arXiv research

Stable algebraic filters improve neural network performance.

We determine the Killing superalgebras underpinning field theories with rigid unextended supersymmetry on Lorentzian four-manifolds by re-interpreting them as filtered deformations of $\mathbb{Z}$-graded subalgebras with maximum odd dimension of the $N{=}1$ Poincaré superalgebra in four dimensions. Part of this calcula…

Researchers study Killing superalgebras in 2D manifolds.

New methods for $Z$-transform inversion and Wiener-Hopf factorization.

Triangular, overlapping Mel-scaled filters ("f-banks") are the current standard input for acoustic models that exploit their input's time-frequency geometry, because they provide a psycho-acoustically motivated time-frequency geometry for a speech signal. F-bank coefficients are provably robust to small deformations in…

We study the algebraic structure of the Killing superalgebra of a supersymmetric background of $11$-dimensional supergravity and show that it is isomorphic to a filtered deformation of a $\mathbb Z$-graded subalgebra of the Poincaré superalgebra. We are able to map the classification problem for highly supersymmetric b…

The paper develops finite knot theory using ropelength-filtered Reidemeister graphs.

Study cohomology rings of Grassmannians using Clifford algebras and symmetric spaces.

Explicit encoding of group actions in deep features makes it possible for convolutional neural networks (CNNs) to handle global deformations of images, which is critical to success in many vision tasks. This paper proposes to decompose the convolutional filters over joint steerable bases across the space and the group …

Extends Killing superalgebras to higher dimensions and signatures.

Many of our core assumptions about how neural networks operate remain empirically untested. One common assumption is that convolutional neural networks need to be stable to small translations and deformations to solve image recognition tasks. For many years, this stability was baked into CNN architectures by incorporat…

Encoding the scale information explicitly into the representation learned by a convolutional neural network (CNN) is beneficial for many computer vision tasks especially when dealing with multiscale inputs. We study, in this paper, a scaling-translation-equivariant (ST-equivariant) CNN with joint convolutions across th…

The first purpose of this note is to comment on a recent article of Bursztyn, Lima and Meinrenken, in which it is proved that if M is a smooth submanifold of a manifold V, then there is a bijection between germs of tubular neighborhoods of M and germs of "Euler-like" vector fields on V. We shall explain how to approach…

Defines Killing (super)algebras for spin manifolds, including gauge transformations.

This study explores Kaluza-Klein reductions of new maximally supersymmetric backgrounds.

This study shows that certain cohomology groups of symplectic manifolds are always even-dimensional.

GNNs generalize CNNs for graph data, showing equivariance and stability.

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of convolutional neural networks. Inspired by recent interest in geometric deep learning, which aims to generalize convolutional neural networks to manifold and graph-structured domains, we define a geometric…

We study symplectic (contact) structures on nilmanifolds that correspond to the filiform Lie algebras - nilpotent Lie algebras of the maximal length of the descending central sequence. We give a complete classification of filiform Lie algebras that possess a basis e_1, ..., e_n, [e_i,e_j]=c_{ij}e_{i{+}j} (N-graded Lie …

Develops formal moduli theory for splitting complex supermanifolds.

We define a hierarchy of special classes of constrained Willmore surfaces by means of the existence of a polynomial conserved quantity of some type, filtered by an integer. Type 1 with parallel top term characterises parallel mean curvature surfaces and, in codimension 1, type 1 characterises constant mean curvature su…

Convolution has been playing a prominent role in various applications in science and engineering for many years. It is the most important operation in convolutional neural networks. There has been a recent growth of interests of research in generalizing convolutions on curved domains such as manifolds and graphs. Howev…

Deep convolutional neural networks have led to breakthrough results in numerous practical machine learning tasks such as classification of images in the ImageNet data set, control-policy-learning to play Atari games or the board game Go, and image captioning. Many of these applications first perform feature extraction …

First steps towards a mathematical theory of deep convolutional neural networks for feature extraction were made---for the continuous-time case---in Mallat, 2012, and Wiatowski and Bölcskei, 2015. This paper considers the discrete case, introduces new convolutional neural network architectures, and proposes a mathemati…

The Euclidean scattering transform was introduced nearly a decade ago to improve the mathematical understanding of the success of convolutional neural networks (ConvNets) in image data analysis and other tasks. Inspired by recent interest in geometric deep learning, which aims to generalize ConvNets to manifold and gra…

This note focuses on some properties and uses of filtered deformations in the context of D=11 supergravity. We define the concept of abstract symbol and give a strong version of the Reconstruction Theorem, namely a bijective correspondence from the space of highly supersymmetric supergravity backgrounds to the space of…

Study on knot types using thickness and length constraints.

Where dealing with temporal sequences it is fair to assume that the same kind of deformations that motivated the development of the Dynamic Time Warp algorithm could be relevant also in the calculation of the dot product ("convolution") in a 1-D convolution layer. In this work a method is proposed for aligning the conv…

NAST generalizes scattering transform for non-stationary time series analysis.

Informative and discriminative feature descriptors play a fundamental role in deformable shape analysis. For example, they have been successfully employed in correspondence, registration, and retrieval tasks. In the recent years, significant attention has been devoted to descriptors obtained from the spectral decomposi…

We recover the classification of the maximally supersymmetric bosonic backgrounds of eleven-dimensional supergravity by Lie algebraic means. We classify all filtered deformations of the $\mathbb Z$-graded subalgebras $\mathfrak{h}=\mathfrak{h}_{-2}\oplus\mathfrak{h}_{-1}\oplus\mathfrak{h}_{0}$ of the Poincaré superalge…

This paper focuses on spectral filters on graphs, namely filters defined as elementwise multiplication in the frequency domain of a graph. In many graph signal processing settings, it is important to transfer a filter from one graph to another. One example is in graph convolutional neural networks (ConvNets), where the…

A new SOHP filter improves trend estimation in economic time series.

Deep density methods improve filtering in high-dimensional systems.

The sophisticated structure of Convolutional Neural Network (CNN) allows for outstanding performance, but at the cost of intensive computation. As significant redundancies inevitably present in such a structure, many works have been proposed to prune the convolutional filters for computation cost reduction. Although ex…

Gradient filters track moving parameters under noisy data and misspecification.

We simplify Bayesian filtering by framing it as optimization, making it practical for high-dimensional systems.

Develops an inverse particle filter for cognitive systems.

Convolutional neural networks (CNNs) achieve state-of-the-art performance in a wide variety of tasks in computer vision. However, interpreting CNNs still remains a challenge. This is mainly due to the large number of parameters in these networks. Here, we investigate the role of compression and particularly pruning fil…

A novel method reduces dimensionality for filtering SRNs with observed variables.

Kernel learning FBSDE filter improves nonlinear filtering efficiency.

New method filters large networks from financial data to reveal key subnetworks.

Many nonlinear extensions of the Kalman filter, e.g., the extended and the unscented Kalman filter, reduce the state densities to Gaussian densities. This approximation gives sufficient results in many cases. However, this filters only estimate states that are correlated with the observation. Therefore, sequential esti…

Paper proves convergence of Kalman filter on Stiefel manifolds with measurement errors.

Improved Kalman filter for non-linear, non-Gaussian data.

This work analyzes the stability of graph filters under large perturbations.

Advances deep network embedding through multi-filtering GCN.

This work preserves linear invariants in ensemble filters for non-Gaussian data assimilation.