG-Pleno VM 1.1).

Detecting anomalous traffic is a critical task for advanced Internet management. The traditional approaches based on Principal Component Analysis (PCA) are effective only when the corruption is caused by small additive i.i.d. Gaussian noise. The recent Direct Robust Matrix Factorization (DRMF) is proven to be more robust and accurate in anomaly detection, but it incurs a high computation cost due to its need of singular value decomposition (SVD) for low-rank matrix approximation and the iterative use of SVD execution to find the final solution. To enable the anomaly detection for large traffic matrix with the use of DRMF, we formulate the low-rank matrix approximation problem as a problem of searching for the subspace to project the traffic matrix with the minimum error. We propose a novel approach, LSH-subspace, for fast low-rank matrix approximation. To facilitate the matrix partition for the quick search of the subspace, we propose several novel techniques: a multi-layer locality sensitive hashing (LSH) table to reorder the OD pairs based on LSH function, a partition principle to guide the partition to minimize the projection error, and a lightweight algorithm to exploit the sparsity of the outlier matrix to update the LSH table at low overhead. Our extensive simulations based on real trace data demonstrate that our LSH-subspace is 3 times faster than DRMF with high anomaly detection accuracy.

We propose new approximate alternating projection methods, based on randomized sketching, for the low-rank nonnegative matrix approximation problem: find a low-rank approximation of a nonnegative matrix that is nonnegative, but whose factors can be arbitrary. We calculate the computational complexities of the proposed methods and evaluate their performance in numerical experiments. The comparison with the known deterministic alternating projection methods shows that the randomized approaches are faster and exhibit similar convergence properties.

Low-rank approximation of large and/or sparse matrices is important in many applications, and the singular value decomposition (SVD) gives the best low-rank approximations with respect to unitarily-invariant norms. In this paper we show that good low-rank approximations can be directly obtained from the Lanczos bidiagonalization process applied to the given matrix without computing any SVD. We also demonstrate that a so-called one-sided reorthogonalization process can be used to maintain an adequate level of orthogonality among the Lanczos vectors and produce accurate low-rank approximations. This technique reduces the computational cost of the Lanczos bidiagonalization process. We illustrate the efficiency and applicability of our algorithm using numerical examples from several applications areas.

Nonnegative matrix factorization (NMF) provides a lower rank approximation of a matrix by a product of two nonnegative factors. NMF has been shown to produce clustering results that are often superior to those by other methods such as K-means. In this paper, we provide further interpretation of NMF as a clustering method and study an extended formulation for graph clustering called Symmetric NMF (SymNMF). In contrast to NMF that takes a data matrix as an input, SymNMF takes a nonnegative similarity matrix as an input, and a symmetric nonnegative lower rank approximation is computed. We show that SymNMF is related to spectral clustering, justify SymNMF as a general graph clustering method, and discuss the strengths and shortcomings of SymNMF and spectral clustering. We propose two optimization algorithms for SymNMF and discuss their convergence properties and computational efficiencies. Our experiments on document clustering, image clustering, and image segmentation support SymNMF as a graph clustering method that captures latent linear and nonlinear relationships in the data.

Subspace algorithms based on higher-order cumulants were developed to achieve high-resolution separation in non-Gaussian processes. However, singular value decomposition (SVD) of a huge matrix is an unavoidable step of these algorithms. The memory space and running time required by the decomposition are super-linear with respect to the size of the matrix, which is prohibitive in terms of practical applications. Thus, in this paper, a fast raypath separation algorithm based on low-rank matrix approximation is proposed in a shallow-water waveguide. The experimental results illustrate that the proposed algorithm dramatically reduces the consumption of time and space, with arbitrarily small error, compared to conventional higher-order cumulant-based algorithms.

A Point Process Framework for Predicting Popularity of Content in Online Social Networks

As an extension of self-exciting Hawkes process, the multivariate Hawkes process models counting processes of different types of random events with mutual excitement. In this paper, we present a perfect sampling algorithm that can generate i.i.d. stationary sample paths of multivariate Hawkes process without any transient bias. In addition, we provide an explicit expression of algorithm complexity in model and algorithm parameters and provide numerical schemes to find the optimal parameter set that minimizes the complexity of the perfect sampling algorithm.

An algorithm for low-rank approximation of bivariate functions using splines

Fixed-precision randomized quaternion singular value decomposition algorithm for low-rank quaternion matrix approximations

In a broad range of computer vision applications, the purpose of Low-rank matrix approximation (LRMA) models is to recover the underlying low-rank matrix from its degraded observation. The latest LRMA methods - Robust Principal Component Analysis (RPCA) resort to using the nuclear norm minimization (NNM) as a convex relaxation of the non-convex rank minimization. However, NNM tends to over-shrink the rank components and treats the different rank components equally, limiting its flexibility in practical applications. We use a more flexible model, namely the Weighted Schatten p-Norm Minimization (WSNM), to generalize the NNM to the Schatten pnorm minimization with weights assigned to different singular values. The proposed WSNM not only gives a better approximation to the original low-rank assumption but also considers the importance of different rank components. In this paper, a comparison of the low-rank recovery performance of two Low-rank matrix approximation (LRMA) algorithms-RPCA and WSNM is brought out on occluded human facial images. The analysis is performed on facial images from the Yale database and a database created by us, where different facial expressions, spectacles, varying illumination account for the facial occlusions. The paper also discusses the prominent trends observed from the experimental results performed through the application of these algorithms. Further, we propose a new method of using the image-data histogram of the sparse images thus obtained to identify the individual in any given image. Extensive experimental results show, both qualitatively and quantitatively, that WSNM surpasses RPCA in its performance more effectively by removing facial occlusions, thus giving recovered low-rank images of higher PSNR and SSIM. As low-rank images sometimes might fail to capture the details of a face adequately, we further propose to use the image- histogram of the sparse images for facial recognition, which is now the order-of-the-day in most of the surveillance and security purposes.