arXiv research

Noiseless linear estimation results are found to be universal across various structured matrices.

New algorithms solve robust MDPs efficiently, significantly faster than existing methods.

Improved sample efficiency in learning sparse Ising models.

Based on a new coupling approach, we prove that the transition step of the Hamiltonian Monte Carlo algorithm is contractive w.r.t. a carefully designed Kantorovich (L1 Wasserstein) distance. The lower bound for the contraction rate is explicit. Global convexity of the potential is not required, and thus multimodal targ…

We address the problem of Compressed Sensing (CS) with side information. Namely, when reconstructing a target CS signal, we assume access to a similar signal. This additional knowledge, the side information, is integrated into CS via L1-L1 and L1-L2 minimization. We then provide lower bounds on the number of measuremen…

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)…

The use of L1 regularisation for sparse learning has generated immense research interest, with successful application in such diverse areas as signal acquisition, image coding, genomics and collaborative filtering. While existing work highlights the many advantages of L1 methods, in this paper we find that L1 regularis…

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…

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

We propose a new deep recurrent neural network (RNN) architecture for sequential signal reconstruction. Our network is designed by unfolding the iterations of the proximal gradient method that solves the l1-l1 minimization problem. As such, our network leverages by design that signals have a sparse representation and t…

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 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…

Feature selection is a technique to screen out less important features. Many existing supervised feature selection algorithms use redundancy and relevancy as the main criteria to select features. However, feature interaction, potentially a key characteristic in real-world problems, has not received much attention. As a…

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…

We theoretically investigate the convergence rate and support consistency (i.e., correctly identifying the subset of non-zero coefficients in the large sample limit) of multiple kernel learning (MKL). We focus on MKL with block-l1 regularization (inducing sparse kernel combination), block-l2 regularization (inducing un…

Paper proposes a new robust LDA method using L1,2-norm ratio minimization.

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 compute approximate solutions to L0 regularized linear regression using L1 regularization, also known as the Lasso, as an initialization step. Our algorithm, the Lass-0 ("Lass-zero"), uses a computationally efficient stepwise search to determine a locally optimal L0 solution given any L1 regularization solution. We …

Solving logistic regression with L1-regularization in distributed settings is an important problem. This problem arises when training dataset is very large and cannot fit the memory of a single machine. We present d-GLMNET, a new algorithm solving logistic regression with L1-regularization in the distributed settings. …

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…

Reweighted l1-algorithms have attracted a lot of attention in the field of applied mathematics. A unified framework of such algorithms has been recently proposed by Zhao and Li. In this paper we construct a few new examples of reweighted l1-methods. These functions are certain concave approximations of the l0-norm func…

This paper analyzes l1-regularized PageRank for local graph clustering, proving its effectiveness and efficiency.

Sparse reconstruction approaches using the re-weighted l1-penalty have been shown, both empirically and theoretically, to provide a significant improvement in recovering sparse signals in comparison to the l1-relaxation. However, numerical optimization of such penalties involves solving problems with l1-norms in the ob…

New theoretical framework improves error rates for sparse learning with convex regularization.

We provide a fast L2-embedding for arbitrary accuracy with applications to regression and L1 tasks.

Hidden Markov Models (HMMs) can be accurately approximated using co-occurrence frequencies of pairs and triples of observations by using a fast spectral method in contrast to the usual slow methods like EM or Gibbs sampling. We provide a new spectral method which significantly reduces the number of model parameters tha…

We consider the l1-regularized Markowitz model, where a l1-penalty term is added to the objective function of the classical mean-variance one to stabilize the solution process, promoting sparsity in the solution. The l1-penalty term can also be interpreted in terms of short sales, on which several financial markets hav…

We prove that L2-Boosting lacks a theoretical property which is central to the behaviour of l1-penalized methods such as basis pursuit and the Lasso: Whereas l1-penalized methods are guaranteed to recover the sparse parameter vector in a high-dimensional linear model under an appropriate restricted nullspace property, …

Recently it has become popular to learn sparse Gaussian graphical models (GGMs) by imposing l1 or group l1,2 penalties on the elements of the precision matrix. Thispenalized likelihood approach results in a tractable convex optimization problem. In this paper, we reinterpret these results as performing MAP estimation u…

Inverse reinforcement learning (IRL) is the problem of finding a reward function that generates a given optimal policy for a given Markov Decision Process. This paper looks at an algorithmic-independent geometric analysis of the IRL problem with finite states and actions. A L1-regularized Support Vector Machine formula…

A new ICA method adds L1-regularization for better interpretability of fMRI data.

PRESTO improves rare event prediction by shrinking towards proportional odds model.

l1 reweighting algorithms are very popular in sparse signal recovery and compressed sensing, since in the practice they have been observed to outperform classical l1 methods. Nevertheless, the theoretical analysis of their convergence is a critical point, and generally is limited to the convergence of the functional to…

RTC-GTNLN model recovers traffic data from missing values and noise.

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…

In this paper, we study Lichnerowicz type estimate for eigenvalues of drifting Laplacian operator and L1 and L2 energy for drifting heat equation on closed manifolds with weighted measure. In some sense, this study is about the eigenvalue estimate on Ricci solitons.

QAPCA uses quantum annealing for robust PCA.

We analyse a linear regression problem with nonconvex regularization called smoothly clipped absolute deviation (SCAD) under an overcomplete Gaussian basis for Gaussian random data. We propose an approximate message passing (AMP) algorithm considering nonconvex regularization, namely SCAD-AMP, and analytically show tha…

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…

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…

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

We consider the class of optimization problems arising from computationally intensive L1-regularized M-estimators, where the function or gradient values are very expensive to compute. A particular instance of interest is the L1-regularized MLE for learning Conditional Random Fields (CRFs), which are a popular class of …

Proposes an L1-regularized functional SVM for binary classification with functional covariates.

Develops efficient method for nonconvex problems using Regula Falsi.

One obstacle that so far prevents the introduction of machine learning models primarily in critical areas is the lack of explainability. In this work, a practicable approach of gaining explainability of deep artificial neural networks (NN) using an interpretable surrogate model based on decision trees is presented. Sim…

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…

In many applications, multivariate samples may harbor previously unrecognized heterogeneity at the level of conditional independence or network structure. For example, in cancer biology, disease subtypes may differ with respect to subtype-specific interplay between molecular components. Then, both subtype discovery and…