arXiv research

Algorithm transforms weakly negative plumbing trees to negative definite ones.

Gaussian belief propagation (GaBP) is an iterative algorithm for computing the mean of a multivariate Gaussian distribution, or equivalently, the minimum of a multivariate positive definite quadratic function. Sufficient conditions, such as walk-summability, that guarantee the convergence and correctness of GaBP are kn…

Models like support vector machines or Gaussian process regression often require positive semi-definite kernels. These kernels may be based on distance functions. While definiteness is proven for common distances and kernels, a proof for a new kernel may require too much time and effort for users who simply aim at prac…

Improved Bayesian learning rule handles positive-definite constraints efficiently.

We introduce a new algorithm, called adaptive sparse backfitting algorithm, for solving high dimensional Sparse Additive Model (SpAM) utilizing symmetric, non-negative definite smoothers. Unlike the previous sparse backfitting algorithm, our method is essentially a block coordinate descent algorithm that guarantees to …

New method classifies manifold-valued data using Riemannian geometry.

Based on differential privacy (DP) framework, we introduce and unify privacy definitions for the multi-armed bandit algorithms. We represent the framework with a unified graphical model and use it to connect privacy definitions. We derive and contrast lower bounds on the regret of bandit algorithms satisfying these def…

In this paper, the Riemannian gradient algorithm and the natural gradient algorithm are applied to solve descent direction problems on the manifold of positive definite Hermitian matrices, where the geodesic distance is considered as the cost function. The first proposed problem is control for positive definite Hermiti…

We consider an online learning process to forecast a sequence of outcomes for nonconvex models. A typical measure to evaluate online learning algorithms is regret but such standard definition of regret is intractable for nonconvex models even in offline settings. Hence, gradient based definition of regrets are common f…

This work surveys algorithmic recourse, aiming to clarify definitions and solutions.

This work addresses the issue of large covariance matrix estimation in high-dimensional statistical analysis. Recently, improved iterative algorithms with positive-definite guarantee have been developed. However, these algorithms cannot be directly extended to use a nonconvex penalty for sparsity inducing. Generally, a…

Paper shows incorrectness of approximate unlearning definitions and challenges exact unlearning verification.

Tool uses text mining to define innovative tech fields from abstracts.

In this paper, we first give a new simple proof to the elimination theorem of definite fold by homotopy for generic smooth maps of manifolds of dimension strictly greater than $2$ into the $2$--sphere or into the real projective plane. Our new proof has the advantage that it is not only constructive, but is also algori…

Signature kernel handles sequential data with theoretical and practical advantages.

The class of Schoenberg transformations, embedding Euclidean distances into higher dimensional Euclidean spaces, is presented, and derived from theorems on positive definite and conditionally negative definite matrices. Original results on the arc lengths, angles and curvature of the transformations are proposed, and v…

We establish a characterization of alternating links in terms of definite spanning surfaces. We apply it to obtain a new proof of Tait's conjecture that reduced alternating diagrams of the same link have the same crossing number and writhe. We also deduce a result of Banks and Hirasawa-Sakuma about Seifert surfaces for…

We study adversarial perturbations when the instances are uniformly distributed over $\{0,1\}^n$. We study both "inherent" bounds that apply to any problem and any classifier for such a problem as well as bounds that apply to specific problems and specific hypothesis classes. As the current literature contains multiple…

This paper proposes a definition of system health in the context of multiple agents optimizing a joint reward function. We use this definition as a credit assignment term in a policy gradient algorithm to distinguish the contributions of individual agents to the global reward. The health-informed credit assignment is t…

Online learning algorithms are designed to learn even when their input is generated by an adversary. The widely-accepted formal definition of an online algorithm's ability to learn is the game-theoretic notion of regret. We argue that the standard definition of regret becomes inadequate if the adversary is allowed to a…

Proposes a score to compare rule-based algorithms' interpretability.

Paper proves MS convergence for radially symmetric kernels with large bandwidths.

