Jensen simplifies machine learning and optimization with an extensible toolkit.
problem Complex machine learning and optimization tasks in production environments.
method Develops a framework for convex functions and optimization algorithms, enabling easy deployment and extension.
result Jensen allows for quick model deployment and extension with minimal code, making machine learning accessible.
Solves convex extension problems for 1-jets and convex hypersurfaces.
problem Finding convex extensions for 1-jets and hypersurfaces.
method Provides necessary and sufficient conditions for convex extensions with C1 smoothness. result Necessary and sufficient conditions for convex extensions of 1-jets and hypersurfaces.
Constructs smooth isometric extensions for submanifolds.
problem Isometric immersions of submanifolds in codimension one.
method Method of convex integration for C1,α extensions. result Constructs C1,α isometric extensions for $α< rac{1}{n(n+1)+1}$. Generalizes Nielsen equivalence theorem to hyperbolic group extensions.
problem Tackles Nielsen equivalence in hyperbolic group extensions.
method Generalizes a theorem by Juan Souto to a broader class of hyperbolic extensions.
result Includes all hyperbolic extensions of surfaces groups and free groups by Out$(F_n).
Locally C1,1 convex extensions for 1-jets defined on subsets of Rn.
problem Finding convex functions with specified values and gradients on subsets of Rn. method Provided necessary and sufficient conditions for existence, and gave an explicit formula for such extensions.
result Explicit formula for Cextrmloc1,1 convex extensions of 1-jets. Paper studies how to combine regret minimizers for solving complex games.
problem Solving large-scale extensive-form games with constraints.
method Derives a calculus for constructing regret minimizers for composite convex sets.
result Local regret minimizers for simpler sets can be combined into an aggregate for composite sets.
Study extends biholomorphisms between convex domains in complex space without boundary constraints.
problem Extending biholomorphisms between convex domains without boundary regularity.
method Combining coarse geometry techniques with dynamical properties of maps in Gromov hyperbolic spaces.
result Proves extensions for biholomorphisms and quasi-isometries between convex domains.
Extends Whitney's theorem for functions on rough boundaries.
problem Global extension of manifold-valued functions on domains with rough boundaries.
method Using locally convex spaces of compactly-supported sections of vector bundles, proving the existence of an extension operator.
result The restriction map from everywhere-defined functions is a submersion, allowing local linear splittings.
Given a monotone convex function on the space of essentially bounded random variables with the Lebesgue property (order continuity), we consider its extension preserving the Lebesgue property to as big solid vector space of random variables as possible. We show that there exists a maximum such extension, with explicit …
Regularized empirical risk minimization with constrained labels (in contrast to fixed labels) is a remarkably general abstraction of learning. For common loss and regularization functions, this optimization problem assumes the form of a mixed integer program (MIP) whose objective function is non-convex. In this form, t…
Solves convex optimization with many constraints in a distributed system.
problem Solving convex optimization problems with many convex constraints in a distributed setting.
method Extension of ADMM to handle arbitrary inequality constraints.
result Inherits convergence guarantees of ADMM and Augmented Lagrangian method.
Two Bartnik mass definitions are shown to be equivalent under convexity conditions.
problem Equivalence of two Bartnik mass definitions for compact Riemannian manifolds.
method Gluing Bartnik extensions and convexity conditions.
result Equivalence of Bartnik mass definitions under convexity conditions.
Convex-constrained sparse additive models improve regression performance.
problem High-dimensional nonparametric regression with shape constraints.
method Sparse difference of convex additive models (SDCAM) with regularization and efficient backfitting algorithm.
result SDCAM estimates functions without smoothness assumptions and outperforms existing methods.
We consider the optimization of active extension portfolios. For this purpose, the optimization problem is rewritten as a stochastic programming model and solved using a clever multi-start local search heuristic, which turns out to provide stable solutions. The heuristic solutions are compared to optimization results o…
Developed a theory of local convexity for second order differential equations on Lie algebroids.
problem Analyzing convexity in differential equations on Lie algebroids.
method Theory development for local convexity of SODEs on Lie algebroids.
result Extensive discussion of homogeneous quadratic SODEs on Lie algebroids.
The paper explores rigidity and flexibility of isometric extensions with critical Hölder exponent.
problem The critical Hölder exponent in isometric extensions and its implications.
method Convex integration and construction of isometric extensions.
result The Hölder exponent $θ_0= rac12$ is critical, with extensions violating the tangential connection for $θ< rac12$.
