arXiv research

Vertex distortion measures how far lattice knots deviate from straight lines.

A l1-norm penalized orthogonal forward regression (l1-POFR) algorithm is proposed based on the concept of leaveone- out mean square error (LOOMSE). Firstly, a new l1-norm penalized cost function is defined in the constructed orthogonal space, and each orthogonal basis is associated with an individually tunable regulari…

Develops efficient method for nonconvex problems using Regula Falsi.

To recover a sparse signal from an underdetermined system, we often solve a constrained L1-norm minimization problem. In many cases, the signal sparsity and the recovery performance can be further improved by replacing the L1 norm with a "weighted" L1 norm. Without any prior information about nonzero elements of the si…

Principal component analysis (PCA) is often used to reduce the dimension of data by selecting a few orthonormal vectors that explain most of the variance structure of the data. L1 PCA uses the L1 norm to measure error, whereas the conventional PCA uses the L2 norm. For the L1 PCA problem minimizing the fitting error of…

We propose a data aggregation-based algorithm with monotonic convergence to a global optimum for a generalized version of the L1-norm error fitting model with an assumption of the fitting function. The proposed algorithm generalizes the recent algorithm in the literature, aggregate and iterative disaggregate (AID), whi…

Proposes an optimization framework for sparse robust subspace estimation.

This paper addresses the problem of sparsity penalized least squares for applications in sparse signal processing, e.g. sparse deconvolution. This paper aims to induce sparsity more strongly than L1 norm regularization, while avoiding non-convex optimization. For this purpose, this paper describes the design and use of…

In this paper, we propose a novel linear discriminant analysis criterion via the Bhattacharyya error bound estimation based on a novel L1-norm (L1BLDA) and L2-norm (L2BLDA). Both L1BLDA and L2BLDA maximize the between-class scatters which are measured by the weighted pairwise distances of class means and meanwhile mini…

As one of the most popular linear subspace learning methods, the Linear Discriminant Analysis (LDA) method has been widely studied in machine learning community and applied to many scientific applications. Traditional LDA minimizes the ratio of squared L2-norms, which is sensitive to outliers. In recent research, many …

Most of the existing methods for sparse signal recovery assume a static system: the unknown signal is a finite-length vector for which a fixed set of linear measurements and a sparse representation basis are available and an L1-norm minimization program is solved for the reconstruction. However, the same representation…

This work proposes an adaptive trace lasso regularized L1-norm based graph cut method for dimensionality reduction of Hyperspectral images, called as `Trace Lasso-L1 Graph Cut' (TL-L1GC). The underlying idea of this method is to generate the optimal projection matrix by considering both the sparsity as well as the corr…

Support Vector Machine (SVM) is an efficient classification approach, which finds a hyperplane to separate data from different classes. This hyperplane is determined by support vectors. In existing SVM formulations, the objective function uses L2 norm or L1 norm on slack variables. The number of support vectors is a me…

Uniform convergence of interpolators proven for Gaussian data.

We study the deformation of the three-dimensional conformal structures by the Ricci flow. We drive the evolution equation of Cotton-York tensor and the L1-norm of it under the Ricci flow. In particular, we investigate the behavior of the L1-norm of the Cotton-York tensor under the Ricci flow on three-dimensional simply…

Study tightens bounds for interpolating noisy data using minimum l1-norm.

We propose two sparsity-aware normalized subband adaptive filter (NSAF) algorithms by using the gradient descent method to minimize a combination of the original NSAF cost function and the l1-norm penalty function on the filter coefficients. This l1-norm penalty exploits the sparsity of a system in the coefficients upd…

A Bayesian approach termed BAyesian Least Squares Optimization with Nonnegative L1-norm constraint (BALSON) is proposed. The error distribution of data fitting is described by Gaussian likelihood. The parameter distribution is assumed to be a Dirichlet distribution. With the Bayes rule, searching for the optimal parame…

Approximate dynamic programming is a popular method for solving large Markov decision processes. This paper describes a new class of approximate dynamic programming (ADP) methods- distributionally robust ADP-that address the curse of dimensionality by minimizing a pessimistic bound on the policy loss. This approach tur…

We present an algorithm for L1-norm kernel PCA and provide a convergence analysis for it. While an optimal solution of L2-norm kernel PCA can be obtained through matrix decomposition, finding that of L1-norm kernel PCA is not trivial due to its non-convexity and non-smoothness. We provide a novel reformulation through …

A typical approach in estimating the learning rate of a regularized learning scheme is to bound the approximation error by the sum of the sampling error, the hypothesis error and the regularization error. Using a reproducing kernel space that satisfies the linear representer theorem brings the advantage of discarding t…

Targeting at sparse learning, we construct Banach spaces B of functions on an input space X with the properties that (1) B possesses an l1 norm in the sense that it is isometrically isomorphic to the Banach space of integrable functions on X with respect to the counting measure; (2) point evaluations are continuous lin…

