相关论文: Convexity Certificates from Hessians

We present recent advances in formal verification and control for autonomous systems with practical safety guarantees enabled by conformal prediction (CP), a statistical tool for uncertainty quantification. This survey is particularly…

The recently introduced notions of ranking functions and closure certificates utilize well-foundedness arguments to facilitate the verification of dynamical systems against $\omega$-regular properties. A ranking function and a closure…

A differentiable function is pseudoconvex if and only if its restrictions over straight lines are pseudoconvex. A differentiable function depending on one variable, defined on some closed interval $[a,b]$ is pseudoconvex if and only if…

Fuzzy Cognitive Maps (FCMs) are considered a soft computing technique combining elements of fuzzy logic and recurrent neural networks. They found multiple application in such domains as modeling of system behavior, prediction of time…

In the problem of high-dimensional convexity testing, there is an unknown set $S \subseteq \mathbb{R}^n$ which is promised to be either convex or $\varepsilon$-far from every convex body with respect to the standard multivariate normal…

Verifying the Second-Order Sufficient Condition (SOSC), thus ensuring a stationary point locally minimizes a given objective function (subject to certain constraints), is an essential component of non-convex computational optimization and…

We investigate the computation of Hessian matrices via Automatic Differentiation, using a graph model and an algebraic model. The graph model reveals the inherent symmetries involved in calculating the Hessian. The algebraic model, based on…

We give sufficient conditions for the expected excess and the upper semideviation of recourse functions to be strongly convex. This is done in the setting of two-stage stochastic programs with complete linear recourse and random right-hand…

Most problems within and beyond the scientific domain can be framed into one of the following three levels of complexity of function approximation. Type 1: Approximate an unknown function given input/output data. Type 2: Consider a…

In this paper, we study the partial differential equation models of neural networks. Neural network can be viewed as a map from a simple base model to a complicate function. Based on solid analysis, we show that this map can be formulated…

This paper presents two methods for approximating a proper subset of the entries of a Hessian using only function evaluations. These approximations are obtained using the techniques called \emph{generalized simplex Hessian} and…

This paper presents necessary, sufficient, and equivalent conditions for the spherical convexity of non-homogeneous quadratic functions. In addition to motivating this study and identifying useful criteria for determining whether such…

Vulnerability detection is a critical problem in software security and attracts growing attention both from academia and industry. Traditionally, software security is safeguarded by designated rule-based detectors that heavily rely on…

How does one verify that the output of a complicated program is correct? One can formally prove that the program is correct, but this may be beyond the power of existing methods. Alternatively one can check that the output produced for a…

The standard paired-sample testing approach in the multidimensional setting applies multiple univariate tests on the individual features, followed by p-value adjustments. Such an approach suffers when the data carry numerous features. A…

A graph $G=(V,E)$ is a geometric intersection graph if every node $v \in V$ is identified with a geometric object of some particular type, and two nodes are adjacent if the corresponding objects intersect. Geometric intersection graph…

Deepfake detection methods have shown promising results in recognizing forgeries within a given dataset, where training and testing take place on the in-distribution dataset. However, their performance deteriorates significantly when…

Let $D$ be a convex subset of a real vector space. It is shown that a radially lower semicontinuous function $f: D\to \mathbf{R}\cup \{+\infty\}$ is convex if and only if for all $x,y \in D$ there exists $\alpha=\alpha(x,y) \in (0,1)$ such…

Computer programs may go wrong due to exceptional behaviors, out-of-bound array accesses, or simply coding errors. Thus, they cannot be blindly trusted. Scientific computing programs make no exception in that respect, and even bring…

Sparse high dimensional graphical model selection is a topic of much interest in modern day statistics. A popular approach is to apply l1-penalties to either (1) parametric likelihoods, or, (2) regularized regression/pseudo-likelihoods,…