arXiv research

The paper examines properties of self-affine Sierpiński sponges using metric invariants.

Constructs infinitely many non-equivalent wild knots in Menger sponge.

We show that every inner metric space X is the metric quotient of a complete R-tree via a free isometric action, which we call the covering R-tree of X. The quotient mapping is a weak submetry (hence, open) and light. In the case of compact 1-dimensional geodesic space X, the free isometric action is via a subgroup of …

Study finds a measure for sponge components of Lalley-Gatzouras type.

Knots can be embedded into fractals like the Menger Sponge and Sierpinski Tetrahedron.

It is shown that if $M$ is a strongly causal free of naked singularities space-time, then its causal structure is completely characterized by a partial order in the space of skies defined by means of a class non-negative Legendrian isotopies. It is also proved that such partial order is determined by the class of futur…

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

We introduce a new structured kernel interpolation (SKI) framework, which generalises and unifies inducing point methods for scalable Gaussian processes (GPs). SKI methods produce kernel approximations for fast computations through kernel interpolation. The SKI framework clarifies how the quality of an inducing point a…

SKI accelerates GP inference with sparse grids to handle higher dimensions.

This paper analyzes error in SKI for Gaussian Processes, providing conditions for linear time inference.

We consider a variant of the classic Ski Rental online algorithm with applications to machine learning. In our variant, we allow the skier access to a black-box machine-learning algorithm that provides an estimate of the probability that there will be at most a threshold number of ski-days. We derive a class of optimal…

Study examines diversification of mid-mountain ski tourism.

A reconstruction theorem in terms of the topology and geometrical structures on the spaces of light rays and skies of a given space-time is discussed. This result can be seen as part of Penrose and Low's programme intending to describe the causal structure of a space-time $M$ in terms of the topological and geometrical…

Efficiently maps indoor magnetic fields with SKI and D-SKI.

New attacks exploit neural network energy and latency, increasing costs by 10-200x.

Regularized spectral methods improve clustering in signed graphs, especially for sparse data.

SoftKI combines SKI and variational methods for scalable GP regression.

Power meters are becoming a widely used tool for measuring training and racing effort in cycling, and are now spreading also to other sports. This means that increasing volumes of data can be collected from athletes, with the aim of helping coaches and athletes analyse and understanding training load, racing efforts, t…

Recent work shows that inference for Gaussian processes can be performed efficiently using iterative methods that rely only on matrix-vector multiplications (MVMs). Structured Kernel Interpolation (SKI) exploits these techniques by deriving approximate kernels with very fast MVMs. Unfortunately, such strategies suffer …

The paper proposes calibration to improve algorithm performance using machine learning predictions.

SKI speeds up Toeplitz Neural Networks by avoiding explicit decay bias and using frequency response.

Kernel-based machine learning approaches are gaining increasing interest for exploring and modeling large dataset in recent years. Gaussian process (GP) is one example of such kernel-based approaches, which can provide very good performance for nonlinear modeling problems. In this work, we first propose a grey-box mode…

We construct a Fredholm module on self-similar sets such as the Cantor dust, the Sierpinski carpet and the Menger sponge. Our construction is a higher dimensional analogue of Connes' combinatorial construction of the Fredholm module on the Cantor set. We also calculate the Dixmier trace of two operators induced by the …

Applying a theorem due to Belopol'ski and Birman, we show that the Laplace-Beltrami operator on 1-forms on ${\bf R}^n$ endowed with an asymptotically Euclidean metric has absolutely continuous spectrum equal to $[0, +\infty)$.

We obtain a nature generalization for an affine Sierpinski carpet and Sierpinski triangle to $n$-dimensional space, by using the generations and characterizations of affinely-equivalent Sierpinski carpet. Exactly, in this paper, a Menger sponge and Sierpinski simplex in $4$-dimensional space could be drawn out clearly …

Optimal hashing embeddings reduce linear least squares solving time.

The set N of all null geodesics of a globally hyperbolic (d+1)-dimensional spacetime (M,g) is naturally a smooth (2d-1)-dimensional contact manifold. The sky of an event is the subset of N defined by all null geodesics through that event, and is an embedded Legendrian submanifold of N diffeomorphic to a (d-1)-dimension…

We introduce a principled and theoretically sound spectral method for $k$-way clustering in signed graphs, where the affinity measure between nodes takes either positive or negative values. Our approach is motivated by social balance theory, where the task of clustering aims to decompose the network into disjoint group…

We define a conformal reference frame, i.e., a special projection of the six-dimensional sky bundle of a Lorentzian manifold (or the five-dimensional twistor space) to a three-dimensional manifold. We construct an example, a conformal compactification, for Minkowski space. Based on the complex structure on the skies, w…

New framework constructs holographic tensor networks using hyperbolic buildings.

Quantum GBS boosts asset clustering for robust statistical arbitrage portfolios.

Let $S$ be a compact, orientable surface of hyperbolic type. Let $(k_+,k_-)$ be a pair of negative numbers and let $(g_+, g_-)$ be a pair of marked metrics over $S$ of constant curvature equal to $k_+$ and $k_-$ respectively. Using a functional introduced by Bonsante, Mondello \& Schlenker, we show that there exists a …

New approach for algorithms that learn predictors to improve performance.

Estimates nonparametric densities from mixed samples.

Researchers compute contact structures for null geodesics on specific spacetimes.

Quandle coloring detects causality in spacetime links.

The linking number $lk$ is defined if link components are zero homologous. Our affine linking invariant $alk$ generalizes $lk$ to the case of linked submanifolds with arbitrary homology classes. We apply $alk$ to the study of causality in Lorentz manifolds. Let $M^m$ be a spacelike Cauchy surface in a globally hyperbol…