In kernel methods, the kernels are often required to be positive definite, which restricts the use of many indefinite kernels. To consider those non-positive definite kernels, in this paper, we aim to build an indefinite kernel learning framework for kernel logistic regression. The proposed indefinite kernel logistic r…

In this paper, we propose a new algorithm for exploratory projection pursuit. The basis of the algorithm is the insight that previous approaches used fairly narrow definitions of interestingness / non interestingness. We argue that allowing these definitions to depend on the problem / data at hand is a more natural app…

This paper generalizes optimization techniques to diffeological spaces.

Paper connects probability density cuts to graph theory eigenfunctions.

This document aims to provide a review on learning with deep generative models (DGMs), which is an highly-active area in machine learning and more generally, artificial intelligence. This review is not meant to be a tutorial, but when necessary, we provide self-contained derivations for completeness. This review has tw…

Positive definite matrices abound in a dazzling variety of applications. This ubiquity can be in part attributed to their rich geometric structure: positive definite matrices form a self-dual convex cone whose strict interior is a Riemannian manifold. The manifold view is endowed with a "natural" distance function whil…

A knot $K$ is definite if $|σ(K)| = 2g(K)$. We prove that the quotient of a definite periodic knot is definite by considering equivariant minimal genus Seifert surfaces.

Proposes a new method to measure and avoid harm in machine learning decisions.

New algorithm accelerates optimization on Riemannian manifolds, including Wasserstein space.

A new optimization algorithm for Gaussian Variational Inference on precision matrices.

We present a grid diagram analogue of Carter, Rieger and Saito's smooth movie theorem. Specifically, we give definitions for grid movies, grid movie isotopies and present a definition of grid planar isotopy as a particular subset of the grid diagram moves: stabilization, destabilization and commutation. We show that gr…

Paper develops PAC verification for hypothesis classes and statistical algorithms.

Birg{é} and Massart proposed in 2001 the slope heuristics as a way to choose optimally from data an unknown multiplicative constant in front of a penalty. It is built upon the notion of minimal penalty, and it has been generalized since to some "minimal-penalty algorithms". This paper reviews the theoretical results ob…

Bayesian method finds voids in galaxy surveys with deep neural networks.

This work introduces oblivious fairness definitions for image generation.

We define the Wirtinger width of a knot. Then we prove the Wirtinger width of a knot equals its Gabai width. The algorithmic nature of the Wirtinger width leads to an efficient technique for establishing upper bounds on Gabai width. As an application, we use this technique to calculate the Gabai width of approximately …

We consider two multi-armed bandit problems with $n$ arms: (i) given an $ε> 0$, identify an arm with mean that is within $ε$ of the largest mean and (ii) given a threshold $μ_0$ and integer $k$, identify $k$ arms with means larger than $μ_0$. Existing lower bounds and algorithms for the PAC framework suggest that both …

We propose definitions of fairness in machine learning and artificial intelligence systems that are informed by the framework of intersectionality, a critical lens arising from the Humanities literature which analyzes how interlocking systems of power and oppression affect individuals along overlapping dimensions inclu…

This work defines observation-specific explanations for black-box models.

A new way to describe correlation matrices makes modeling easier.

Dictionary leaning (DL) and dimensionality reduction (DR) are powerful tools to analyze high-dimensional noisy signals. This paper presents a proposal of a novel Riemannian joint dimensionality reduction and dictionary learning (R-JDRDL) on symmetric positive definite (SPD) manifolds for classification tasks. The joint…

We introduce a graphical framework for multiple instance learning (MIL) based on Markov networks. This framework can be used to model the traditional MIL definition as well as more general MIL definitions. Different levels of ambiguity -- the portion of positive instances in a bag -- can be explored in weakly supervise…

A new geometric definition of integration for differential forms.

This short note highlights some links between two lines of research within the emerging topic of trustworthy machine learning: differential privacy and robustness to adversarial examples. By abstracting the definitions of both notions, we show that they build upon the same theoretical ground and hence results obtained …

The paper introduces Robust Correlated Equilibrium for games with time-varying costs and proposes an algorithm to achieve it.

Brain decoding is a popular multivariate approach for hypothesis testing in neuroimaging. It is well known that the brain maps derived from weights of linear classifiers are hard to interpret because of high correlations between predictors, low signal to noise ratios, and the high dimensionality of neuroimaging data. T…