Let C be a compact convex subset of Rn, f:C→R be a convex function, and m∈{1,2,...,∞}. Assume that, along with f, we are given a family of polynomials satisfying Whitney's extension condition for Cm, and thus that there exists F∈Cm(Rn) such that F=f on $…
We introduce and study a new class of $\eps$-convex bodies (extending the class of convex bodies) in metric and normed linear spaces. We analyze relations between characteristic properties of convex bodies, demonstrate how $\eps$-convex bodies connect with some classical results of Convex Geometry, as Helly theorem, an…
Survey on extending rigidity theorems to Riemannian manifolds.
problem Extending classical rigidity theorems to Riemannian manifolds.
method Review and extension of existing rigidity theorems.
result Rigidity results for convex hypersurfaces of homogeneous 3-manifolds.
Paper extends Green-Osher inequality for convex bodies at dilation position.
problem Extending Green-Osher inequality for specific geometric configurations.
method Analyzes strictly convex bodies at dilation position and derives necessary and sufficient conditions.
result Establishes extended Green-Osher inequality with conditions for equality.
We extend the idea and techniques in \cite{Miao} to study variational effect of the boundary geometry on the ADM mass of an asymptotically flat manifold. We show that, for a Lipschitz asymptotically flat metric extension of a bounded Riemannian domain with quasi-convex boundary, if the boundary mean curvature of the ex…
Kernelized convex clustering handles non-linear and non-convex data.
problem Lack of effective clustering methods for non-linear and non-convex data.
method Kernelized convex clustering in RKHS.
result Superior performance compared to state-of-the-art techniques.
Study nilpotent Lie algebras with Ricci negative solvable extensions.
problem Understanding curvature behavior in Ricci negative solvmanifolds.
method Characterize derivations of nilpotent Lie algebras leading to Ricci negative solvable extensions.
result Found unexpected behaviors in nilpotent Lie algebras with Ricci negative solvable extensions.
We prove that the torsion of any closed space curve which bounds a simply connected locally convex surface vanishes at least 4 times. This answers a question of Rosenberg related to a problem of Yau on characterizing the boundary of positively curved disks in Euclidean space. Furthermore, our result generalizes the 4 v…
We investigate the structure of good deal bounds, which are subintervals of a no-arbitrage pricing bound, for financial market models with convex constraints as an extension of Arai and Fukasawa (2014). The upper and lower bounds of a good deal bound are naturally described by a convex risk measure. We call such a risk…
We study relations of some classes of k-convex, k-visible bodies in Euclidean spaces. We introduce and study \textrm{circular projections} in normed linear spaces and classes of bodies related with families of such maps, in particular, \textrm{k-circular convex} and \textrm{k-circular visible} ones. Investigati…
Let C be a subset of Rn (not necessarily convex), f:C→R be a function, and G:C→Rn be a uniformly continuous function, with modulus of continuity ω. We provide a necessary and sufficient condition on f, G for the existence of a convex function F∈C1,ω(Rn) …
Relative to the large literature on upper bounds on complexity of convex optimization, lesser attention has been paid to the fundamental hardness of these problems. Given the extensive use of convex optimization in machine learning and statistics, gaining an understanding of these complexity-theoretic issues is importa…
Sparse estimation methods are aimed at using or obtaining parsimonious representations of data or models. They were first dedicated to linear variable selection but numerous extensions have now emerged such as structured sparsity or kernel selection. It turns out that many of the related estimation problems can be cast…
Study spherical convex bodies using Lp-floating areas and curvature entropy.
problem Analogous isoperimetric inequalities for spherical convex bodies.
method Introduced Lp-floating areas and curvature entropy for spherical convex bodies. result Established isoperimetric inequalities and dual isoperimetric inequalities.
Geodesically convex functions are continuous on Riemannian manifolds.
problem Continuity of geodesically convex functions on Riemannian manifolds.
method Proof of continuity using geodesic convexity and addressing a gap in existing proof.
result All geodesically convex functions are continuous in the interior of their domain on Riemannian manifolds.
