arXiv research

Quantum kernel machines need to use more complex kernels to fully exploit their potential.

Mapper and Ball Mapper tools for complex data analysis.

Algorithm optimizes cascaded functions with known structure.

Extends complex manifold structures to line bundles, revealing new projective manifolds.

Develops vector-valued RKBS for neural networks and operators.

Let $M$ be a complete Riemannian manifold and let $Ω^*(M)$ denote the space of differential forms on $M$. Let $d:Ω^*(M) \to Ω^{*+1}(M)$ be the exterior differential operator and let $\Del=dd^*+d^*d$ be the Laplacian. We establish a sufficient condition for the Schroedinger operator $H=\Del+V(x)$ (where the potential $V…

In recent years, a rapidly growing literature has focussed on the construction of wavelet systems to analyze functions defined on the sphere. Our purpose in this paper is to generalize these constructions to situations where sections of line bundles, rather than ordinary scalar-valued functions, are considered. In part…

We consider the problem of learning a vector-valued function f in an online learning setting. The function f is assumed to lie in a reproducing Hilbert space of operator-valued kernels. We describe two online algorithms for learning f while taking into account the output structure. A first contribution is an algorithm,…

Positive definite operator-valued kernels generalize the well-known notion of reproducing kernels, and are naturally adapted to multi-output learning situations. This paper addresses the problem of learning a finite linear combination of infinite-dimensional operator-valued kernels which are suitable for extending func…

The Cheap Gradient Principle (Griewank 2008) --- the computational cost of computing the gradient of a scalar-valued function is nearly the same (often within a factor of $5$) as that of simply computing the function itself --- is of central importance in optimization; it allows us to quickly obtain (high dimensional) …

Sharp inequality for $p$-harmonic maps with new optimal constant.

This work evaluates deep generative models using RD curves, providing a more comprehensive quality assessment.

Study first-order locally convex Lie algebroids in Bastiani calculus.

Tree-based algorithm for functional data analysis reduces generalization error.

Paper introduces a new kernel model for PSD-valued functions with theoretical guarantees and applications.

Proves properties of neural network basins of attraction and their expressiveness.

We formulate a private learning model to study an intrinsic tradeoff between privacy and query complexity in sequential learning. Our model involves a learner who aims to determine a scalar value, $v^*$, by sequentially querying an external database and receiving binary responses. In the meantime, an adversary observes…

This paper considers a distributed reinforcement learning problem in which a network of multiple agents aim to cooperatively maximize the globally averaged return through communication with only local neighbors. A randomized communication-efficient multi-agent actor-critic algorithm is proposed for possibly unidirectio…

Enhances reward specification in RL with a novel language-based approach.

In this paper, we study the possibility of inferring early warning indicators (EWIs) for periods of extreme bitcoin price volatility using features obtained from Bitcoin daily transaction graphs. We infer the low-dimensional representations of transaction graphs in the time period from 2012 to 2017 using Bitcoin blockc…

Differential privacy mechanism design has traditionally been tailored for a scalar-valued query function. Although many mechanisms such as the Laplace and Gaussian mechanisms can be extended to a matrix-valued query function by adding i.i.d. noise to each element of the matrix, this method is often suboptimal as it for…

We address the problem of classifying discrete differential-geometric Poisson brackets (dDGPBs) of any fixed order on target space of dimension 1. It is proved that these Poisson brackets (PBs) are in one-to-one correspondence with the intersection points of certain projective hypersurfaces. In addition, they can be re…

We investigate invariants of compact hyperk{ä}hler manifolds introduced by Rozansky and Witten: they associate an invariant to each graph homology class. It is obtained by using the graph to perform contractions on a power of the curvature tensor and then integrating the resulting scalar-valued function over the manifo…

STRIDE improves explainable AI by efficiently decomposing feature interactions without subset enumeration.

The paper bounds solutions to complex optimization problems with uncertain data.

Suppose that $Ω$ is the open region in $\mathbb{R}^n$ above a Lipschitz graph and let $d$ denote the exterior derivative on $\mathbb{R}^n$. We construct a convolution operator $T$ which preserves support in $\bar{Ω$}, is smoothing of order 1 on the homogeneous function spaces, and is a potential map in the sense that …

We study $n$-dimensional area-minimizing currents $T$ in $\mathbb{R}^{n+1},$ with boundary $\partial T$ satisfying two properties: $\partial T$ is locally a finite sum of $(n-1)$-dimensional $C^{1,α}$ orientable submanifolds which only meet tangentially and with same orientation, for some $α\in (0,1]$; $\partial T$ has…

A new framework learns differentiable structured losses from data.

This paper aims at formulating the issue of ranking multivariate unlabeled observations depending on their degree of abnormality as an unsupervised statistical learning task. In the 1-d situation, this problem is usually tackled by means of tail estimation techniques: univariate observations are viewed as all the more …

Stein variational gradient descent (SVGD) is a particle-based inference algorithm that leverages gradient information for efficient approximate inference. In this work, we enhance SVGD by leveraging preconditioning matrices, such as the Hessian and Fisher information matrix, to incorporate geometric information into SV…

Study improves self-normalized bounds for vector-valued processes beyond sub-Gaussianity.

Deep Gaussian processes on manifolds improve performance on complex data.

This paper optimizes sampling for least-squares approximation.

Gaussian processes adapted for Riemannian manifolds using gauge-independent kernels.

Gradient boosted decision trees are a popular machine learning technique, in part because of their ability to give good accuracy with small models. We describe two extensions to the standard tree boosting algorithm designed to increase this advantage. The first improvement extends the boosting formalism from scalar-val…

Study on neural networks' sample complexity with one hidden layer.

Python package for ordinal regression using gradient boosting.

New sample complexity bounds for linear predictors and neural networks, focusing on initialization.

Wasserstein Policy Learning for Distributional Outcomes

Efficient inference for adaptive data with directional stability condition.

Gaussian processes adapted for non-Euclidean spaces enhance decision-making.

The notion of integrability will often extend from systems with scalar-valued fields to systems with algebra-valued fields. In such extensions the properties of, and structures on, the algebra play a central role in ensuring integrability is preserved. In this paper a new theory of Frobenius-algebra valued integrable s…

This paper shows connections between two complex mathematical theories are equivalent.

We study the query complexity of a learner-private sequential learning problem, motivated by the privacy and security concerns due to eavesdropping that arise in practical applications such as pricing and Federated Learning. A learner tries to estimate an unknown scalar value, by sequentially querying an external datab…

Proposes a new way to represent and analyze Pareto front surfaces.

We develop a variant of multiclass logistic regression that is significantly more robust to noise. The algorithm has one weight vector per class and the surrogate loss is a function of the linear activations (one per class). The surrogate loss of an example with linear activation vector $\mathbf{a}$ and class $c$ has t…

We present Vector-Space Markov Random Fields (VS-MRFs), a novel class of undirected graphical models where each variable can belong to an arbitrary vector space. VS-MRFs generalize a recent line of work on scalar-valued, uni-parameter exponential family and mixed graphical models, thereby greatly broadening the class o…

Study vector-valued robust control under uncertainty.