arXiv research

An elementary family of local Hamiltonians $H_{\c ,\ell}, \ell = 1,2,3, ldots$, is described for a $2-$dimensional quantum mechanical system of spin $={1/2}$ particles. On the torus, the ground state space $G_{\circ,\ell}$ is $(\log)$ extensively degenerate but should collapse under $ł$perturbation" to an anyonic syste…

A new PCA method using T$\ell_1$-norm outperforms existing methods.

We provide recovery guarantees for compressible signals that have been corrupted with noise and extend the framework introduced in \cite{bafna2018thwarting} to defend neural networks against $\ell_0$-norm, $\ell_2$-norm, and $\ell_{\infty}$-norm attacks. Our results are general as they can be applied to most unitary tr…

Paper improves $\ell^{0}$-SSC for noisy data by proving SDP and proposing Noisy-DR-$\ell^{0}$-SSC.

A pseudo-length function defined on an arbitrary group $G = (G,\cdot,e, (\,)^{-1})$ is a map $\ell: G \to [0,+\infty)$ obeying $\ell(e)=0$, the symmetry property $\ell(x^{-1}) = \ell(x)$, and the triangle inequality $\ell(xy) \leqslant \ell(x) + \ell(y)$ for all $x,y \in G$. We consider pseudo-length functions which sa…

Sparse clustering, which aims to find a proper partition of an extremely high-dimensional data set with redundant noise features, has been attracted more and more interests in recent years. The existing studies commonly solve the problem in a framework of maximizing the weighted feature contributions subject to a $\ell…

New surfaces near a sphere violate Minkowski inequality.

This paper addresses how well we can recover a data matrix when only given a few of its elements. We present a randomized algorithm that element-wise sparsifies the data, retaining only a few its elements. Our new algorithm independently samples the data using sampling probabilities that depend on both the squares ($\e…

Researchers redefine $\ell^\infty$-cohomology for groups and spaces, linking it to amenability, hyperbolicity, and algorithmic undecidability.

Generalized distance-squared mappings are quadratic mappings of $\mathbb{R}^m$ into $\mathbb{R}^\ell$ of special type. In the case that matrices $A$ constructed by coefficients of generalized distance-squared mappings of $\mathbb{R}^2$ into $\mathbb{R}^\ell$ ($\ell \geq3$) are full rank, the generalized distance-square…

The paper analyzes $\ell_1$-LinR for Ising model selection using statistical mechanics.

Fix a prime number ell. In this paper we develop the theory of relative pro-ell completion of discrete and profinite groups -- a natural generalization of the classical notion of pro-ell completion -- and show that the pro-ell completion of the Torelli group does not inject into the relative pro-ell completion of the c…

In this paper, we discuss the statistical properties of the $\ell_q$ optimization methods $(0<q\leq 1)$, including the $\ell_q$ minimization method and the $\ell_q$ regularization method, for estimating a sparse parameter from noisy observations in high-dimensional linear regression with either a deterministic or rando…

The paper constructs stable minimal hypersurfaces with specific singularities.

Enhances robustness of AT frameworks to multiple perturbations without increasing training complexity.

In this paper, we propose $\ell_p$-norm regularized models to seek near-optimal sparse portfolios. These sparse solutions reduce the complexity of portfolio implementation and management. Theoretical results are established to guarantee the sparsity of the second-order KKT points of the $\ell_p$-norm regularized models…

Improved approximation for socially fair clustering with $\ell_p$-objective.

State-of-the-art subspace clustering methods are based on expressing each data point as a linear combination of other data points while regularizing the matrix of coefficients with $\ell_1$, $\ell_2$ or nuclear norms. $\ell_1$ regularization is guaranteed to give a subspace-preserving affinity (i.e., there are no conne…

Suppose M is a non-compact connected smooth n-manifold. Let D(M) denote the group of diffeomorphisms of M endowed with the compact-open C^\infty-topology and D^c(M) denote the subgroup consisting of diffeomorphisms of M with compact support. Let D(M)_0 and D^c(M)_0 be the connected components of id_M in D(M) and D^c(M)…

This work provides efficient algorithms for approximating ℓ_p sensitivities and related statistics.

We investigate the difference between using an $\ell_1$ penalty versus an $\ell_1$ constraint in generalized eigenvalue problems, such as principal component analysis and discriminant analysis. Our main finding is that an $\ell_1$ penalty may fail to provide very sparse solutions; a severe disadvantage for variable sel…

Paper optimizes sparse feature selection for cancer detection using GSVP and SVM.

