arXiv research

Characterizes values at infinity for real polynomial maps with 2D fibers.

Extends Tanimoto kernel to real-valued functions.

Study optimality conditions for interval-valued optimization problems on Riemannian manifolds.

Study introduces indecomposability for varifolds, leading to geometric consequences.

A dimension allowing in particular to state necessary and sufficient conditions of the Morse-Sard Theorem for real valued functions is introduced.

The paper develops axioms for uniquely decomposing functions with real arguments.

The Novikov complex of a circle-valued Morse function is constructed algebraically from the Morse-Smale complex of the restriction to a fundamental domain of the real-valued Morse function on the pullback infinite cyclic cover.

Complex-valued neural networks are not a new concept, however, the use of real-valued models has often been favoured over complex-valued models due to difficulties in training and performance. When comparing real-valued versus complex-valued neural networks, existing literature often ignores the number of parameters, r…

Real valued homomorphisms on the algebra of smooth functions on a differential space are described. The concept of generators of this algebra is emphasized in this description.

The classical Kaehler potential is a real-valued function (KP) such that one can determine a Kaehler (symplectic) structure by differentiating KP. We define a mirror Kaehler potential on Calabi-Yau 3-folds, a real-valued function (MKP) such that one can determine a complex structure by differentiating MKP.

Study robust regression learning under adversarial attacks.

Constructs explicit $p$-harmonic functions on Grassmannians and flag manifolds.

In this note we prove the a pointwise ergodic theorem for functions taking values in a separable complete CAT(0)-space, analogous to Lindenstrauss' pointwise ergodic theorem for real-valued integrable functions on a probability space subject to a probability-preserving action of an amenable l.c.s.c. group, where in the…

The study analyzes decision trees on real and categorical features, deriving bounds on their VC dimension and proposing improved pruning algorithms.

In this paper, we address the stability of a broad class of discrete-time hypercomplex-valued Hopfield-type neural networks. To ensure the neural networks belonging to this class always settle down at a stationary state, we introduce novel hypercomplex number systems referred to as real-part associative hypercomplex nu…

FuBIF enhances AD by using real-valued functions for more flexible anomaly detection.

The paper uses complex-valued functions to simplify plane differential geometry and kinematics.

Necessary and sufficient condition is given for a set $A\subset R^1$ to be a subset of the critical values set for a $C^k$ function $f:R^m \to R^1$.

We study regularity properties of the dynamic value functions of primal and dual problems of optimal investing for utility functions defined on the whole real line. Relations between decomposition terms of value processes of primal and dual problems and between optimal solutions of basic and conditional utility maximiz…

Proposes a method to handle missing inputs in Bayesian optimization.

The construction of synthetic complex-valued signals from real-valued observations is an important step in many time series analysis techniques. The most widely used approach is based on the Hilbert transform, which maps the real-valued signal into its quadrature component. In this paper, we define a probabilistic gene…

The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…

Smooth manifolds have functions with exactly two critical values.

Whitney type examples of maps $f\in C^k(\real^m,\real^n)$ for a maximal possible real $k$, and multidimensional space-filling curves with special properties are constructed.

Researchers develop neural networks for approximating functions in Banach spaces.

We prove that a real-valued function (that is not assumed to be continuous) on a real analytic manifold is analytic whenever all its restrictions to analytic submanifolds homeomorphic to the 2-sphere are analytic. This is a real analog for the classical theorem of Hartogs that a function on a complex manifold is comple…

In a complete Riemannian manifold $(M, g)$ if the hessian of a real valued function satisfies some suitable conditions then it restricts the geometry of $(M, g)$. In this paper we characterize all compact rank-1 symmetric spaces, as those Riemannian manifolds $(M, g)$ admitting a real valued function $u$ such that the …

These lecture notes provide a self-contained introduction to the mathematical methods required in a Bachelor degree programme in Business, Economics, or Management. In particular, the topics covered comprise real-valued vector and matrix algebra, systems of linear algebraic equations, Leontief's stationary input-output…

We generalize the Omori-Yau almost maximum principle of the Laplace-Beltrami operator on a complete Riemannian manifold $M$ to a second-order linear semi-elliptic operator $L$ with bounded coefficients and no zeroth order term. Using this result, we prove some Liouville-type theorems for a real-valued $C^{2}$ function …

Estimates fat-shattering dimension of aggregated function classes.

FFBO optimizes functions as inputs and outputs, improving on existing BO methods.

We give an algorithmically efficient version of the learner-to-compression scheme conversion in Moran and Yehudayoff (2016). In extending this technique to real-valued hypotheses, we also obtain an efficient regression-to-bounded sample compression converter. To our knowledge, this is the first general compressed regre…

Usually, complex-valued RKHS are presented as an straightforward application of the real-valued case. In this paper we prove that this procedure yields a limited solution for regression. We show that another kernel, here denoted as pseudo kernel, is needed to learn any function in complex-valued fields. Accordingly, we…

In this article we have studied some properties of subharmonic functions in a strongly symmetric Riemannian manifold with a pole. As a generalization of polynomial growth of a function we have introduced the notion of polynomial growth of some degree of a function with respect to a real function and proved that any non…

Complex-valued neural networks can approximate any continuous function.

We exploit the Cartan-Kähler theory to prove the local existence of real analytic quaternionic contact structures for any prescribed values of the respective curvature functions and their covariant derivatives at a given point on a manifold. We show that, in a certain sense, the different real analytic quaternionic con…

Bayesian Optimization (BO) methods are useful for optimizing functions that are expen- sive to evaluate, lack an analytical expression and whose evaluations can be contaminated by noise. These methods rely on a probabilistic model of the objective function, typically a Gaussian process (GP), upon which an acquisition f…

We propose a plan online and learn offline (POLO) framework for the setting where an agent, with an internal model, needs to continually act and learn in the world. Our work builds on the synergistic relationship between local model-based control, global value function learning, and exploration. We study how local traj…

A new method for incorporating preferences in multi-objective Bayesian optimization.

We prove implicit function theorems for mappings on topological vector spaces over valued fields. In the real and complex cases, we obtain implicit function theorems for mappings from arbitrary (not necessarily locally convex) topological vector spaces to Banach spaces.

NP-PROV separates mean and variance spaces to improve function uncertainty.

Generalizes Fenchel conjugation to nonlinear functions on arbitrary sets.

The paper uses transfinite induction to prove existence in analysis.

Survey of complex-valued neural networks for improved performance.

We present the FuSSO, a functional analogue to the LASSO, that efficiently finds a sparse set of functional input covariates to regress a real-valued response against. The FuSSO does so in a semi-parametric fashion, making no parametric assumptions about the nature of input functional covariates and assuming a linear f…

We analyze the Gambler's problem, a simple reinforcement learning problem where the gambler has the chance to double or lose the bets until the target is reached. This is an early example introduced in the reinforcement learning textbook by Sutton and Barto (2018), where they mention an interesting pattern of the optim…

A new method for interpreting AI models using Shapley value for functional data.

We prove the following theorem. Let $U$ be a pseudo-harmonic function on a surface $M^2$. For a real valued continuous function $V : M^2 \to {\mathbb R}$ to be a conjugate pseudo-harmonic function of $U$ on $M^2$ it is necessary and sufficient that $V$ is open on level sets of $U$.