Convex optimization refines neural network training, improving model performance and reducing hyperparameter sensitivity.
problem Training deep neural networks using non-convex optimization methods often leads to suboptimal solutions and requires extensive tuning.
method Formulate neural network training as convex programs with regularization terms, leveraging sparse recovery models and semi-infinite programming theory.
result Convex models can achieve global optima and outperform traditional non-convex methods, with improved robustness to hyperparameters.
We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric homeomorphism φ, we study the relation between the width of the convex hull of th…
New analysis of Langevin Monte Carlo via convex optimization.
problem Sampling from logconcave smooth and non-smooth target distributions.
method Formulation as a convex optimization problem, analysis using convex optimization techniques.
result Non-asymptotic analysis of Unadjusted Langevin Algorithm and new sampling methods.
We prove that any smooth Riemannian manifold of non-negative scalar curvature and with a strictly mean convex and compact boundary component can be (C^2) extended beyond the component to have non-negative scalar curvature and to enjoy anyone of the following three types of (new) boundary: strictly convex, totally geode…
Most learning methods with rank or sparsity constraints use convex relaxations, which lead to optimization with the nuclear norm or the ℓ1-norm. However, several important learning applications cannot benefit from this approach as they feature these convex norms as constraints in addition to the non-convex rank a…
Deep learning models generalize by extending decision boundaries outside the convex hull of training data.
problem Understanding how deep learning models generalize beyond their training data.
method Investigation of decision boundaries inside and outside the convex hull of training sets, using various neural network architectures and training regimes.
result Over-parameterization is necessary for deep learning models to extend decision boundaries outside the convex hull of their training data.
New method finds near-optimal solutions for non-convex optimization problems.
problem Finding near-optimal solutions for non-convex optimization problems.
method Riemannian stochastic recursive momentum method
result Achieves a near-optimal complexity of ildeO(ε−3). Extends techniques to show existence of all cusp types in convex projective manifolds.
problem Existence of all cusp types in convex projective manifolds.
method Extension of techniques by Ballas-Marquis.
result Existence of all cusp types in all dimensions except diagonalizable.
Study derivations for nilpotent Lie algebras with negative Ricci curvature.
problem Characterize derivations leading to solvable extensions with negative Ricci curvature.
method Investigate the space of diagonalizable derivations for specific Lie algebras.
result Prove conjecture about derivations in dimension 5 and for Heisenberg and standard filiform Lie algebras.
In this paper, we study open complete metric spaces with non-negative curvature. Among other things, we establish an extension of Perelman's soul theorem for possibly singular spaces: "Let X be a complete, non-compact, finite dimensional Alexandrov space with non-negative curvature. Suppose that X has no boundary and h…
A new convex loss function optimizes set predictions with balanced size and coverage.
problem Optimizing set predictions with balanced size and coverage.
method Proposes a convex loss function using Choquet integrals for nondecreasing subset-valued functions.
result Optimal trade-offs between conditional probabilistic coverage and set size.
The use of convex regularizers allows for easy optimization, though they often produce biased estimation and inferior prediction performance. Recently, nonconvex regularizers have attracted a lot of attention and outperformed convex ones. However, the resultant optimization problem is much harder. In this paper, for a …
Faster, better sparse model estimation for large datasets.
problem Sparse model estimation for large datasets with millions of samples and features.
method Coordinate descent, working sets, Anderson acceleration.
result Significantly faster and more efficient than state-of-the-art algorithms.
The paper refines and generalizes worst-case law invariant convex risk measures.
problem Developing robust convex risk measures under uncertainty sets.
method Generalizing closed forms for worst-case law invariant convex risk measures with uncertainty sets based on norms and moment constraints.
result Explicit closed forms for convex risk measures are developed and assessed through numerical simulations.
Novel neural network models using convex optimization for improved training.
problem Improving the training of neural networks.
method Representing activation functions as argmin of convex optimization problems, applying block-coordinate descent methods.
result Proposed models provide excellent initial guesses and avenues for extensions.
In this survey, we report on the state of the art of some of the fundamental problems in the Lie theory of Lie groups modeled on locally convex spaces, such as integrability of Lie algebras, integrability of Lie subalgebras to Lie subgroups, and integrability of Lie algebra extensions to Lie group extensions. We furthe…
Paper introduces ICGNs to model convex gradients.
problem Modeling convex gradients efficiently.
method Integrates Jacobian-vector product in a neural network.
result Single layer ICGN outperforms single layer ICNN in fitting.