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Graphical models express conditional independence relationships among variables. Although methods for vector-valued data are well established, functional data graphical models remain underdeveloped. We introduce a notion of conditional…

In this work we use the tensorial language developed in [8] and [9] to differentiate functions of eigenvalues of symmetric matrices. We describe the formulae for the k-th derivative of such functions in two cases. The first case concerns…

Gaussian Process (GP) models are a powerful tool in probabilistic machine learning with a solid theoretical foundation. Thanks to current advances, modeling complex data with GPs is becoming increasingly feasible, which makes them an…

Situations of a functional predictor paired with a scalar response are increasingly encountered in data analysis. Predictors are often appropriately modeled as square integrable smooth random functions. Imposing minimal assumptions on the…

Motivated by extending the functional stochastic calculus, to important functionals to which it does not apply, a notion of functional derivative along a curve is introduced. This new setting is developed by incorporating path-dependent…

The paper introduces a generalization for known probabilistic models such as log-linear and graphical models, called here multiplicative models. These models, that express probabilities via product of parameters are shown to capture…

We give formulae for first and second derivatives of generalized eigenvalues/eigenvectors of symmetric matrices and generalized singular values/singular vectors of rectangular matrices when the matrices are linear or nonlinear functions of…

Gaussian graphical models are parametric statistical models for jointly normal random variables whose dependence structure is determined by a graph. In previous work, we introduced trek separation, which gives a necessary and sufficient…

We determine the number of functionally independent components of tensors involving higher-order derivatives of a Riemannian metric.

Probabilistic independence can dramatically simplify the task of eliciting, representing, and computing with probabilities in large domains. A key technique in achieving these benefits is the idea of graphical modeling. We survey existing…

We consider the notion of the matrix (tensor) distribution of a measurable function of several variables. On the one hand, it is an invariant of this function with respect to a certain group of transformations of variables; on the other…

The simple product formulae for derivatives of scalar functions raised to different powers are generalized for functions which take values in the set of symmetric positive definite matrices. These formulae are fundamental in derivation of…

We consider the problem of constructing nonparametric undirected graphical models for high-dimensional functional data. Most existing statistical methods in this context assume either a Gaussian distribution on the vertices or linear…

In this work, multi-variable derivative-free optimization algorithms for unconstrained optimization problems are developed. A novel procedure for approximating the gradient of multi-variable objective functions based on non-commutative maps…

Multivariate functions emerge naturally in a wide variety of data-driven models. Popular choices are expressions in the form of basis expansions or neural networks. While highly effective, the resulting functions tend to be hard to…

Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically enormous sizes of datasets for reliable conclusions. We develop an approach based on partial derivatives,…

Undirected graphical models are widely used to model the conditional independence structure of vector-valued data. However, in many modern applications, for example those involving EEG and fMRI data, observations are more appropriately…

The necessary and sufficient conditions for a function to be totally or partially separable are derived. It is shown that a function is totally separable if and only if each component of the gradient vector of depends only on the…

Graphical models are commonly used to represent conditional dependence relationships between variables. There are multiple methods available for exploring them from high-dimensional data, but almost all of them rely on the assumption that…

Derivative polynomials in two variables are defined by repeated differentiation of the tangent and secant functions. We establish the connections between the coefficients of these derivative polynomials and the numbers of interior and left…