arXiv research

Method provides bounds for sparse PCA and nuclear norm problems.

Paper derives estimates for complex Hessian equations on Hermitian manifolds.

SOC-ICNN expands neural network representational capacity by using conic optimization.

Paper proposes a method to find approximate SOSP for nonconvex conic optimization problems.

We consider a smooth Euclidean solid cone endowed with a smooth homogeneous density function used to weight Euclidean volume and hypersurface area. By assuming convexity of the cone and a curvature-dimension condition we prove that the unique compact, orientable, second order minima of the weighted area under variation…

Associative submanifolds $A$ in nearly parallel $G_2$-manifolds $Y$ are minimal 3-submanifolds in spin 7-manifolds with a real Killing spinor. The Riemannian cone over $Y$ has the holonomy group contained in ${\rm Spin(7)}$ and the Riemannian cone over $A$ is a Cayley submanifold. Infinitesimal deformations of associat…

The SCMU algorithm computes cone factorizations for symmetric cones, improving upon existing methods.

Improved Compressed Sensing by optimizing sparse solutions with mixed integer programming.

We study the problem of existence of regions separating a given amount of volume with the least possible perimeter inside a Euclidean cone. Our main result shows that nonexistence for a given volume implies that the isoperimetric profile of the cone coincides with the one of the half-space. This allows us to give some …

Compact hypersurfaces minimize area in convex cones with free boundary.

We present a quantum interior-point method (IPM) for second-order cone programming (SOCP) that runs in time $\widetilde{O} \left( n\sqrt{r} \frac{ζκ}{δ^2} \log \left(1/ε\right) \right)$ where $r$ is the rank and $n$ the dimension of the SOCP, $δ$ bounds the distance of intermediate solutions from the cone boundary, $ζ$…

Paper introduces structured sparsity estimators for Generalized Linear Models.

New algorithm for online optimization over symmetric cones, unifying previous methods.

We develop a second-order model for limit order books in a single scaling regime.

New geometric proof of convex function differentiability and approximation.

This paper mainly aims to establish the well-posedness on time interval $[0,\varepsilon^{-\frac{1}{2}}T]$ of the classical initial problem for the bosonic membrane in the light cone gauge. Here $\varepsilon$ is the small parameter measures the nonlinear effects. In geometric, the bosonic membrane are timelike submanifo…

This paper proves a rigidity result for annuli in $RCD(K, N)$-spaces.

We use geometric measure theory to introduce the notion of asymptotic cones associated with a singular subspace of a Riemannian manifold. This extends the classical notion of asymptotic directions usually defined on smooth submanifolds. We get a simple expression of these cones for polyhedra in E^3, as well as converge…

The paper classifies periodic solitons in curve flows on the light-cone.

Solves Merton's investment-consumption problem with certainty equivalent approach.

The paper proves Schauder estimates on cone products and characterizes harmonic functions.

New method learns BN structures from data efficiently.

A new model captures forward curve dynamics with stochastic volatility.

Paper introduces STSL, a second-order Tweedie sampler for efficient posterior sampling in inverse problems.

In this paper we derive a second order approximation for an infinite dimensional limit order book model, in which the dynamics of the incoming order flow is allowed to depend on the current market price as well as on a volume indicator (e.g.~the volume standing at the top of the book). We study the fluctuations of the …

The paper debiases mini-batch approximations in deep learning for more accurate optimization and uncertainty quantification.

New algorithm finds approximate stationary points in non-convex optimization.

A new method for optimizing deep neural networks using TKFAC.

Paper examines risk measure expansions under FGM dependence, improving accuracy at extreme levels.

Second-order economic theory considers new variables to improve price volatility predictions.

We prove that Wilson loop expectation values for arbitrary simple closed contours obey an area law up to second order in perturbative two-dimensional Yang-Mills theory. Our analysis occurs within a general family of axial-like gauges, which include and interpolate between holomorphic gauge and the Wu-Mandelstam-Liebran…

Unified framework for hard affine SDP constraints in vRKHSs.

Nearly Kähler manifolds are the Riemannian 6-manifolds admitting real Killing spinors. Equivalently, the Riemannian cone over a nearly Kähler manifold has holonomy contained in G2. In this paper we study the deformation theory of nearly Kähler manifolds, showing that it is obstructed in general. More precisely, we show…

We study the problem of removable singularities for degenerate elliptic equations. Let F be a fully nonlinear second-order partial differential subequation of degenerate elliptic type on a manifold X. We study the question: Which closed subsets E in X have the property that every F-subharmonic function (subsolution) on…

A new method improves ICA performance by approximating MDI.

The Einstein equations in wave map gauge are a geometric second order system for a Lorentzian metric. To study existence of solutions of this hyperbolic quasi diagonal system with initial data on a characteristic cone which are not zero in a neighbourhood of the vertex one can appeal to theorems due to Cagnac and Dossa…

Natural gradient descent, which preconditions a gradient descent update with the Fisher information matrix of the underlying statistical model, is a way to capture partial second-order information. Several highly visible works have advocated an approximation known as the empirical Fisher, drawing connections between ap…

We consider the problem of decomposing a multivariate polynomial as the difference of two convex polynomials. We introduce algebraic techniques which reduce this task to linear, second order cone, and semidefinite programming. This allows us to optimize over subsets of valid difference of convex decompositions (dcds) a…

Proves approximation and interpolation for regular immersions directed by algebraically elliptic cones.

Advanced optimization algorithms such as Newton method and AdaGrad benefit from second order derivative or second order statistics to achieve better descent directions and faster convergence rates. At their heart, such algorithms need to compute the inverse or inverse square root of a matrix whose size is quadratic of …

Enhances SMC² with Hessian info for more efficient posterior approximation.

Proposes second-order influence functions for identifying influential groups in test-time predictions.

Proposes a Doubly Robust mean-CVaR portfolio method to improve investment risk management.

SOAR improves deep networks' robustness against adversarial examples.

Investigates how options can control systemic risk in portfolios.

We propose convex relaxations for convolutional neural nets with one hidden layer where the output weights are fixed. For convex activation functions such as rectified linear units, the relaxations are convex second order cone programs which can be solved very efficiently. We prove that the relaxation recovers the glob…

The quantification of diversification benefits due to risk aggregation plays a prominent role in the (regulatory) capital management of large firms within the financial industry. However, the complexity of today's risk landscape makes a quantifiable reduction of risk concentration a challenging task. In the present pap…

Paper approximates Kähler metrics with cone singularities near a hypersurface.