arXiv research

Counterexamples show failure of uniform laws of large numbers for subdifferentials.

This work establishes uniform convergence of subdifferentials in stochastic optimization.

Paper explores subdifferential chain rules for matrix factorization and related machine learning models.

Study on tensor nuclear norm's decomposability and subdifferential.

The paper tackles finding stationary points in stochastic convex optimization problems.

We investigate the notion of H-subdifferential and H-normal map of a function on the Heisenberg group, based on its sub-Riemannian structure. In particular, a characterization of the convexity of a function is given via the nonemptiness of the H-subdifferential at every point.

The paper defines subdifferentials on Hadamard manifolds and identifies conditions for Fenchel conjugate equality.

We prove that every function $f:\mathbb{R}^n\to \mathbb{R}$ satisfies that the image of the set of critical points at which the function $f$ has Taylor expansions of order $n-1$ and non-empty subdifferentials of order $n$ is a Lebesgue-null set. As a by-product of our proof, for the proximal subdifferential $\partial_{…

New framework for studying eigenvalue functionals of metrics.

Mirror flows converge to a limiting flow with a convex potential.

New algorithms optimize spectral risk measures, improving interpolation between average and worst-case performance.

Screening rules help identify active sets in optimization problems.

Characterizes infinite harmonic maps using 1-currents.

We establish some perturbed minimization principles, and we develop a theory of subdifferential calculus, for functions defined on Riemannian manifolds. Then we apply these results to show existence and uniqueness of viscosity solutions to Hamilton-Jacobi equations defined on Riemannian manifolds.

Recently, based on the idea of randomizing space theory, random convex analysis has been being developed in order to deal with the corresponding problems in random environments such as analysis of conditional convex risk measures and the related variational problems and optimization problems. Random convex analysis is …

Study proves convergence of subgradients for optimal transport-based objectives.

We propose a new class of convex penalty functions, called \emph{variational Gram functions} (VGFs), that can promote pairwise relations, such as orthogonality, among a set of vectors in a vector space. These functions can serve as regularizers in convex optimization problems arising from hierarchical classification, m…

The concept of subdifferentiability is studied in the context of $C^1$ Finsler manifolds (modeled on a Banach space with a Lipschitz $C^1$ bump function). A class of Hamilton-Jacobi equations defined on $C^1$ Finsler manifolds is studied and several results related to the existence and uniqueness of viscosity solutions…

SGD avoids critical points on weakly convex functions.

In this paper we study integer multiplicity rectifiable currents carried by the subgradient (subdifferential) graphs of semi-convex functions on a $n$-dimensional convex domain, and show a weak continuity theorem with respect to pointwise convergence for such currents. As an application, the $k$-Hessian measures are ca…

Given a real-valued function defined on the Heisenberg group, we provide a definition of abstract convexity and Fenchel transform that takes into account the sub-Riemannian structure of the group. In our main result, we prove that, likewise the Euclidean case, a convex function can be characterized via its iterated Fen…

New examples of sub-Riemannian structures satisfying Minimizing Sard conjecture found.

Generalized matrix-fractional (GMF) functions are a class of matrix support functions introduced by Burke and Hoheisel as a tool for unifying a range of seemingly divergent matrix optimization problems associated with inverse problems, regularization and learning. In this paper we dramatically simplify the support func…

We show how risk measures originally defined in a model free framework in terms of acceptance sets and reference assets imply a meaningful underlying probability structure. Hereafter we construct a maximal domain of definition of the risk measure respecting the underlying ambiguity profile. We particularly emphasise li…

The subdifferential of convex functions of the singular spectrum of real matrices has been widely studied in matrix analysis, optimization and automatic control theory. Convex analysis and optimization over spaces of tensors is now gaining much interest due to its potential applications to signal processing, statistics…

We introduce a proximal subdifferential and develop a calculus for nonsmooth functions defined on any Riemannian manifold $M$. We give several applications of this theory, concerning: 1) differentiability and geometrical properties of the distance function to a closed subset $C$ of $M$; 2) solvability and implicit func…

Unified framework for pattern recovery in penalized and thresholded estimation.

A new screening rule improves SLOPE efficiency for high-dimensional data.

We propose an approach to multivariate nonparametric regression that generalizes reduced rank regression for linear models. An additive model is estimated for each dimension of a $q$-dimensional response, with a shared $p$-dimensional predictor variable. To control the complexity of the model, we employ a functional fo…

In this paper we propose a novel gradient algorithm to learn a policy from an expert's observed behavior assuming that the expert behaves optimally with respect to some unknown reward function of a Markovian Decision Problem. The algorithm's aim is to find a reward function such that the resulting optimal policy matche…

We consider the problem of estimating an unknown signal $x_0$ from noisy linear observations $y = Ax_0 + z\in R^m$. In many practical instances, $x_0$ has a certain structure that can be captured by a structure inducing convex function $f(\cdot)$. For example, $\ell_1$ norm can be used to encourage a sparse solution. T…

Study normal curves in sub-Finsler Lie groups with specific norms, focusing on branching and face stability.

Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.

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) …

Study shows convergence of stochastic gradient method for unregularized Wasserstein optimization.

We consider the issue of solution uniqueness for portfolio optimization problem and its inverse for asset returns with a finite number of possible scenarios. The risk is assessed by deviation measures introduced by [Rockafellar et al., Mathematical Programming, Ser. B, 108 (2006), pp. 515-540] instead of variance as in…

Given a real-valued function $c$ defined on the cartesian product of a generic Carnot group $\G$ and the first layer $V_1$ of its Lie algebra, we introduce a notion of $c$ horizontal convex ($c$ H-convex) function on $\G$ as the supremum of a suitable family of affine functions; this family is defined pointwisely, and …

Adaptive learning method for stochastic programs with latent uncertainty.

DFR reduces the computational cost of sparse-group lasso and adaptive sparse-group lasso.

The study analyzes robustness of estimators in linear models with adversarial errors.

We extend the well-known BFGS quasi-Newton method and its memory-limited variant LBFGS to the optimization of nonsmooth convex objectives. This is done in a rigorous fashion by generalizing three components of BFGS to subdifferentials: the local quadratic model, the identification of a descent direction, and the Wolfe …

We study the problem of corrupted sensing, a generalization of compressed sensing in which one aims to recover a signal from a collection of corrupted or unreliable measurements. While an arbitrary signal cannot be recovered in the face of arbitrary corruption, tractable recovery is possible when both signal and corrup…

DCCNNs reduce computational overhead and ambiguity in convolutional neural networks.

The abstract discusses convergence properties of Lipschitz functions and sets defined by equations.

Sign-RIP improves robust low-rank matrix recovery by preserving norms even with corrupted measurements.

A fast sketching algorithm solves regularized least squares problems efficiently.

The paper describes flows of MMD functionals with distance kernel and quantile functions.

In this paper we study general Schatten-$p$ quasi-norm (SPQN) regularized matrix minimization problems. In particular, we first introduce a class of first-order stationary points for them, and show that the first-order stationary points introduced in [11] for an SPQN regularized $vector$ minimization problem are equiva…