Tangencies and Polynomial Optimization

Given a polynomial function $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ and a unbounded basic closed semi-algebraic set $S \subset \mathbb{R}^n,$ in this paper we show that the conditions listed below are characterized exactly in terms of the so-called {\em tangency variety} of $f$ on $S$: (i) The $f$ is bounded from below on $S;$ (ii) The $f$ attains its infimum on $S;$ (iii) The sublevel set $\{x \in S \ | \ f(x) \le \lambda\}$ for $\lambda \in \mathbb{R}$ is compact; (iv) The $f$ is coercive on $S.$ Besides, we also provide some stability criteria for boundedness and coercivity of $f$ on $S.$Comment: a minor change in the introductio

L\'evy-driven causal CARMA random fields

We introduce L\'evy-driven causal CARMA random fields on $\mathbb{R}^d$, extending the class of CARMA processes. The definition is based on a system of stochastic partial differential equations which generalize the classical state-space representation of CARMA processes. The resulting CARMA model differs fundamentally from the isotropic CARMA random field of Brockwell and Matsuda. We show existence of the model under mild assumptions and examine some of its features including the second-order structure and path properties. In particular, we investigate the sampling behavior and formulate conditions for the causal CARMA random field to be an ARMA random field when sampled on an equidistant lattice.Comment: 27 page

Compactness criteria for real algebraic sets and newton polyhedra

Let $f \colon \mathbb{R}^n \rightarrow \mathbb{R}$ be a polynomial and $\mathcal{Z}(f)$ its zero set. In this paper, in terms of the so-called Newton polyhedron of $f,$ we present a necessary criterion and a sufficient condition for the compactness of $\mathcal{Z}(f).$ From this we derive necessary and sufficient criteria for the stable compactness of $\mathcal{Z}(f).

Optimality conditions for minimizers at infinity in polynomial programming

In this paper we study necessary optimality conditions for the optimization problem $$\textrm{infimum}f_0(x) \quad \textrm{ subject to } \quad x \in S,$$ where $f_0 \colon \mathbb{R}^n \rightarrow \mathbb{R}$ is a polynomial function and $S \subset \mathbb{R}^n$ is a set defined by polynomial inequalities. Assume that the problem is bounded below and has the Mangasarian--Fromovitz property at infinity. We first show that if the problem does {\em not} have an optimal solution, then a version at infinity of the Fritz-John optimality conditions holds. From this we derive a version at infinity of the Karush--Kuhn--Tucker optimality conditions. As applications, we obtain a Frank--Wolfe type theorem which states that the optimal solution set of the problem is nonempty provided the objective function $f_0$ is convenient. Finally, in the unconstrained case, we show that the optimal value of the problem is the smallest critical value of some polynomial. All the results are presented in terms of the Newton polyhedra of the polynomials defining the problem

On types of degenerate critical points of real polynomial functions

In this paper, we consider the problem of identifying the type (local minimizer, maximizer or saddle point) of a given isolated real critical point $c$, which is degenerate, of a multivariate polynomial function $f$. To this end, we introduce the definition of faithful radius of $c$ by means of the curve of tangency of $f$. We show that the type of $c$ can be determined by the global extrema of $f$ over the Euclidean ball centered at $c$ with a faithful radius.We propose algorithms to compute a faithful radius of $c$ and determine its type.Comment: 16 pages, 2 figure

Autocovariance Varieties of Moving Average Random Fields

We study the autocovariance functions of moving average random fields over the integer lattice $\mathbb{Z}^d$ from an algebraic perspective. These autocovariances are parametrized polynomially by the moving average coefficients, hence tracing out algebraic varieties. We derive dimension and degree of these varieties and we use their algebraic properties to obtain statistical consequences such as identifiability of model parameters. We connect the problem of parameter estimation to the algebraic invariants known as euclidean distance degree and maximum likelihood degree. Throughout, we illustrate the results with concrete examples. In our computations we use tools from commutative algebra and numerical algebraic geometry.Comment: 20 pages, 5 tables, 2 figure

Dialogue Act Segmentation for Vietnamese Human-Human Conversational Texts

Dialog act identification plays an important role in understanding conversations. It has been widely applied in many fields such as dialogue systems, automatic machine translation, automatic speech recognition, and especially useful in systems with human-computer natural language dialogue interfaces such as virtual assistants and chatbots. The first step of identifying dialog act is identifying the boundary of the dialog act in utterances. In this paper, we focus on segmenting the utterance according to the dialog act boundaries, i.e. functional segments identification, for Vietnamese utterances. We investigate carefully functional segment identification in two approaches: (1) machine learning approach using maximum entropy (ME) and conditional random fields (CRFs); (2) deep learning approach using bidirectional Long Short-Term Memory (LSTM) with a CRF layer (Bi-LSTM-CRF) on two different conversational datasets: (1) Facebook messages (Message data); (2) transcription from phone conversations (Phone data). To the best of our knowledge, this is the first work that applies deep learning based approach to dialog act segmentation. As the results show, deep learning approach performs appreciably better as to compare with traditional machine learning approaches. Moreover, it is also the first study that tackles dialog act and functional segment identification for Vietnamese.Comment: 6 pages, 2 figure

On the B +AFw−toJ/+AFw−PsiK∗ +AFw-to J/+AFw-Psi K^* Decay

We calculate the process- and polarization-dependent nonfactorizable terms $+AFw-widetilde{a}_{+AFw-lambda}$ of the B$+AFw-to$ J$/+AFw-Psi {+AFw-rm K^*}$ decay within the QCD-improved factorization approach. The longitudinal part $+AFw-widetilde{a}_{0}$ is infrared convergent and large enough to agree with recent experimental data, provided that the B-K$^*$ form factors $A_1(m^2_{+AFw-Psi})$ and $A_2(m^2_{+AFw-Psi})$ satisfy some constraints met by many (but not all) models. The transverse parts $+AFw-widetilde{a}_{+AFw-pm}$ on the other hand are infrared divergent, the procedure used to handle such divergence is discussed in relation with the B$+AFw-to$ J$/+AFw-Psi {+AFw-rm K}$ case in which the same problem arises. Our nonzero phases of the helicity amplitudes are consistent with experimental data recently measured for the first time by the BaBar collaboration.Comment: 12 pages, no figure

ConeRANK: Ranking as Learning Generalized Inequalities

We propose a new data mining approach in ranking documents based on the concept of cone-based generalized inequalities between vectors. A partial ordering between two vectors is made with respect to a proper cone and thus learning the preferences is formulated as learning proper cones. A pairwise learning-to-rank algorithm (ConeRank) is proposed to learn a non-negative subspace, formulated as a polyhedral cone, over document-pair differences. The algorithm is regularized by controlling the `volume' of the cone. The experimental studies on the latest and largest ranking dataset LETOR 4.0 shows that ConeRank is competitive against other recent ranking approaches

On tangent cones at infinity of algebraic varieties

In this paper, we establish the following version at infinity of Whitney's theorem [6, 7]: Geometric and algebraic tangent cones at infinity of complex algebraic varieties coincide. The proof of this fact is based on a geometric characterization of geometric tangent cones at infinity and using the global \L ojasiewicz inequality with explicit exponents for complex algebraic varieties. We also show that tangent cones at infinity of complex algebraic varieties can be computed using Gr\"obner bases.Comment: 9 pages, Section 3 re-written, one reference added, acknowledgment updated with thanks to Prof. Gert-Martin Greue