New algorithm solves $\ell_0$-norm constrained multilinear logistic regression for tensor data.

The paper analyzes kNN density estimation's convergence rates under different conditions.

Efficiently performs robust and sparse kernel regression.

In this paper we consider the problem of grouped variable selection in high-dimensional regression using $\ell_1-\ell_q$ regularization ($1\leq q \leq \infty$), which can be viewed as a natural generalization of the $\ell_1-\ell_2$ regularization (the group Lasso). The key condition is that the dimensionality $p_n$ can…

Proposes a new method for joint sample and feature selection in multi-view data.

We study the robustness properties of $\ell_1$ norm minimization for the classical linear regression problem with a given design matrix and contamination restricted to the dependent variable. We perform a fine error analysis of the $\ell_1$ estimator for measurements errors consisting of outliers coupled with noise. We…

AL$\ell_0$CORE tensor decomposition reduces computational cost for sparse count data.

Study homogeneous Einstein metrics on specific non-Kähler C-spaces.

In this paper, we study the Lévy-Milman concentration phenomenon of 1-Lipschitz maps into infinite dimensional metric spaces. Our main theorem asserts that the concentration to an infinite dimensional $\ell^p$-ball with the $\ell^q$-distance function for $1\leq p<q\leq +\infty$ is equivalent to the concentration to the…

New MPNNs match 2-WL, faster distinguishing graphs.

Safe screening rules reduce computation time in logistic regression with $\ell_0-\ell_2$ regularization.

Support selection and eventwise decoupling for simultaneous bets proven.

Let $n$ be a positive integer, and let $\ell>1$ be square-free odd. We classify the set of equivariant homeomorphism classes of free $C_\ell$-actions on the product $S^1 \times S^n$ of spheres, up to indeterminacy bounded in $\ell$. The description is expressed in terms of number theory. The techniques are various appl…

Signal estimation problems with smoothness and sparsity priors can be naturally modeled as quadratic optimization with $\ell_0$-"norm" constraints. Since such problems are non-convex and hard-to-solve, the standard approach is, instead, to tackle their convex surrogates based on $\ell_1$-norm relaxations. In this paper…

In many applications, high-dimensional data points can be well represented by low-dimensional subspaces. To identify the subspaces, it is important to capture a global and local structure of the data which is achieved by imposing low-rank and sparseness constraints on the data representation matrix. In low-rank sparse …

Constructs generalized Frobenius manifolds for specific Weyl groups.

AdamW optimizes a constrained loss with $\ell_\infty$ norm constraint.

Extending work of Kapouleas and Yang, for any integers $N \geq 2$, $k, \ell \geq 1$, and $m$ sufficiently large, we apply gluing methods to construct in the round $3$-sphere a closed embedded minimal surface that has genus $k\ell m^2(N-1)+1$ and is invariant under a $D_{km} \times D_{\ell m}$ subgroup of $O(4)$, where …

Decomposes string links in a surface into prime components.

A new algorithm speeds up EEG source localization using $\ell_1$ regularization.

We investigate the learning rate of multiple kernel learning (MKL) with $\ell_1$ and elastic-net regularizations. The elastic-net regularization is a composition of an $\ell_1$-regularizer for inducing the sparsity and an $\ell_2$-regularizer for controlling the smoothness. We focus on a sparse setting where the total …

Let $t_1,\ldots,t_n$ be $\ell$-group terms in the variables $X_1,\ldots,X_m$. Let $\hat t_1,\ldots,\hat t_n$ be their associated piecewise homogeneous linear functions. Let $G$ be the $\ell$-group generated by $\hat t_1, \ldots,\hat t_n$ in the free $m$-generator $\ell$-group $\mathcal A_m.$ We prove: (i) the problem …

The paper studies the asymptotic behavior of adversarial training under $\ell_\infty$-perturbation.

Averages are invariants defined on the $\ell^1$ cohomology of Lie groups. We prove that they vanish for abelian and Heisenberg groups. This result completes work by other authors and allows to show that the $\ell^1$ cohomology vanishes in these cases.

Safe screening rules reduce $\ell_0$-regression computation by fixing 76% of variables.

Formulae for special almost-complex structures on Vogan diagrams.