arXiv research

The paper sets sample complexity bounds for learning high-dimensional simplices in noisy data.

Similar simplices can be inscribed in most smoothly embedded spheres.

Paper finds sample complexity for learning high-dimensional simplices from noisy data.

High-dimensional data and high-dimensional representations of reality are inherent features of modern Artificial Intelligence systems and applications of machine learning. The well-known phenomenon of the "curse of dimensionality" states: many problems become exponentially difficult in high dimensions. Recently, the ot…

We study the sample complexity of learning a high-dimensional simplex from a set of points uniformly sampled from its interior. Learning of simplices is a long studied problem in computer science and has applications in computational biology and remote sensing, mostly under the name of `spectral unmixing'. We theoretic…

A new GP inference method using simplices for high-dimensional data.

The study identifies spurious correlations in high-dimensional regression and quantifies their impact.

We present new findings in regard to data analysis in very high dimensional spaces. We use dimensionalities up to around one million. A particular benefit of Correspondence Analysis is its suitability for carrying out an orthonormal mapping, or scaling, of power law distributed data. Power law distributed data are foun…

The paper develops formulas for hyperbolic simplices based on edge lengths.

Paper introduces semi-supervised linear extremile regression for high-dimensional data.

Estimates dimensions of maximal simplices for rational and irrational trees in Outer space.

New approach uses loss functions to extend data depth for anomaly detection.

The paper establishes conditions for Riemannian connections and semi-simplicity of Lie algebras using spray structures.

It is proved that the volume of spherical or hyperbolic simplices, when considered as a function of the dihedral angles, can be extended continuously to degenerated simplices.

A feature-weighted mean shift algorithm improves clustering in high-dimensional data.

Geodesic simplices in pseudo-hyperbolic space get a cohomological treatment.

Label noise SGD converges to a simple model with a single linear feature.

This paper proposes an online tree-based Bayesian approach for reinforcement learning. For inference, we employ a generalised context tree model. This defines a distribution on multivariate Gaussian piecewise-linear models, which can be updated in closed form. The tree structure itself is constructed using the cover tr…

The paper analyzes phase transitions in transfer learning for perceptrons.

A new method improves robustness and efficiency of Bayesian LOO-CV.

Study PL bordism theories with quantitative bounds on filling simplices.

Framework reduces simplicity bias in NNs, improving OOD generalization and robustness.

An increasing amount of collected data are high-dimensional multi-way arrays (tensors), and it is crucial for efficient learning algorithms to exploit this tensorial structure as much as possible. The ever-present curse of dimensionality for high dimensional data and the loss of structure when vectorizing the data moti…

New framework shows $C^*$-simplicity for groups without certain subalgebras.

In this article, we prove a theorem comparing the dihedral angles of simplices in the hyperbolic, spherical and Euclidean geometries.

One of the fundamental problems in machine learning is the estimation of a probability distribution from data. Many techniques have been proposed to study the structure of data, most often building around the assumption that observations lie on a lower-dimensional manifold of high probability. It has been more difficul…

We study a natural intrinsic definition of geometric simplices in Riemannian manifolds of arbitrary dimension $n$, and exploit these simplices to obtain criteria for triangulating compact Riemannian manifolds. These geometric simplices are defined using Karcher means. Given a finite set of vertices in a convex set on t…

Letter analyzes training dynamics of a nonlinear contrastive learning model in high dimensions.

Research reveals simplicity bias in random logistic map, impacting data analysis and forecasting.

We propose Deep Closed-Form Subspace Clustering (DCFSC), a new embarrassingly simple model for subspace clustering with learning non-linear mapping. Compared with the previous deep subspace clustering (DSC) techniques, our DCFSC does not have any parameters at all for the self-expressive layer. Instead, DCFSC utilizes …

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

We study prismatics sets analogously to simplical sets except that realization involves prisms, i.e., products of simplices rather than just simplices. Particular examples are the prismatic subdivision of a simplicial set S and the prismatic star of S. Both have the same homotopy type as S and in particular the latter …

Simplicial sets deformation retract onto transverse simplices.

We generalize the very well known boundary operator of the ordinary singular homology theory, defined in many books about algebraic topology. We describe a variant of this ordinary simplicial boundary operator where the usual boundary (n-1)-simplices of each n-simplex are replaced by combinations of internal (n-1)- sim…

We present a procedure for effective estimation of entropy and mutual information from small-sample data, and apply it to the problem of inferring high-dimensional gene association networks. Specifically, we develop a James-Stein-type shrinkage estimator, resulting in a procedure that is highly efficient statistically …

The global minimum-variance portfolio is a typical choice for investors because of its simplicity and broad applicability. Although it requires only one input, namely the covariance matrix of asset returns, estimating the optimal solution remains a challenge. In the presence of high-dimensionality in the data, the samp…

The paper explores how simplicity leads to better out-of-distribution generalization in models.

The study reveals simplicity bias in neural networks leading to better compositional mappings.

Study on simplicity of Lie skew braces, proving new results for compact cases.

Triangulations of R^n have at least tensor rank of determinant simplices.

We propose a high dimensional classification method that involves nonparametric feature augmentation. Knowing that marginal density ratios are the most powerful univariate classifiers, we use the ratio estimates to transform the original feature measurements. Subsequently, penalized logistic regression is invoked, taki…

Simplicial learning improves classification by generating compact sparse representations.

Two-layer networks favor simple features, especially in complex datasets.

Adam avoids simplicity bias in neural networks, leading to better generalization.

A new reinforcement learning method reduces action complexity for robust control.

We give several new criteria to judge whether a simple convex polytope in a Euclidean space is combinatorially equivalent to a product of simplices. These criteria are mixtures of combinatorial, geometrical and topological conditions that are inspired by the ideas from toric topology.

Neural nets learn simple distributions first, then more complex ones.

The paper proves eigenvalues are simple for specific operators on bundles.