arXiv research

The paper shows that g-convex functions on manifolds are sparse.

Geodesic convexity generalizes the notion of (vector space) convexity to nonlinear metric spaces. But unlike convex optimization, geodesically convex (g-convex) optimization is much less developed. In this paper we contribute to the understanding of g-convex optimization by developing iteration complexity analysis for …

New methods optimize functions on hyperbolic and spherical spaces, matching Euclidean rates up to logarithmic factors.

This is an essay on potential theory for geometric plurisubharmonic functions. It begins with a given closed subset G of the Grassmann bundle $G(p,TX)$ of tangent $p$-planes to a riemannian manifold $X$. This determines a nonlinear partial differential equation which is convex but never uniformly elliptic (p < dim X). …

Lower bounds for geodesically convex optimization show curvature negatively impacts complexity.

New method for optimization on Hadamard manifolds with curvature-independent guarantees.

We consider the problem of finding model-independent bounds on the price of an Asian option, when the call prices at the maturity date of the option are known. Our methods differ from most approaches to model-independent pricing in that we consider the problem as a dynamic programming problem, where the controlled proc…

New method optimizes on curved manifolds without curvature dependence.

Developing efficient and guaranteed nonconvex algorithms has been an important challenge in modern machine learning. Algorithms with good empirical performance such as stochastic gradient descent often lack theoretical guarantees. In this paper, we analyze the class of homotopy or continuation methods for global optimi…

Stochastic gradient descent (SGD) is the workhorse of modern machine learning. Sometimes, there are many different potential gradient estimators that can be used. When so, choosing the one with the best tradeoff between cost and variance is important. This paper analyzes the convergence rates of SGD as a function of ti…

Stochastic optimization algorithms update models with cheap per-iteration costs sequentially, which makes them amenable for large-scale data analysis. Such algorithms have been widely studied for structured sparse models where the sparsity information is very specific, e.g., convex sparsity-inducing norms or $\ell^0$-n…

Graphs are naturally sparse objects that are used to study many problems involving networks, for example, distributed learning and graph signal processing. In some cases, the graph is not given, but must be learned from the problem and available data. Often it is desirable to learn sparse graphs. However, making a grap…

Study exact minimax rates for density estimation over convex classes, extending previous work.

In classification, the de facto method for aggregating individual losses is the average loss. When the actual metric of interest is 0-1 loss, it is common to minimize the average surrogate loss for some well-behaved (e.g. convex) surrogate. Recently, several other aggregate losses such as the maximal loss and average t…

Recursive experts improve dynamic sequential learning systems.

This paper tackles convex-submodular minimax problems in mixed continuous-discrete domains.

New neural network models for complex functional data analysis.

Distance function to a finite set is a topological Morse function.

Introduces new weighted floating functions and affine surface areas.

Develops methods for selecting and estimating smooth functional coefficients in high-dimensional multivariate functional data.

Neural networks can approximate functionals on RKHS with error bounds.

FFBO optimizes functions as inputs and outputs, improving on existing BO methods.

Chirped sinosoids and interferometric phase plots are functions that are not periodic, but are the composition of a smooth function and a periodic function. These functions functions factor into a pair of maps: from their domain to a circle, and from a circle to their codomain. One can easily imagine replacing the circ…

The Fridman function is bounded by the injectivity radius for certain hyperbolic manifolds.

Optimally estimates a functional using nuisance function tuning and sample splitting.

The paper proves isoparametric functions on Finsler space forms under specific conditions.

Paper introduces a nonparametric functional graphical model for random functions.

Robustifies elicitable functionals to handle small distribution misspecifications.

The paper characterizes strong Hamel functions using symmetries and proves their preservation properties.

Study biharmonic functions on vector bundles with spherical symmetry.

Two new methods improve forecasting of functional time series data.

This paper introduces the concept of functional current as a mathematical framework to represent and treat functional shapes, i.e. sub-manifold supported signals. It is motivated by the growing occurrence, in medical imaging and computational anatomy, of what can be described as geometrico-functional data, that is a da…

Study stabilizers of smooth functions on surfaces, focusing on Morse-Bott functions.

The paper connects convex functions to p-subharmonic functions and proves their equivalence.

A new deep neural network tackles nonlinear functional regression with improved dimensionality reduction.

New model for network analysis using functional data.

The study finds a special type of smooth function on connected sums of manifolds.

Function trees simplify complex ML models for better understanding.

We study functions whose truncations are convex or quasiconvex.

NeuTSFlow models continuous functions behind time series forecasting.

New spectral functionals for Dirac operators with inner fluctuations computed.

Regularizers change the geometric properties of loss functions in neural networks.

A new model approximates complex functions in parameter space.

The paper generalizes inequalities on almost Kähler manifolds.

As a generalization of geodesic function, in the present paper, we introduce the notion of geodesic $\varphi$-convex function and deduce some basic properties of $\varphi$-convex function and geodesic $\varphi$-convex function. We also introduce the concept of geodesic $\varphi$-convex set and $\varphi$-epigraph and in…

The diameter function is a topological Morse function.

Proposes an L1-regularized functional SVM for binary classification with functional covariates.

Paper defines Farey Recursive Functions and explores their properties.