arXiv research

A short proof for a theorem about composite knots.

Paper simplifies DP composition for adaptive privacy budgets, enabling better privacy and accuracy in deep learning.

In this paper, we mainly prove a theorem with a corollary establishing two characterizations of the Calabi composition of hyperbolic hyperspheres, where the second characterization (i.e., the corollary) has been given via a dual correspondence theorem earlier but now we would like to use a very direct method. Note that…

Sharp privacy bounds for sequential analysis of sensitive data.

New conditions for weighted composition operators in group homomorphisms.

We present an enhanced prime decomposition theorem for knots that gives the isotopy classes of composite knots that can be constructed from a given list of prime factors (allowing for the mirroring and orientation reversing for each factor). Underlying the theorem is an algebraic construction that also allows for the c…

The purpose of this erratum is to fill a gap in the proof of the `Composite Braid Theorem' in the manuscript "Studying Links Via Closed Braids IV: Composite Links and Split Links (SLVCB-IV)", Inventiones Math, \{bf 102\} Fasc. 1 (1990), 115-139. The statement of the theorem is unchanged. The gap occurs on page 135, lin…

This paper studies neural network operators and their convergence properties.

Connected sum and trivalent vertex sum are natural operations on genus 2 spatial graphs and, as with knots, tunnel number behaves in interesting ways under these operations. We prove sharp Scharlemann-Schultens type bounds for the tunnel number of a composite genus 2 spatial graph. For the tunnel number of a composite …

We study b-arc foliation change and exchange move of open book foliations which generalize the corresponding operations in braid foliation theory. We also define a bypass move as an analogue of Honda's bypass attachment operation. As applications, we study how open book foliations change under a stabilization of the op…

New filters match advanced composition for adaptive privacy, with practical constants.

Paper extends FFT-based differential privacy method to heterogeneous compositions.

Study improves HOMFLY polynomial coefficients for positive braid links.

We show that deep networks are better than shallow networks at approximating functions that can be expressed as a composition of functions described by a directed acyclic graph, because the deep networks can be designed to have the same compositional structure, while a shallow network cannot exploit this knowledge. Thu…

The paper describes fitting submanifolds to data using Sussmann's orbit theorem.

We prove a classification theorem for conformal maps with respect to the control distance generated by a system of diagonal vector fields. It turns out that all such maps can be obtained as compositions of suitable dilations, inversions and isometries. We also classify all umbilical surfaces of the underlying metric.

We provide a methodology, resilient feature engineering, for creating adversarially resilient classifiers. According to existing work, adversarial attacks identify weakly correlated or non-predictive features learned by the classifier during training and design the adversarial noise to utilize these features. Therefore…

Paper proposes iLPA for solving DC composite optimization problems, with applications to matrix completion with outliers.

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…

Improves privacy guarantees by analyzing randomness in privacy-preserving mechanisms.

We describe pairs (p,n) such that n-dimensional affine space is fibered by pairwise skew p-dimensional affine subspaces. The problem is closely related with the theorem of Adams on vector fields on spheres and the Hurwitz-Radon theory of composition of quadratic forms.

New method for causal inference with complex treatment compositions.

Let $N$ (resp., $U$) be a manifold (resp., an open subset of $\mathbb{R}^m$). Let $f:N\to U$ and $F:U\to \mathbb{R}^\ell$ be an immersion and a $C^{\infty}$ mapping, respectively. Generally, the composition $F\circ f$ does not necessarily yield a mapping transverse to a given subfiber-bundle of $J^1(N,\mathbb{R}^\ell)$…

Differential privacy has seen remarkable success as a rigorous and practical formalization of data privacy in the past decade. This privacy definition and its divergence based relaxations, however, have several acknowledged weaknesses, either in handling composition of private algorithms or in analyzing important primi…

L. Capogna and M. Cowling showed that if $φ$ is 1-quasiconformal on an open subset of a Carnot group G, then composition with $φ$ preserves Q-harmonic functions, where Q denotes the homogeneous dimension of G. Then they combine this with a regularity theorem for Q-harmonic functions to show that $φ$ is in fact $C^\inft…

In this paper, we develop a convolutional neural network model to predict the mechanical properties of a two-dimensional checkerboard composite quantitatively. The checkerboard composite possesses two phases, one phase is soft and ductile while the other is stiff and brittle. The ground-truth data used in the training …

Simpler proof for non-basic sets in 2D.

The main goal of the present paper is two-fold. First we extend the theory of toroidal embeddings introduced by Kempf, Knudsen, Mumford and Saint-Donat to the class of toroidal varieties with stratifications (which is the main body of the paper). Second we give a proof of the following weak factorization theorem as an …

Base of fibered correspondence is arbitrary correspondence. Fibered correspondence is interesting when we consider relationship between different bundles. However composition of fibered correspondences may not always be defined. Reduced fibered correspondence is defined only between fibers over the same point of base. …

Under Markovian assumptions, we leverage a Central Limit Theorem (CLT) for the empirical measure in the test statistic of the composite hypothesis Hoeffding test so as to establish weak convergence results for the test statistic, and, thereby, derive a new estimator for the threshold needed by the test. We first show t…

Unified analysis of KL divergence using shifted composition for sampling.

New proof of generalized Chow-Rashevskii theorem for non-linear systems.

New method shows hidden state can significantly improve differential privacy in SGD.

Paper generalizes paracomposition and change of variables for paradifferential operators.

Proves geodesic connections on 2-torus without invariant tori.

This work is an analytical and numerical study of the composition of several fractals into one and of the relation between the composite dimension and the dimensions of the component fractals. In the case of composition of standard IFS with segments of equal size, the composite dimension can be expressed as a function …

New geometric approach for analyzing compositional data like gut microbiomes.

Study on deep neural networks using branching processes and Mehler's formula.

We establish conditions for compositional generalization in machine learning.

This paper proves hyperbolicity of virtual knot compositions.

Develops methods for causal inference in compositional data using instrumental variables.

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertic…

X-ray transform on H-type groups solved, revealing function injectivity.

The p-index improves investment performance for NYSE stocks but not for SSE stocks.

Model predicts composite structures assembly quality with input uncertainty.

This paper introduces compositional data analysis for financial ratios, improving industry-level analysis.

Study challenges neural models in compositional learning tasks.

SCL discovers compositional structures in analogical reasoning tasks.