arXiv research

Piecewise flat approximations for curvature in Euclidean and non-Euclidean spaces.

Discretizations of the mean curvature and extrinsic curvature components are constructed on piecewise flat simplicial manifolds, giving approximations for smooth curvature values in a mostly mesh-independent way. These constructions are given in combinatoric form in terms of the extrinsic hinge angles, the intrinsic st…

This paper improves Koopman operator approximations by pruning subspaces in RKHS.

In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $φ_0$; we refer the piece-wised elastica $Λ_{φ_0}$-elastica or $Λ$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly du…

We propose random hinge forests, a simple, efficient, and novel variant of decision forests. Importantly, random hinge forests can be readily incorporated as a general component within arbitrary computation graphs that are optimized end-to-end with stochastic gradient descent or variants thereof. We derive random hinge…

Motivated by the hinge structure present in protein chains and other molecular conformations, we study the singularities of certain maps associated to body-and-hinge and panel-and-hinge chains. These are sequentially articulated systems where two consecutive rigid pieces are connected by a hinge, that is, a codimension…

Hinge-FM2I fills missing data in time series with high accuracy.

Support vector machines (SVMs) naturally embody sparseness due to their use of hinge loss functions. However, SVMs can not directly estimate conditional class probabilities. In this paper we propose and study a family of coherence functions, which are convex and differentiable, as surrogates of the hinge function. The …

New loss function calibrates WW-hinge loss for multiclass SVM.

New loss function improves convergence rate for neural networks.

We study body-and-hinge and panel-and-hinge chains in R^d, with two marked points: one on the first body, the other on the last. For a general chain, the squared distance between the marked points gives a Morse-Bott function on a torus configuration space. Maximal configurations, when the distance between the two marke…

JoVA combines two VAEs to learn user and item representations for better recommendation.

Classification is the most important process in data analysis. However, due to the inherent non-convex and non-smooth structure of the zero-one loss function of the classification model, various convex surrogate loss functions such as hinge loss, squared hinge loss, logistic loss, and exponential loss are introduced. T…

Gradient descent is a simple and widely used optimization method for machine learning. For homogeneous linear classifiers applied to separable data, gradient descent has been shown to converge to the maximal margin (or equivalently, the minimal norm) solution for various smooth loss functions. The previous theory does …

Paper proposes a boosting method with fast learning rates and early stopping.

New approach uses hinge loss for iterative regularization in classification.

We propose a new algorithm to incorporate class conditional information into the critic of GANs via a multi-class generalization of the commonly used Hinge loss that is compatible with both supervised and semi-supervised settings. We study the compromise between training a state of the art generator and an accurate cla…

The top-$k$ error is often employed to evaluate performance for challenging classification tasks in computer vision as it is designed to compensate for ambiguity in ground truth labels. This practical success motivates our theoretical analysis of consistent top-$k$ classification. Surprisingly, it is not rigorously und…

Proposes hinge-Wasserstein to improve uncertainty estimation in regression tasks.

The paper studies consistency of surrogate loss procedures under constrained classifiers.

New framework enhances neural network robustness against adversarial attacks.

A new procedure for learning cost-sensitive SVM(CS-SVM) classifiers is proposed. The SVM hinge loss is extended to the cost sensitive setting, and the CS-SVM is derived as the minimizer of the associated risk. The extension of the hinge loss draws on recent connections between risk minimization and probability elicitat…

Distance/Similarity learning is a fundamental problem in machine learning. For example, kNN classifier or clustering methods are based on a distance/similarity measure. Metric learning algorithms enhance the efficiency of these methods by learning an optimal distance function from data. Most metric learning methods nee…

Study efficient learning of halfspaces with constant noise tolerance.

Learning with non-modular losses is an important problem when sets of predictions are made simultaneously. The main tools for constructing convex surrogate loss functions for set prediction are margin rescaling and slack rescaling. In this work, we show that these strategies lead to tight convex surrogates iff the unde…

Introduces Soft-SVM for binary classification bridging logistic and SVM.

New loss functions improve extreme classification with missing labels.

Partial monitoring is a general model for sequential learning with limited feedback formalized as a game between two players. In this game, the learner chooses an action and at the same time the opponent chooses an outcome, then the learner suffers a loss and receives a feedback signal. The goal of the learner is to mi…

We propose the Margin Adaptation for Generative Adversarial Networks (MAGANs) algorithm, a novel training procedure for GANs to improve stability and performance by using an adaptive hinge loss function. We estimate the appropriate hinge loss margin with the expected energy of the target distribution, and derive princi…

New method learns distances and similarities robustly from noisy data.

We give elementary constructions for Satake-Furstenberg, Martin and Karpelevich boundaries of symmetric spaces. We also consruct some "new" boundaries

Defines Kahler angle for a broader context.

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

AUC (area under ROC curve) is an important evaluation criterion, which has been popularly used in many learning tasks such as class-imbalance learning, cost-sensitive learning, learning to rank, etc. Many learning approaches try to optimize AUC, while owing to the non-convexity and discontinuousness of AUC, almost all …

Study angle structures on pseudo 3-manifolds, proving existence for some cases.

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflect…

Introduces a new geometry based on difference angles, showing unique properties.

The paper analyzes the dynamics of a simple neural network using a mean-field approach.

Stiefel-Whitney classes of moment-angle manifolds are trivial.

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…

Discrete forms of the scalar, sectional and Ricci curvatures are constructed on simplicial piecewise flat triangulations of smooth manifolds, depending directly on the simplicial structure and a choice of dual tessellation. This is done by integrating over volumes which include appropriate samplings of hinges for each …

We present an efficient second-order algorithm with $\tilde{O}(\frac{1}η\sqrt{T})$ regret for the bandit online multiclass problem. The regret bound holds simultaneously with respect to a family of loss functions parameterized by $η$, for a range of $η$ restricted by the norm of the competitor. The family of loss funct…

We derive the fast convergence rates of a deep neural network (DNN) classifier with the rectified linear unit (ReLU) activation function learned using the hinge loss. We consider three cases for a true model: (1) a smooth decision boundary, (2) smooth conditional class probability, and (3) the margin condition (i.e., t…

Uniqueness of quasi-roots explored in right-angled Artin groups.

We develop efficient algorithms to train $\ell_1$-regularized linear classifiers with large dimensionality $d$ of the feature space, number of classes $k$, and sample size $n$. Our focus is on a special class of losses that includes, in particular, the multiclass hinge and logistic losses. Our approach combines several…

This note generalizes the visual angle to convex sets in 3D space.

Study proves existence of weak mean curvature flow with contact angle.

Improved volume estimates for right-angled polyhedra in hyperbolic space.