arXiv research

Quantum machine learning models can approximate any continuous function.

Effective feature representation is key to the predictive performance of any algorithm. This paper introduces a meta-procedure, called Non-Euclidean Upgrading (NEU), which learns feature maps that are expressive enough to embed the universal approximation property (UAP) into most model classes while only outputting fea…

Path signatures adapted for Lie groups improve action recognition in computer vision.

Unified theorem for deep and shallow joint-equivariant machines.

Max-margin classifiers' behavior is studied in high dimensions with non-Gaussian features.

We study the approximation properties of random ReLU features through their reproducing kernel Hilbert space (RKHS). We first prove a universality theorem for the RKHS induced by random features whose feature maps are of the form of nodes in neural networks. The universality result implies that the random ReLU features…

We introduce a new feature map for barcodes that arise in persistent homology computation. The main idea is to first realize each barcode as a path in a convenient vector space, and to then compute its path signature which takes values in the tensor algebra of that vector space. The composition of these two operations …

Quantum kernels can be efficiently embedded into classical feature spaces.

Lower bound proves ridgeless regression performs poorly near interpolation threshold.

Random feature models approximate functions in Banach spaces efficiently.

Identifying small subsets of features that are relevant for prediction and/or classification tasks is a central problem in machine learning and statistics. The feature selection task is especially important, and computationally difficult, for modern datasets where the number of features can be comparable to, or even ex…

New method creates universal perturbations to fool neural network interpretations.

New conditions ensure deep neural networks can approximate any function on non-Euclidean spaces.

New maps connect universal circles to ideal sphere for hyperbolic manifolds.

Signature portfolios approximate optimal wealth in non-Markovian markets.

We present effective methods to compute equivariant harmonic maps from the universal cover of a surface into a nonpositively curved space. By discretizing the theory appropriately, we show that the energy functional is strongly convex and derive convergence of the discrete heat flow to the energy minimizer, with explic…

Dimension reduction is the process of embedding high-dimensional data into a lower dimensional space to facilitate its analysis. In the Euclidean setting, one fundamental technique for dimension reduction is to apply a random linear map to the data. This dimension reduction procedure succeeds when it preserves certain …

Deep random feature models are analyzed for their performance with exact asymptotic expressions.

Constructs a universal Cannon-Thurston map for a new curve complex.

The feature map obtained from the denoising autoencoder (DAE) is investigated by determining transportation dynamics of the DAE, which is a cornerstone for deep learning. Despite the rapid development in its application, deep neural networks remain analytically unexplained, because the feature maps are nested and param…

Unified method for CNNs to approximate equivariant maps across various groups.

We give chain homotopy maps of Khovanov-type link homology of a universal differential. The universal differential, discussed by Mikhail Khovanov, Marco Mackaay, Paul Turner and Pedro Vaz, contains the original Khovanov's differential and Lee's differential. We also consider the conditions of any differential ensuring …

In this work we show that, using the eigen-decomposition of the adjacency matrix, we can consistently estimate feature maps for latent position graphs with positive definite link function $κ$, provided that the latent positions are i.i.d. from some distribution F. We then consider the exploitation task of vertex classi…

Universal approximation theorem for differentiable maps on infinite-dimensional manifolds

Paper relaxes symmetry conditions for universal feature selection in noisy data.

The family of image visibility graphs (IVGs) have been recently introduced as simple algorithms by which scalar fields can be mapped into graphs. Here we explore the usefulness of such operator in the scenario of image processing and image classification. We demonstrate that the link architecture of the image visibilit…

Gaussian equivalence fails for simple polynomial embeddings in quadratic scaling RF models.

The paper defines a universal Teichmüller space for PGL_d(R) and proves its properties.

The paper generalizes equivariant neural networks on homogeneous spaces to the non-linear setting.

Assume that all spaces and maps are localised at a fixed prime $p$. We study the possibility of generating a universal space $U(X)$ from a space $X$ which is universal in the category of homotopy associative, homotopy commutative H-spaces in the sense that any map f:X->Y to a homotopy associative, homotopy commutative …

New theory maps spin structures on surfaces, generating key elements.

Generalizes neural network approximation to infinite-dimensional manifolds and derivatives.

Implementing $k$-NN classification using Gromov--Wasserstein distances

The study proves Gaussian universality of deep random features learning.

NOs can learn any finite collection of classes in functional data.

We propose a stochastic map model of economic dynamics. In the last decade, an array of observations in economics has been investigated in the econophysics literature, a major example being the universal features of inequality in terms of income and wealth. Another area of inquiry is the formation of opinion in a socie…

This study approximates neural network features for modeling relations and attention mechanisms.

We designed a machine learning algorithm that identifies patterns between ESG profiles and financial performances for companies in a large investment universe. The algorithm consists of regularly updated sets of rules that map regions into the high-dimensional space of ESG features to excess return predictions. The fin…

UniFeat is an open-source Java tool for feature selection.

Model approximates continuous functions in 1-Wasserstein space.

Mathematical proof of S-duality and universal isometries in q-map spaces.

We give a new and simple proof for the computation of the oriented and the unoriented fold cobordism groups of Morse functions on surfaces. We also compute similar cobordism groups of Morse functions based on simple stable maps of 3-manifolds into the plane. Furthermore, we show that certain cohomology classes associat…

LUNA improves linear attention for long sequences without sacrificing accuracy.

Narasihman and Ramanan proved that an arbitrary connection in a vector bundle over a base space B can be obtained as the pull-back (via a correctly chosen classifying map from B into the appropriate Grassmannian) of the universal connection in the universal bundle over the Grassmannian. The purpose of this paper is to …

The aim of this paper is to introduce a group containing the mapping class groups of all genus zero surfaces. Roughly speaking, such a group is intended to be a discrete analogue of the diffeomorphism group of the circle. One defines indeed a {\it universal mapping class group of genus zero}, denoted $\B$. The latter i…

Introduces a new method for symplectic reduction along submanifolds.

Universal approximation for ODENet and ResNet with a single activation function.

We introduce a topological object, called hairy Cantor set, which in many ways enjoys the universal features of objects like Jordan curve, Cantor set, Cantor bouquet, hairy Jordan curve, etc. We give an axiomatic characterisation of hairy Cantor sets, and prove that any two such objects in the plane are ambiently homeo…