arXiv research

Paper optimizes hyperspherical prototypes for better class separation.

This paper introduces hyperspherical prototype networks, which unify classification and regression with prototypes on hyperspherical output spaces. For classification, a common approach is to define prototypes as the mean output vector over training examples per class. Here, we propose to use hyperspheres as output spa…

Sparse prototypes improve clustering of high-dimensional directional data.

Convolution as inner product has been the founding basis of convolutional neural networks (CNNs) and the key to end-to-end visual representation learning. Benefiting from deeper architectures, recent CNNs have demonstrated increasingly strong representation abilities. Despite such improvement, the increased depth and l…

This work optimizes alignment and uniformity of features on a hypersphere for better downstream performance.

The Variational Auto-Encoder (VAE) is one of the most used unsupervised machine learning models. But although the default choice of a Gaussian distribution for both the prior and posterior represents a mathematically convenient distribution often leading to competitive results, we show that this parameterization fails …

We describe extrinsic hyperspheres and totally geodesic hypersurfaces in manifolds with special holonomy. In particular we prove the nonexistence of extrinsic hyperspheres in quaternion-Kaehler manifolds. We develop a new approach to extrinsic hyperspheres based on the classification of special Killing forms.

In this paper we study $\widetilde{J}$-tangent affine hyperspheres, where $\widetilde{J}$ is the canonical para-complex structure on $\mathbb{R}^{2n+2}$. The main purpose of this paper is to give a classification of $\widetilde{J}$-tangent affine hyperspheres of an arbitrary dimension with an involutive distribution $\…

Learning suitable latent representations for observed, high-dimensional data is an important research topic underlying many recent advances in machine learning. While traditionally the Gaussian normal distribution has been the go-to latent parameterization, recently a variety of works have successfully proposed the use…

Many contemporary statistical learning methods assume a Euclidean feature space. This paper presents a method for defining similarity based on hyperspherical geometry and shows that it often improves the performance of support vector machine compared to other competing similarity measures. Specifically, the idea of usi…

A new loss function HUG decouples and generalizes neural collapse.

Method generates prototypes from small datasets for efficient learning.

Constructs hyperspheres with prescribed mean curvature in Euclidean space.

In this paper, we explicitly construct the Calabi composition of multiple affine hyperspheres possibly including some points viewing as 0-dimensional hypersheres. Then we compute all the basic affine invariants of the composed affine hyperspheres, proving that the composed affine hypersphere is symmetric one if and onl…

The study characterizes kernel spaces on hyperspheres, impacting cubature algorithms.

Study on hyperspheres in 4-spaces as special Riemannian manifolds.

In this paper, we study locally strongly convex affine hyperspheres in the unimodular affine space $\mathbb{R}^{n+1}$ which, as Riemannian manifolds, are locally isometric to the Riemannian product of two Riemannian manifolds both possessing constant sectional curvatures. As the main result, a complete classification o…

We propose infinite mixture prototypes to adaptively represent both simple and complex data distributions for few-shot learning. Our infinite mixture prototypes represent each class by a set of clusters, unlike existing prototypical methods that represent each class by a single cluster. By inferring the number of clust…

The paper projects unknown manifolds onto hyperspheres for efficient function approximation.

This paper proves the manifold hypothesis for lower embedding dimensions using osculating hyperspheres.

OPT framework improves neural network generalization by learning an orthogonal transformation.

Small hypersphere is unstable in both 4-harmonic and ES-4-harmonic settings.

A holomorphic representation formula for special parabolic hyperspheres is given.

Prototypal analysis is introduced to overcome two shortcomings of archetypal analysis: its sensitivity to outliers and its non-locality, which reduces its applicability as a learning tool. Same as archetypal analysis, prototypal analysis finds prototypes through convex combination of the data points and approximates th…

Upper estimate of Dirac eigenvalue linked to hyperspherical radius.

ProtoryNet interprets text sequences using prototype trajectories for better understanding.

Optimal prototypes found for challenging pathological geometries.

The emergence of deep learning networks raises a need for explainable AI so that users and domain experts can be confident applying them to high-risk decisions. In this paper, we leverage data from the latent space induced by deep learning models to learn stereotypical representations or "prototypes" during training to…

PTBCC improves accuracy in multi-class annotation aggregation by learning from prototype confusion matrices.

Pantypes improve prototypical models by capturing diverse input distributions.

We present the Bayesian Case Model (BCM), a general framework for Bayesian case-based reasoning (CBR) and prototype classification and clustering. BCM brings the intuitive power of CBR to a Bayesian generative framework. The BCM learns prototypes, the "quintessential" observations that best represent clusters in a data…

We propose prototypical networks for the problem of few-shot classification, where a classifier must generalize to new classes not seen in the training set, given only a small number of examples of each new class. Prototypical networks learn a metric space in which classification can be performed by computing distances…

DPTA improves CIL by adapting PTMs with dual prototypes.

SPOT uses optimal transport to select important prototypes.

We show how in many cases the algebraic number of immersed hyperspheres of constant (and prescribed) curvature may be related to the Euler Characteristic of the ambient space.

Researchers prove constant mean curvature graphs in hyperbolic 3-space for specific domains.

Advances AT with HE to improve model robustness.

Prototype model improves model auditing and understanding.

We discuss two kinds of almost contact metric structures on a one-parameter family of totally umbilical hyperspheres in the nearly Kaehler unit 6-sphere.

We show that the natural S^1-bundle over a projective special Kaehler manifold carries the geometry of a proper affine hypersphere endowed with a Sasakian structure. The construction generalizes the geometry of the Hopf-fibration $\Sr^{2n+1} \longrightarrow \CP^n$ in the context of projective special Kaehler manifolds.…

A new method for speaker recognition on hyperspheres improves on PLDA's limitations.

An affine hypersurface M is said to admit a pointwise symmetry, if there exists a subgroup G of Aut(T_p M) for all p in M, which preserves (pointwise) the affine metric h, the difference tensor K and the affine shape operator S. Here, we consider 3-dimensional indefinite affine hyperspheres, i.e. S= H Id (and thus S is…

A new neural network method improves interpretability and detection of outliers.

Deep neural networks are widely used for classification. These deep models often suffer from a lack of interpretability -- they are particularly difficult to understand because of their non-linear nature. As a result, neural networks are often treated as "black box" models, and in the past, have been trained purely to …

The study characterizes quadrics among affine hyperspheres based on centroid collinearity of sections.

A new learning method using hyperbolic geometry for class labels.

We introduce and study the equiaffine symmetric {\bf hyperspheres}. For the first step we consider the locally strongly convex ones. In fact, by the idea used by Naitoh, we provide in this paper a direct proof of the complete classification for those affine symmetric hyperspheres. Then, via an earlier result of the fir…

This work studies the chord length distribution, in the case where both ends lie on a $N$-dimensional hypersphere ($N \geq 2$). Actually, after connecting this distribution to the recently estimated surface of a hyperspherical cap \cite{SLi11}, closed-form expressions of both the probability density function and the cu…