It is well known that quantile regression model minimizes the portfolio extreme risk, whenever the attention is placed on the estimation of the response variable left quantiles. We show that, by considering the entire conditional distribution of the dependent variable, it is possible to optimize different risk and perf…

In this work we investigate to which extent one can recover class probabilities within the empirical risk minimization (ERM) paradigm. The main aim of our paper is to extend existing results and emphasize the tight relations between empirical risk minimization and class probability estimation. Based on existing literat…

Network anomaly detection is still a vibrant research area. As the fast growth of network bandwidth and the tremendous traffic on the network, there arises an extremely challengeable question: How to efficiently and accurately detect the anomaly on multiple traffic? In multi-task learning, the traffic consisting of flo…

QAPCA uses quantum annealing for robust PCA.

Noise injection before gradient steps helps in regularization for neural networks.

We study rank-1 {L1-norm-based TUCKER2} (L1-TUCKER2) decomposition of 3-way tensors, treated as a collection of $N$ $D \times M$ matrices that are to be jointly decomposed. Our contributions are as follows. i) We prove that the problem is equivalent to combinatorial optimization over $N$ antipodal-binary variables. ii)…

New method for efficient proximal mapping of 1-path-norm in shallow networks.

So-called sparse estimators arise in the context of model fitting, when one a priori assumes that only a few (unknown) model parameters deviate from zero. Sparsity constraints can be useful when the estimation problem is under-determined, i.e. when number of model parameters is much higher than the number of data point…

Improved greedy 2-coordinate updates for optimization problems with constraints.

New variational model preserves image contrasts and features using Weingarten map minimization.

For supervised and unsupervised learning, positive definite kernels allow to use large and potentially infinite dimensional feature spaces with a computational cost that only depends on the number of observations. This is usually done through the penalization of predictor functions by Euclidean or Hilbertian norms. In …

Smoothed analysis shows that many classes become learnable from positive-only samples.

Eye Movement analysis with Hidden Markov Models (EMHMM) is a method for modeling eye fixation sequences using hidden Markov models (HMMs). In this report, we run a simulation study to investigate the estimation error for learning HMMs with variational Bayesian inference, with respect to the number of sequences and the …

Iterative algorithms are ubiquitous in the field of data mining. Widely known examples of such algorithms are the least mean square algorithm, backpropagation algorithm of neural networks. Our contribution in this paper is an improvement upon this iterative algorithms in terms of their respective performance metrics an…

It was shown recently that the $K$ L1-norm principal components (L1-PCs) of a real-valued data matrix $\mathbf X \in \mathbb R^{D \times N}$ ($N$ data samples of $D$ dimensions) can be exactly calculated with cost $\mathcal{O}(2^{NK})$ or, when advantageous, $\mathcal{O}(N^{dK - K + 1})$ where $d=\mathrm{rank}(\mathbf …

A key element of understanding the efficacy of overparameterized neural networks is characterizing how they represent functions as the number of weights in the network approaches infinity. In this paper, we characterize the norm required to realize a function $f:\mathbb{R}^d\rightarrow\mathbb{R}$ as a single hidden-lay…

New L0 norm added to TDA for market analysis.

Estimates hybrid dynamical systems with polynomial expansions and Markovian switching.

We propose $\ell_1$ norm regularized quadratic surface support vector machine models for binary classification in supervised learning. We establish their desired theoretical properties, including the existence and uniqueness of the optimal solution, reduction to the standard SVMs over (almost) linearly separable data s…

The problem of biclustering consists of the simultaneous clustering of rows and columns of a matrix such that each of the submatrices induced by a pair of row and column clusters is as uniform as possible. In this paper we approximate the optimal biclustering by applying one-way clustering algorithms independently on t…

We study a family of sparse estimators defined as minimizers of some empirical Lipschitz loss function -- which include the hinge loss, the logistic loss and the quantile regression loss -- with a convex, sparse or group-sparse regularization. In particular, we consider the L1 norm on the coefficients, its sorted Slope…

Solving l1 regularized optimization problems is common in the fields of computational biology, signal processing and machine learning. Such l1 regularization is utilized to find sparse minimizers of convex functions. A well-known example is the LASSO problem, where the l1 norm regularizes a quadratic function. A multil…

There are different problems for resolution of complex LC-MS or GC-MS data, such as the existence of embedded chromatographic peaks, continuum background and overlapping in mass channels for different components. These problems cause rotational ambiguity in recovered profiles calculated using multivariate curve resolut…

RODEO speeds up MRI and CT image reconstruction from sparse data.

New method for sparse data using L1-NMF with improved sparsity control.

Develops high-dimensional measurement error models for non-linear loss functions.