arXiv research

Three new efficient algorithms project vectors onto weighted l1 ball.

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…

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

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.

The paper proposes a method to solve L1 regression with fewer labels using Lewis weights.

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…

In this paper, we give a new sharp generalization bound of lp-MKL which is a generalized framework of multiple kernel learning (MKL) and imposes lp-mixed-norm regularization instead of l1-mixed-norm regularization. We utilize localization techniques to obtain the sharp learning rate. The bound is characterized by the d…

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…

Uniform convergence of interpolators proven for Gaussian data.

The paper proves new inequalities on the unit ball in higher dimensions.

Stable capillary surfaces in weighted balls are disks.

New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.

We give a fast oblivious L2-embedding of $A\in \mathbb{R}^{n x d}$ to $B\in \mathbb{R}^{r x d}$ satisfying $(1-\varepsilon)\|A x\|_2^2 \le \|B x\|_2^2 <= (1+\varepsilon) \|Ax\|_2^2.$ Our embedding dimension $r$ equals $d$, a constant independent of the distortion $\varepsilon$. We use as a black-box any L2-embedding $Π…

Paper proves Faber-Krahn inequalities for weighted Laplacian eigenvalues.

The paper proves cohomology vanishing for a specific type of minimal submanifolds in a weighted Euclidean ball.

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 …

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…

In seeking for sparse and efficient neural network models, many previous works investigated on enforcing L1 or L0 regularizers to encourage weight sparsity during training. The L0 regularizer measures the parameter sparsity directly and is invariant to the scaling of parameter values, but it cannot provide useful gradi…

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

Study on optimal ReLU networks with weight decay for interpolation.

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…

The paper sets up eigenvalue comparison theorems for specific Laplacians on manifolds.

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 illumination bodies defined for ball-convex shapes, proving convexity and establishing surface area measures.

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…

This paper studies ordered weighted L1 (OWL) norm regularization for sparse estimation problems with strongly correlated variables. We prove sufficient conditions for clustering based on the correlation/colinearity of variables using the OWL norm, of which the so-called OSCAR is a particular case. Our results extend pr…

The use of machine-learning in neuroimaging offers new perspectives in early diagnosis and prognosis of brain diseases. Although such multivariate methods can capture complex relationships in the data, traditional approaches provide irregular (l2 penalty) or scattered (l1 penalty) predictive pattern with a very limited…

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…

Paper refines Alesker-Bernig-Schuster theorem, proving Hodge-Riemann relations for Euclidean balls.

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…

Graphical estimation of count time series dependencies.

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…

We study Deraux's non arithmetic orbifold ball quotient surfaces obtained as birational transformations of a quotient $X$ of a particular Abelian surface $A$. Using the fact that $A$ is the Jacobian of the Bolza genus $2$ curve, we identify $X$ as the weighted projective plane $\mathbb{P}(1,3,8)$. We compute the equati…

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. …

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…

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…

No radial balanced metrics found on Kepler manifold unit ball with mild boundary conditions.

We obtain upper estimates for the bottom (that is, greatest lower bound) of the essential spectrum of weighted Laplacian operator of a weighted manifold under assumptions of the volume growth of their geodesic balls and spheres. Furthermore, we find examples where the equality occurs in the estimates obtained. As a con…

This paper approximates 1-Wasserstein distance using tree-based embedding.

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

Regularization improves stability and consistency of sparse autoencoders.